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"f638f0a6-7fb0-5443-88ba-1cc74229b280" +version = "2.0.2+0" + +[[deps.libinput_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"] +git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce" +uuid = "36db933b-70db-51c0-b978-0f229ee0e533" +version = "1.18.0+0" + +[[deps.libpng_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] +git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c" +uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" +version = "1.6.38+0" + +[[deps.libvorbis_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] +git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" +uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" +version = "1.3.7+1" + +[[deps.mtdev_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11" +uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e" +version = "1.1.6+0" + +[[deps.nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" +version = "1.48.0+0" + +[[deps.p7zip_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" +version = "17.4.0+0" + +[[deps.x264_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" +uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" +version = "2021.5.5+0" + +[[deps.x265_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" +uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "3.5.0+0" + +[[deps.xkbcommon_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.4.1+1" diff --git a/v2.1.0/assets/Project.toml b/v2.1.0/assets/Project.toml new file mode 100644 index 0000000..b59f8c8 --- /dev/null +++ b/v2.1.0/assets/Project.toml @@ -0,0 +1,40 @@ +[deps] +DiffEqGPU = "071ae1c0-96b5-11e9-1965-c90190d839ea" +DiffEqNoiseProcess = "77a26b50-5914-5dd7-bc55-306e6241c503" +DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa" +Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f" +Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4" +ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210" +IntegralsCuba = "e00cd5f1-6337-4131-8b37-28b2fe4cd6cb" +KernelDensity = "5ab0869b-81aa-558d-bb23-cbf5423bbe9b" +MCMCChains = "c7f686f2-ff18-58e9-bc7b-31028e88f75d" +Optimization = "7f7a1694-90dd-40f0-9382-eb1efda571ba" +OptimizationMOI = "fd9f6733-72f4-499f-8506-86b2bdd0dea1" +OptimizationNLopt = "4e6fcdb7-1186-4e1f-a706-475e75c168bb" +OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +SciMLExpectations = "afe9f18d-7609-4d0e-b7b7-af0cb72b8ea8" +SciMLSensitivity = "1ed8b502-d754-442c-8d5d-10ac956f44a1" +StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd" +StochasticDiffEq = "789caeaf-c7a9-5a7d-9973-96adeb23e2a0" +Turing = "fce5fe82-541a-59a6-adf8-730c64b5f9a0" + +[compat] +DifferentialEquations = "7" +DiffEqGPU = "2" +DiffEqNoiseProcess = "5" +Distributions = "0.25" +Documenter = "1" +ForwardDiff = "0.10" +IntegralsCuba = "0.3" +KernelDensity = "0.6" +MCMCChains = "6" +Optimization = "3" +OptimizationMOI = "0.1" +OptimizationNLopt = "0.1" +OrdinaryDiffEq = "6" +Plots = "1" +SciMLSensitivity = "7" +StatsPlots = "0.15" +StochasticDiffEq = "6" +Turing = "0.29" diff --git a/v2.1.0/assets/documenter.js b/v2.1.0/assets/documenter.js new file mode 100644 index 0000000..7002e25 --- /dev/null +++ b/v2.1.0/assets/documenter.js @@ -0,0 +1,874 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'minisearch': 'https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); +}); + +$(document).on("click", ".docs-article-toggle-button", function () { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'minisearch'], function($, minisearch) { + +// In general, most search related things will have "search" as a prefix. +// To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +let results = []; +let timer = undefined; + +let data = documenterSearchIndex["docs"].map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; +}); + +// list below is the lunr 2.1.3 list minus the intersect with names(Base) +// (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) +// ideally we'd just filter the original list but it's not available as a variable +const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", +]); + +let index = new minisearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with search results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + boost: { title: 100 }, + fuzzy: 2, + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + } + + return word ?? null; + }, + tokenize: (string) => string.split(/[\s\-\.]+/), + }, +}); + +index.addAll(data); + +let filters = [...new Set(data.map((x) => x.category))]; +var modal_filters = make_modal_body_filters(filters); +var filter_results = []; + +$(document).on("keyup", ".documenter-search-input", function (event) { + // Adding a debounce to prevent disruptions from super-speed typing! + debounce(() => update_search(filter_results), 300); +}); + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + $(this).removeClass("search-filter-selected"); + } else { + $(this).addClass("search-filter-selected"); + } + + // Adding a debounce to prevent disruptions from crazy clicking! + debounce(() => get_filters(), 300); +}); + +/** + * A debounce function, takes a function and an optional timeout in milliseconds + * + * @function callback + * @param {number} timeout + */ +function debounce(callback, timeout = 300) { + clearTimeout(timer); + timer = setTimeout(callback, timeout); +} + +/** + * Make/Update the search component + * + * @param {string[]} selected_filters + */ +function update_search(selected_filters = []) { + let initial_search_body = ` +
Type something to get started!
+ `; + + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + results = index.search(querystring, { + filter: (result) => { + // Filtering results + if (selected_filters.length === 0) { + return result.score >= 1; + } else { + return ( + result.score >= 1 && selected_filters.includes(result.category) + ); + } + }, + }); + + let search_result_container = ``; + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + results.forEach(function (result) { + if (result.location) { + // Checking for duplication of results for the same page + if (!links.includes(result.location)) { + search_results += make_search_result(result, querystring); + count++; + } + + links.push(result.location); + } + }); + + let result_count = `
${count} result(s)
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + filter_results = []; + modal_filters = make_modal_body_filters(filters, filter_results); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(initial_search_body); + } +} + +/** + * Make the modal filter html + * + * @param {string[]} filters + * @param {string[]} selected_filters + * @returns string + */ +function make_modal_body_filters(filters, selected_filters = []) { + let str = ``; + + filters.forEach((val) => { + if (selected_filters.includes(val)) { + str += `${val}`; + } else { + str += `${val}`; + } + }); + + let filter_html = ` +
+ Filters: + ${str} +
+ `; + + return filter_html; +} + +/** + * Make the result component given a minisearch result data object and the value of the search input as queryString. + * To view the result object structure, refer: https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ +function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`\\b${querystring}\\b`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`\\b${querystring}\\b`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${result.title}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; +} + +/** + * Get selected filters, remake the filter html and lastly update the search modal + */ +function get_filters() { + let ele = $(".search-filters .search-filter-selected").get(); + filter_results = ele.map((x) => $(x).text().toLowerCase()); + modal_filters = make_modal_body_filters(filters, filter_results); + update_search(filter_results); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+`; + +let initial_search_body = ` +
Type something to get started!
+`; + +let search_modal_footer = ` + +`; + +$(document.body).append( + ` + + ` +); + +document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); +}); + +document.querySelector(".close-search-modal").addEventListener("click", () => { + closeModal(); +}); + +$(document).on("click", ".search-result-link", function () { + closeModal(); +}); + +document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; +}); + +// Functions to open and close a modal +function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); +} + +function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); +} + +document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = 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#1abc9c}html.theme--documenter-dark a.box:active{box-shadow:inset 0 1px 2px rgba(10,10,10,0.2),0 0 0 1px #1abc9c}html.theme--documenter-dark .button{background-color:#282f2f;border-color:#4c5759;border-width:1px;color:#375a7f;cursor:pointer;justify-content:center;padding-bottom:calc(0.5em - 1px);padding-left:1em;padding-right:1em;padding-top:calc(0.5em - 1px);text-align:center;white-space:nowrap}html.theme--documenter-dark .button strong{color:inherit}html.theme--documenter-dark .button .icon,html.theme--documenter-dark .button .icon.is-small,html.theme--documenter-dark .button #documenter .docs-sidebar form.docs-search>input.icon,html.theme--documenter-dark #documenter .docs-sidebar .button form.docs-search>input.icon,html.theme--documenter-dark .button .icon.is-medium,html.theme--documenter-dark .button .icon.is-large{height:1.5em;width:1.5em}html.theme--documenter-dark .button .icon:first-child:not(:last-child){margin-left:calc(-0.5em - 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.button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark 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.button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content 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.button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark 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.button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table 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html.theme--documenter-dark .navbar-dropdown,html.theme--documenter-dark .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--documenter-dark .navbar-dropdown.is-right{left:auto;right:0}html.theme--documenter-dark .navbar-divider{display:block}html.theme--documenter-dark .navbar>.container .navbar-brand,html.theme--documenter-dark .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--documenter-dark .navbar>.container .navbar-menu,html.theme--documenter-dark .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--documenter-dark .navbar.is-fixed-bottom-desktop,html.theme--documenter-dark .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--documenter-dark 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a.navbar-item.is-active:not(:focus):not(:hover),html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--documenter-dark .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--documenter-dark .pagination{font-size:1rem;margin:-.25rem}html.theme--documenter-dark .pagination.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--documenter-dark .pagination.is-medium{font-size:1.25rem}html.theme--documenter-dark .pagination.is-large{font-size:1.5rem}html.theme--documenter-dark .pagination.is-rounded .pagination-previous,html.theme--documenter-dark #documenter .docs-sidebar 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+ Maintainer: @highlightjs/core-team + Website: https://highlightjs.org/ + License: see project LICENSE + Touched: 2021 +*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold} diff --git a/v2.1.0/assets/themeswap.js b/v2.1.0/assets/themeswap.js new file mode 100644 index 0000000..9f5eebe --- /dev/null +++ b/v2.1.0/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. 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Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/v2.1.0/index.html b/v2.1.0/index.html new file mode 100644 index 0000000..36adcdc --- /dev/null +++ b/v2.1.0/index.html @@ -0,0 +1,492 @@ + +Home · SciMLExpectations.jl
GitHub

SciMLExpectations.jl: Expected Values of Simulations and Uncertainty Quantification

SciMLExpectations.jl is a package for quantifying the uncertainties of simulations by calculating the expectations of observables with respect to input uncertainties. Its goal is to make it fast and easy to compute solution moments in a differentiable way, to enable fast optimization under uncertainty.

Warn

This package is still under heavy construction. Use at your own risk!

Installation

To install SciMLExpectations.jl, use the Julia package manager:

using Pkg
+Pkg.add("SciMLExpectations")

Contributing

Reproducibility

The documentation of this SciML package was built using these direct dependencies,
Status `~/work/SciMLExpectations.jl/SciMLExpectations.jl/docs/Project.toml`
+  [071ae1c0] DiffEqGPU v2.5.0
+  [77a26b50] DiffEqNoiseProcess v5.19.0
+⌃ [0c46a032] DifferentialEquations v7.10.0
+  [31c24e10] Distributions v0.25.102
+  [e30172f5] Documenter v1.1.0
+  [f6369f11] ForwardDiff v0.10.36
+  [e00cd5f1] IntegralsCuba v0.3.0
+  [5ab0869b] KernelDensity v0.6.7
+  [c7f686f2] MCMCChains v6.0.3
+  [7f7a1694] Optimization v3.19.1
+  [fd9f6733] OptimizationMOI v0.1.15
+  [4e6fcdb7] OptimizationNLopt v0.1.8
+  [1dea7af3] OrdinaryDiffEq v6.57.0
+  [91a5bcdd] Plots v1.39.0
+  [afe9f18d] SciMLExpectations v2.1.0 `~/work/SciMLExpectations.jl/SciMLExpectations.jl`
+  [1ed8b502] SciMLSensitivity v7.40.0
+  [f3b207a7] StatsPlots v0.15.6
+⌃ [789caeaf] StochasticDiffEq v6.62.0
+  [fce5fe82] Turing v0.29.1
+Info Packages marked with ⌃ have new versions available and may be upgradable.
and using this machine and Julia version.
Julia Version 1.9.3
+Commit bed2cd540a1 (2023-08-24 14:43 UTC)
+Build Info:
+  Official https://julialang.org/ release
+Platform Info:
+  OS: Linux (x86_64-linux-gnu)
+  CPU: 2 × Intel(R) Xeon(R) CPU E5-2673 v4 @ 2.30GHz
+  WORD_SIZE: 64
+  LIBM: libopenlibm
+  LLVM: libLLVM-14.0.6 (ORCJIT, broadwell)
+  Threads: 1 on 2 virtual cores
A more complete overview of all dependencies and their versions is also provided.
Status `~/work/SciMLExpectations.jl/SciMLExpectations.jl/docs/Manifest.toml`
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diff --git a/v2.1.0/manual/algorithms/index.html b/v2.1.0/manual/algorithms/index.html new file mode 100644 index 0000000..50af29e --- /dev/null +++ b/v2.1.0/manual/algorithms/index.html @@ -0,0 +1,2 @@ + +Expectation Algorithms · SciMLExpectations.jl diff --git a/v2.1.0/manual/problem/index.html b/v2.1.0/manual/problem/index.html new file mode 100644 index 0000000..897a5be --- /dev/null +++ b/v2.1.0/manual/problem/index.html @@ -0,0 +1,4 @@ + +ExpectationProblem · SciMLExpectations.jl
GitHub

ExpectationProblem

SciMLExpectations.ExpectationProblem — Type
ExpectationProblem(S, g, h, pdist, params, nout)
+ExpectationProblem(g, pdist, params; nout = 1)
+ExpectationProblem(sm::SystemMap, g, h, d; nout = 1)

Defines ∫ g(S(h(x,u0,p)))*f(x)dx

Arguments

Let 𝕏 = uncertainty space, 𝕌 = Initial condition space, ℙ = model parameter space

  • S: 𝕌 × ℙ → 𝕌 also known as system map.
  • g: 𝕌 × ℙ → ℝⁿᵒᵘᵗ also known as the observables or output function.
  • h: 𝕏 × 𝕌 × ℙ → 𝕌 × ℙ also known as covariate function.
  • pdf(d,x): 𝕏 → ℝ the uncertainty distribution of the initial states.
  • params
  • nout
source
SciMLExpectations.SystemMap — Type
SystemMap(prob, args...; kwargs...)

Representation of a system solution map for a given prob::DEProblem. args and kwargs are forwarded to the equation solver.

source
SciMLExpectations.ProcessNoiseSystemMap — Type
ProcessNoiseSystemMap(prob, n, args...; kwargs...)

Representation of a system solution map for a given prob::SDEProblem. args and kwargs are forwarded to the equation solver. n is the number of terms in the Kosambi–Karhunen–Loève representation of the process noise.

source
SciMLExpectations.GenericDistribution — Type

GenericDistribution(d, ds...)

Defines a generic distribution that just wraps functions for pdf function, rand and bounds. User can use this for define any arbitrary joint pdf. Included b/c Distributions.jl Product method of mixed distributions are type instable.

source
diff --git a/v2.1.0/manual/solve/index.html b/v2.1.0/manual/solve/index.html new file mode 100644 index 0000000..9237513 --- /dev/null +++ b/v2.1.0/manual/solve/index.html @@ -0,0 +1,4 @@ + +Solving Expectation Problems · SciMLExpectations.jl
GitHub

Solving Expectation Problems

CommonSolve.solve — Method
solve(exprob::ExpectationProblem, expalg::MonteCarlo)

Solve an ExpectationProblem using Monte Carlo integration.

source
CommonSolve.solve — Method
solve(exprob::ExpectationProblem, expalg::Koopman;
+      maxiters = 1000000, batch = 0, quadalg = HCubatureJL(),
+      ireltol = 1e-2, iabstol = 1e-2, kwargs...)

Solve an ExpectationProblem using Koopman integration.

source
diff --git a/v2.1.0/search_index.js b/v2.1.0/search_index.js new file mode 100644 index 0000000..3efa3a5 --- /dev/null +++ b/v2.1.0/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"tutorials/introduction/#An-Introduction-to-Expectations-via-SciMLExpectations.jl","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"","category":"section"},{"location":"tutorials/introduction/#System-Model","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"System Model","text":"","category":"section"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"First, let's consider the following linear model.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"u = p u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"f(u, p, t) = p .* u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"We then wish to solve this model on the timespan t=0.0 to t=10.0, with an initial condition u0=10.0 and parameter p=-0.3. 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Recall, that u0_dist = [Uniform(-10.0,10.0)], while p = [-0.3]. From this specification, the expectation is solved as","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"mathbbEleftgleft(Sleft(hleft(xu_0pright)right)right)vert xsim textgdright","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"where Pf is the “push-forward” density of the initial joint pdf f on initial conditions and parameters.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Alternatively, we could solve the same problem using the Koopman() algorithm.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"sol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Being that this system is linear, we can analytically compute the solution as a deterministic ODE with its initial condition set to the expectation of the initial condition, i.e.,","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"e^ptmathbbEleftu_0right","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"exp(p[1] * 4.0) * mean(u0_dist[1])","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"We see that for this case, the Koopman() algorithm produces a more accurate solution than MonteCarlo(). Not only is it more accurate, it is also much faster","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"@time solve(exprob, MonteCarlo(100000))","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"@time solve(exprob, Koopman())","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Changing the distribution, we arrive at","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"u0_dist = [Uniform(0.0, 10.0)]\ngd = GenericDistribution(u0_dist...)\nexprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))\nsol = solve(exprob, MonteCarlo(100000))\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"and","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"u0_dist = [Uniform(0.0, 10.0)]\ngd = GenericDistribution(u0_dist...)\nexprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"where the analytical solution is","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"exp(p[1] * 4.0) * mean(u0_dist[1])","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Note that the Koopman() algorithm doesn't currently support infinite or semi-infinite integration domains, where the integration domain is determined by the extrema of the given distributions. So, trying to use a Normal distribution will produce NaN","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"u0_dist = [Normal(3.0, 2.0)]\ngd = GenericDistribution(u0_dist...)\nexprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Here, the analytical solution is","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"exp(p[1] * 4.0) * mean(u0_dist[1])","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Using a truncated distribution will alleviate this problem. However, there is another gotcha. If a large majority of the probability mass of the distribution exists in a small region in the support, then the adaptive methods used to solve the expectation can “miss” the non-zero portions of the distribution and erroneously return 0.0.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"u0_dist = [truncated(Normal(3.0, 2.0), -1000, 1000)]\ngd = GenericDistribution(u0_dist...)\nexprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"whereas truncating at pm 4sigma produces the correct result","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"u0_dist = [truncated(Normal(3.0, 2.0), -5, 11)]\ngd = GenericDistribution(u0_dist...)\nexprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"If a large truncation is required, it is best practice to center the distribution on the truncated interval. This is because many of the underlying quadrature algorithms use the center of the interval as an evaluation point.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"u0_dist = [truncated(Normal(3.0, 2.0), 3 - 1000, 3 + 1000)]\ngd = GenericDistribution(u0_dist...)\nexprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/#Vector-Valued-Functions","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"Vector-Valued Functions","text":"","category":"section"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"ExpectationProblem can also handle vector-valued functions. Simply pass the vector-valued function and set the nout kwarg to the length of the vector the function returns.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Here, we demonstrate this by computing the expectation of u at t=4.0s and t=6.0s","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"g(sol, p) = [sol(4.0)[1], sol(6.0)[1]]\nexprob = ExpectationProblem(sm, g, h, gd; nout = 2)\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"with analytical solution","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"exp.(p .* [4.0, 6.0]) * mean(u0_dist[1])","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"this can be used to compute the expectation at a range of times simultaneously","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"saveat = tspan[1]:0.5:tspan[2]\ng(sol, p) = sol[1, :]\nprob = ODEProblem(f, u0, tspan, p, saveat = saveat)\nsm = SystemMap(prob, Tsit5())\nexprob = ExpectationProblem(sm, g, h, gd; nout = length(saveat))\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"We can then plot these values along with the analytical solution","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"plot(t -> exp(p[1] * t) * mean(u0_dist[1]), tspan..., xlabel = \"t\", label = \"analytical\")\nscatter!(collect(saveat), sol.u, marker = :o, label = nothing)","category":"page"},{"location":"tutorials/introduction/#Benefits-of-Using-Vector-Valued-Functions","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"Benefits of Using Vector-Valued Functions","text":"","category":"section"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"In the above examples, we used vector-valued expectation calculations to compute the various expectations required. Alternatively, one could simply compute multiple scalar-valued expectations. However, in most cases it is more efficient to use the vector-valued form. This is especially true when the ODE to be solved is computationally expensive.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"To demonstrate this, let's compute the expectation of x, x^2, and x^3 using both approaches while counting the number of times g() is evaluated. This is the same as the number of simulation runs required to arrive at the solution. First, consider the scalar-valued approach. Here, we follow the same method as before, but we add a counter to our function evaluation that stores the number of function calls for each expectation calculation to an array.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"function g(sol, p, power, counter)\n counter[power] += 1\n sol(4.0)[1]^power\nend\ncounter = [0, 0, 0]\ng(sol, p, power) = g(sol, p, power, counter)\ng(sol, p) = g(sol, p, 1)\nexprob = ExpectationProblem(sm, g, h, gd; nout = 1)\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"g(sol, p) = g(sol, p, 2)\nexprob = ExpectationProblem(sm, g, h, gd; nout = 1)\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"g(sol, p) = g(sol, p, 3)\nexprob = ExpectationProblem(sm, g, h, gd; nout = 1)\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"counter","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Leading to a total of j sum(counters) function evaluations.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Now, let's compare this to the vector-valued approach","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"function g(sol, p, counter)\n counter[1] += 1\n v = sol(4.0)[1]\n [v, v^2, v^3]\nend\ncounter = [0]\ng(sol, p) = g(sol, p, counter)\nexprob = ExpectationProblem(sm, g, h, gd; nout = 3)\nsol = solve(exprob, Koopman())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"counter","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"This is j round(counter[1]/sum(counters)*100,digits=2)% the number of simulations required when using scalar-valued expectations. Note how the number of evaluations used in the vector-valued form is equivalent to the maximum number of evaluations for the 3 scalar-valued expectation calls.","category":"page"},{"location":"tutorials/introduction/#Higher-Order-Moments","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"Higher-Order Moments","text":"","category":"section"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Leveraging this vector-valued capability, we can also efficiently compute higher-order central moments.","category":"page"},{"location":"tutorials/introduction/#Variance","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"Variance","text":"","category":"section"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"The variance, or 2nd central moment, of a random variable X is defined as","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"mathrmVarleft(Xright)=mathbbEleftleft(X-muright)^2right","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"where","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"mu = mathbbEleftXright","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"The expression for the variance can be expanded to","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"mathrmVarleft(Xright)=mathbbEleftX^2right-mathbbEleftXright^2","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Using this, we define a function that returns the expectations of X and X^2 as a vector-valued function and then compute the variance from these","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"function g(sol, p)\n x = sol(4.0)[1]\n [x, x^2]\nend\nexprob = ExpectationProblem(sm, g, h, gd; nout = 2)\nsol = solve(exprob, Koopman())\nsol.u\nmean_g = sol.u[1]\nvar_g = sol.u[2] - mean_g^2","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"For a linear system, we can propagate the variance analytically as","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"e^2ptmathrmVarleft(u_0right)","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"exp(2 * p[1] * 4.0) * var(u0_dist[1])","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"This can be computed at multiple time instances as well","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"saveat = tspan[1]:0.5:tspan[2]\ng(sol, p) = [sol[1, :]; sol[1, :] .^ 2]\nprob = ODEProblem(f, u0, tspan, p, saveat = saveat)\nsm = SystemMap(prob, Tsit5())\nexprob = ExpectationProblem(sm, g, h, gd; nout = length(saveat) * 2)\nsol = solve(exprob, Koopman())\nsol.u\n\nμ = sol.u[1:length(saveat)]\nσ = sqrt.(sol.u[(length(saveat) + 1):end] - μ .^ 2)\n\nplot(t -> exp(p[1] * t) * mean(u0_dist[1]), tspan...,\n ribbon = t -> -sqrt(exp(2 * p[1] * t) * var(u0_dist[1])),\n label = \"Analytical Mean, 1 std bounds\")\nscatter!(collect(saveat), μ, marker = :x, yerror = σ, c = :black,\n label = \"Koopman Mean, 1 std bounds\")","category":"page"},{"location":"tutorials/introduction/#Skewness","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"Skewness","text":"","category":"section"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"A similar approach can be used to compute skewness","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"function g(sol, p)\n x = sol(4.0)[1]\n [x, x^2, x^3]\nend\nexprob = ExpectationProblem(sm, g, h, gd; nout = 3)\nsol = solve(exprob, Koopman())\nsol.u\nmean_g = sol.u[1]\nvar_g = sol.u[2] - mean_g^2\nskew_g = (sol.u[3] - 3.0 * mean_g * var_g - mean_g^3) / var_g^(3 / 2)","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"As the system is linear, we expect the skewness to be unchanged from the initial distribution. Because the distribution is a truncated Normal distribution centered on the mean, the true skewness is 0.0.","category":"page"},{"location":"tutorials/introduction/#Batch-Mode","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"Batch-Mode","text":"","category":"section"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"It is also possible to solve the various simulations in parallel by using the batch kwarg and a batch-mode supported quadrature algorithm via the quadalg kwarg. To view the list of batch compatible quadrature algorithms, refer to Integrals.jl. Note: Batch-mode operation is built on top of DifferentialEquation.jl's EnsembleProblem. See the EnsembleProblem documentation for additional options.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"The default quadrature algorithm to solve ExpectationProblem does not support batch-mode evaluation. So, we first load dependencies for additional quadrature algorithms","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"using IntegralsCuba","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"We then solve our expectation as before, using a batch=10 multi-thread parallelization via EnsembleThreads() of Cuba's SUAVE algorithm. We also introduce additional uncertainty in the model parameter.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"u0_dist = truncated(Normal(3.0, 2.0), -5, 11)\np_dist = truncated(Normal(-0.7, 0.1), -1, 0)\ngd = GenericDistribution(u0_dist, p_dist)\ng(sol, p) = sol(6.0)[1]\nh(x, u, p) = [x[1]], [x[2]]\nsm = SystemMap(prob, Tsit5(), EnsembleThreads())\nexprob = ExpectationProblem(sm, g, h, gd; nout = 1)\n# batchmode = EnsembleThreads() #where to pass this?\nsol = solve(exprob, Koopman(), batch = 10, quadalg = CubaSUAVE())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Now, let's compare the performance of the batch and non-batch modes","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"@time solve(exprob, Koopman(), batch = 10, quadalg = CubaSUAVE())","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"solve(exprob, Koopman(), batch = 0, quadalg = CubaSUAVE())\n@time solve(exprob, Koopman(), batch = 0, quadalg = CubaSUAVE())","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"It is also possible to parallelize across the GPU. However, one must be careful of the limitations of ensemble solutions with the GPU. Please refer to DiffEqGPU.jl for details.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Here we load DiffEqGPU and modify our problem to use Float32 and to put the ODE in the required GPU form","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"note: Note\n","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"Switch EnsembleCPUArray() to EnsembleGPUArray() to make this example work on your GPU. Currently, these docs are built on a machine without a GPU.","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"using DiffEqGPU\n\nfunction f(du, u, p, t)\n @inbounds begin du[1] = p[1] * u[1] end\n nothing\nend\n\nu0 = Float32[10.0]\np = Float32[-0.3]\ntspan = (0.0f0, 10.0f0)\nprob = ODEProblem(f, u0, tspan, p)\n\ng(sol, p) = sol(6.0)[1]\n\nu0_dist = truncated(Normal(3.0f0, 2.0f0), -5.0f0, 11.0f0)\np_dist = truncated(Normal(-0.7f0, 0.1f0), -1.0f0, 0.0f0)\ngd = GenericDistribution(u0_dist, p_dist)\ng(sol, p) = sol(6.0f0)[1]\nh(x, u, p) = [x[1]], [x[2]]\nprob = ODEProblem(f, u0, tspan, p)\nsm = SystemMap(prob, Tsit5(), EnsembleCPUArray())\nexprob = ExpectationProblem(sm, g, h, gd; nout = 1)\nsol = solve(exprob, Koopman(), batch = 10, quadalg = CubaSUAVE())\nsol.u","category":"page"},{"location":"tutorials/introduction/","page":"An Introduction to Expectations via SciMLExpectations.jl","title":"An Introduction to Expectations via SciMLExpectations.jl","text":"The performance gains realized by leveraging batch GPU processing is problem-dependent. In this case, the number of batch evaluations required to overcome the overhead of using the GPU exceeds the number of simulations required to converge to the quadrature solution.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/#optimization_under_uncertainty","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"","category":"section"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"This tutorial showcases how to leverage the efficient Koopman expectation method from SciMLExpectations to perform optimization under uncertainty. We demonstrate this by using a bouncing ball model with an uncertain model parameter. We also demonstrate its application to problems with probabilistic constraints, in particular a special class of constraints called chance constraints.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/#System-Model","page":"Optimization Under Uncertainty","title":"System Model","text":"","category":"section"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"First let's consider a 2D bouncing ball, where the states are the horizontal position x, horizontal velocity dotx, vertical position y, and vertical velocity doty. This model has two system parameters, acceleration due to gravity and coefficient of restitution (models energy loss when the ball impacts the ground). We can simulate such a system using ContinuousCallback as","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"using DifferentialEquations, Plots\n\nfunction ball!(du, u, p, t)\n du[1] = u[2]\n du[2] = 0.0\n du[3] = u[4]\n du[4] = -p[1]\nend\n\nground_condition(u, t, integrator) = u[3]\nground_affect!(integrator) = integrator.u[4] = -integrator.p[2] * integrator.u[4]\nground_cb = ContinuousCallback(ground_condition, ground_affect!)\n\nu0 = [0.0, 2.0, 50.0, 0.0]\ntspan = (0.0, 50.0)\np = [9.807, 0.9]\n\nprob = ODEProblem(ball!, u0, tspan, p)\nsol = solve(prob, Tsit5(), callback = ground_cb)\nplot(sol, vars = (1, 3), label = nothing, xlabel = \"x\", ylabel = \"y\")","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"For this particular problem, we wish to measure the impact distance from a point y=25 on a wall at x=25. So, we introduce an additional callback that terminates the simulation on wall impact.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"stop_condition(u, t, integrator) = u[1] - 25.0\nstop_cb = ContinuousCallback(stop_condition, terminate!)\ncbs = CallbackSet(ground_cb, stop_cb)\n\ntspan = (0.0, 1500.0)\nprob = ODEProblem(ball!, u0, tspan, p)\nsol = solve(prob, Tsit5(), callback = cbs)\nplot(sol, vars = (1, 3), label = nothing, xlabel = \"x\", ylabel = \"y\")","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"To help visualize this problem, we plot as follows, where the star indicates a desired impact location","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"rectangle(xc, yc, w, h) = Shape(xc .+ [-w, w, w, -w] ./ 2.0, yc .+ [-h, -h, h, h] ./ 2.0)\n\nbegin\n plot(sol, vars = (1, 3), label = nothing, lw = 3, c = :black)\n xlabel!(\"x [m]\")\n ylabel!(\"y [m]\")\n plot!(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)\n scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)\n ylims!(0.0, 50.0)\nend","category":"page"},{"location":"tutorials/optimization_under_uncertainty/#Considering-Uncertainty","page":"Optimization Under Uncertainty","title":"Considering Uncertainty","text":"","category":"section"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"We now wish to introduce uncertainty in p[2], the coefficient of restitution. This is defined via a continuous univariate distribution from Distributions.jl. We can then run a Monte Carlo simulation of 100 trajectories via the EnsembleProblem interface.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"using Distributions\n\ncor_dist = truncated(Normal(0.9, 0.02), 0.9 - 3 * 0.02, 1.0)\ntrajectories = 100\n\nprob_func(prob, i, repeat) = remake(prob, p = [p[1], rand(cor_dist)])\nensemble_prob = EnsembleProblem(prob, prob_func = prob_func)\nensemblesol = solve(ensemble_prob, Tsit5(), EnsembleThreads(), trajectories = trajectories,\n callback = cbs)\n\nbegin # plot\n plot(ensemblesol, vars = (1, 3), lw = 1)\n xlabel!(\"x [m]\")\n ylabel!(\"y [m]\")\n plot!(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)\n scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)\n plot!(sol, vars = (1, 3), label = nothing, lw = 3, c = :black, ls = :dash)\n xlims!(0.0, 27.5)\nend","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"Here, we plot the first 350 Monte Carlo simulations along with the trajectory corresponding to the mean of the distribution (dashed line).","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"We now wish to compute the expected squared impact distance from the star. This is called an “observation” of our system or an “observable” of interest.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"We define this observable as","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"obs(sol, p) = abs2(sol[3, end] - 25)","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"With the observable defined, we can compute the expected squared miss distance from our Monte Carlo simulation results as","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"mean_ensemble = mean([obs(sol, p) for sol in ensemblesol])","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"Alternatively, we can use the Koopman() algorithm in SciMLExpectations.jl to compute this expectation much more efficiently as","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"using SciMLExpectations\ngd = GenericDistribution(cor_dist)\nh(x, u, p) = u, [p[1]; x[1]]\nsm = SystemMap(prob, Tsit5(), callback = cbs)\nexprob = ExpectationProblem(sm, obs, h, gd; nout = 1)\nsol = solve(exprob, Koopman(), ireltol = 1e-5)\nsol.u","category":"page"},{"location":"tutorials/optimization_under_uncertainty/#Optimization-Under-Uncertainty","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"","category":"section"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"We now wish to optimize the initial position (x_0y_0) and horizontal velocity (dotx_0) of the system to minimize the expected squared miss distance from the star, where x_0inleft-1000right, y_0inleft13right, and dotx_0inleft1050right. We will demonstrate this using a gradient-based optimization approach from NLopt.jl using ForwardDiff.jl AD through the expectation calculation.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"using Optimization, OptimizationNLopt, OptimizationMOI\nmake_u0(θ) = [θ[1], θ[2], θ[3], 0.0]\nfunction 𝔼_loss(θ, pars)\n prob = ODEProblem(ball!, make_u0(θ), tspan, p)\n sm = SystemMap(prob, Tsit5(), callback = cbs)\n exprob = ExpectationProblem(sm, obs, h, gd; nout = 1)\n sol = solve(exprob, Koopman(), ireltol = 1e-5)\n sol.u\nend\nopt_f = OptimizationFunction(𝔼_loss, Optimization.AutoForwardDiff())\nopt_ini = [-1.0, 2.0, 50.0]\nopt_lb = [-100.0, 1.0, 10.0]\nopt_ub = [0.0, 3.0, 50.0]\nopt_prob = OptimizationProblem(opt_f, opt_ini; lb = opt_lb, ub = opt_ub)\noptimizer = OptimizationMOI.MOI.OptimizerWithAttributes(NLopt.Optimizer,\n \"algorithm\" => :LD_MMA)\nopt_sol = solve(opt_prob, optimizer)\nminx = opt_sol.u","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"Let's now visualize 100 Monte Carlo simulations","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"ensembleprob = EnsembleProblem(remake(prob, u0 = make_u0(minx)), prob_func = prob_func)\nensemblesol = solve(ensembleprob, Tsit5(), EnsembleThreads(), trajectories = 100,\n callback = cbs)\n\nbegin\n plot(ensemblesol, vars = (1, 3), lw = 1, alpha = 0.1)\n plot!(solve(remake(prob, u0 = make_u0(minx)), Tsit5(), callback = cbs),\n vars = (1, 3), label = nothing, c = :black, lw = 3, ls = :dash)\n xlabel!(\"x [m]\")\n ylabel!(\"y [m]\")\n plot!(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)\n scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)\n ylims!(0.0, 50.0)\n xlims!(minx[1], 27.5)\nend","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"Looks pretty good! But, how long did it take? Let's benchmark.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"@time solve(opt_prob, optimizer)","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"Not bad for bound constrained optimization under uncertainty of a hybrid system!","category":"page"},{"location":"tutorials/optimization_under_uncertainty/#Probabilistic-Constraints","page":"Optimization Under Uncertainty","title":"Probabilistic Constraints","text":"","category":"section"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"With this approach, we can also consider probabilistic constraints. Let us now consider a wall at x=20 with height 25.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"constraint = [20.0, 25.0]\nbegin\n plot(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)\n xlabel!(\"x [m]\")\n ylabel!(\"y [m]\")\n plot!([constraint[1], constraint[1]], [0.0, constraint[2]], lw = 5, c = :black,\n label = nothing)\n scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)\n ylims!(0.0, 50.0)\n xlims!(minx[1], 27.5)\nend","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"We now wish to minimize the same loss function as before, but introduce an inequality constraint such that the solution must have less than a 1% chance of colliding with the wall at x=20. This class of probabilistic constraints is called a chance constraint.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"To do this, we first introduce a new callback and solve the system using the previous optimal solution","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"constraint_condition(u, t, integrator) = u[1] - constraint[1]\nfunction constraint_affect!(integrator)\n integrator.u[3] < constraint[2] ? terminate!(integrator) : nothing\nend\nconstraint_cb = ContinuousCallback(constraint_condition, constraint_affect!,\n save_positions = (true, false));\nconstraint_cbs = CallbackSet(ground_cb, stop_cb, constraint_cb)\n\nensemblesol = solve(ensembleprob, Tsit5(), EnsembleThreads(), trajectories = 500,\n callback = constraint_cbs)\n\nbegin\n plot(ensemblesol, vars = (1, 3), lw = 1, alpha = 0.1)\n plot!(solve(remake(prob, u0 = make_u0(minx)), Tsit5(), callback = constraint_cbs),\n vars = (1, 3), label = nothing, c = :black, lw = 3, ls = :dash)\n\n xlabel!(\"x [m]\")\n ylabel!(\"y [m]\")\n plot!(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)\n plot!([constraint[1], constraint[1]], [0.0, constraint[2]], lw = 5, c = :black)\n scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)\n ylims!(0.0, 50.0)\n xlims!(minx[1], 27.5)\nend","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"That doesn't look good!","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"We now need a second observable for the system. To compute a probability of impact, we use an indicator function for if a trajectory impacts the wall. In other words, this functions returns 1 if the trajectory hits the wall and 0 otherwise.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"function constraint_obs(sol, p)\n sol((constraint[1] - sol[1, 1]) / sol[2, 1])[3] <= constraint[2] ? one(sol[1, end]) :\n zero(sol[1, end])\nend","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"Using the previously computed optimal initial conditions, let's compute the probability of hitting this wall","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"sm = SystemMap(remake(prob, u0 = make_u0(minx)), Tsit5(), callback = cbs)\nexprob = ExpectationProblem(sm, constraint_obs, h, gd; nout = 1)\nsol = solve(exprob, Koopman(), ireltol = 1e-5)\nsol.u","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"We then set up the constraint function for NLopt just as before.","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"function 𝔼_constraint(res, θ, pars)\n prob = ODEProblem(ball!, make_u0(θ), tspan, p)\n sm = SystemMap(prob, Tsit5(), callback = cbs)\n exprob = ExpectationProblem(sm, constraint_obs, h, gd; nout = 1)\n sol = solve(exprob, Koopman(), ireltol = 1e-5)\n res .= sol.u\nend\nopt_lcons = [-Inf]\nopt_ucons = [0.01]\noptimizer = OptimizationMOI.MOI.OptimizerWithAttributes(NLopt.Optimizer,\n \"algorithm\" => :LD_MMA)\nopt_f = OptimizationFunction(𝔼_loss, Optimization.AutoForwardDiff(), cons = 𝔼_constraint)\nopt_prob = OptimizationProblem(opt_f, opt_ini; lb = opt_lb, ub = opt_ub, lcons = opt_lcons,\n ucons = opt_ucons)\nopt_sol = solve(opt_prob, optimizer)\nminx2 = opt_sol.u","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"The probability of impacting the wall is now","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"container = zeros(1)\n𝔼_constraint(container, minx2, nothing)\nλ = container[1]","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"We can check if this is within tolerance by","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"λ <= 0.01 + 1e-5","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"Again, we plot some Monte Carlo simulations from this result as follows","category":"page"},{"location":"tutorials/optimization_under_uncertainty/","page":"Optimization Under Uncertainty","title":"Optimization Under Uncertainty","text":"ensembleprob = EnsembleProblem(remake(prob, u0 = make_u0(minx2)), prob_func = prob_func)\nensemblesol = solve(ensembleprob, Tsit5(), EnsembleThreads(),\n trajectories = 500, callback = constraint_cbs)\n\nbegin\n plot(ensemblesol, vars = (1, 3), lw = 1, alpha = 0.1)\n plot!(solve(remake(prob, u0 = make_u0(minx2)), Tsit5(), callback = constraint_cbs),\n vars = (1, 3), label = nothing, c = :black, lw = 3, ls = :dash)\n plot!([constraint[1], constraint[1]], [0.0, constraint[2]], lw = 5, c = :black)\n\n xlabel!(\"x [m]\")\n ylabel!(\"y [m]\")\n plot!(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)\n scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)\n ylims!(0.0, 50.0)\n xlims!(minx[1], 27.5)\nend","category":"page"},{"location":"tutorials/gpu_bayesian/#GPU-Accelerated-Data-Driven-Bayesian-Uncertainty-Quantification-with-Koopman-Operators","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"","category":"section"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"What if you have data and a general model and would like to evaluate the probability that the fitted model outcomes would have had a given behavior? The purpose of this tutorial is to demonstrate a fast workflow for doing exactly this. It composes together a few different pieces of the SciML ecosystem:","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Parameter estimation with uncertainty with Bayesian differential equations by integrating the differentiable differential equation solvers with the Turing.jl library.\nFast calculation of probabilistic estimates of differential equation solutions with parametric uncertainty using the Koopman expectation.\nGPU-acceleration of batched differential equation solves.","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Let's dive right in.","category":"page"},{"location":"tutorials/gpu_bayesian/#Bayesian-Parameter-Estimation-with-Uncertainty","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"Bayesian Parameter Estimation with Uncertainty","text":"","category":"section"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Let's start by importing all the necessary libraries:","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"using OrdinaryDiffEq\nusing Turing, MCMCChains, Distributions\nusing KernelDensity\nusing SciMLExpectations, IntegralsCuba\nusing DiffEqGPU\nusing Plots, StatsPlots\nusing Random;\nRandom.seed!(1);","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"For this tutorial, we will use the Lotka-Volterra equation:","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"function lotka_volterra(du, u, p, t)\n @inbounds begin\n x = u[1]\n y = u[2]\n α = p[1]\n β = p[2]\n γ = p[3]\n δ = p[4]\n du[1] = (α - β * y) * x\n du[2] = (δ * x - γ) * y\n end\nend\np = [1.5, 1.0, 3.0, 1.0]\nu0 = [1.0, 1.0]\nprob1 = ODEProblem(lotka_volterra, u0, (0.0, 10.0), p)\nsol = solve(prob1, Tsit5())\nplot(sol)","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"From the Lotka-Volterra equation, we will generate a dataset with known parameters:","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"sol1 = solve(prob1, Tsit5(), saveat = 0.1)","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Now let's assume our dataset should have noise. We can add this noise in and plot the noisy data against the generating set:","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"odedata = Array(sol1) + 0.8 * randn(size(Array(sol1)))\nplot(sol1, alpha = 0.3, legend = false);\nscatter!(sol1.t, odedata');","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Now let's assume that all we know is the data odedata and the model form. What we want to do is use the data to inform us of the parameters, but also get a probabilistic sense of the uncertainty around our parameter estimate. This is done via Bayesian estimation. For a full look at Bayesian estimation of differential equations, look at the Bayesian differential equation tutorial from Turing.jl.","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Following that tutorial, we choose a set of priors and perform NUTS sampling to arrive at the MCMC chain:","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Turing.setadbackend(:forwarddiff)\n\n@model function fitlv(data, prob1)\n σ ~ InverseGamma(2, 3) # ~ is the tilde character\n α ~ truncated(Normal(1.5, 0.5), 1.0, 2.0)\n β ~ truncated(Normal(1.2, 0.5), 0.5, 1.5)\n γ ~ truncated(Normal(3.0, 0.5), 2, 4)\n δ ~ truncated(Normal(1.0, 0.5), 0.5, 1.5)\n\n p = [α, β, γ, δ]\n prob = remake(prob1, p = p)\n predicted = solve(prob, Tsit5(), saveat = 0.1)\n\n for i in 1:length(predicted)\n data[:, i] ~ MvNormal(predicted[i], σ)\n end\nend\n\nmodel = fitlv(odedata, prob1)\n\n# This next command runs 3 independent chains without using multithreading.\nchain = mapreduce(c -> sample(model, NUTS(0.45), 1000), chainscat, 1:3)","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"This chain gives a discrete approximation to the probability distribution of our desired quantities. We can plot the chains to see these distributions in action:","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"plot(chain)","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Great! From our data, we have arrived at a probability distribution for our parameter values.","category":"page"},{"location":"tutorials/gpu_bayesian/#Evaluating-Model-Hypotheses-with-the-Koopman-Expectation","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"Evaluating Model Hypotheses with the Koopman Expectation","text":"","category":"section"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Now, let's try to ask a question: what is the expected value of x (the first term in the differential equation) at time t=10 given the known uncertainties in our parameters? This is a good tutorial question because all other probabilistic statements can be phrased similarly. Asking a question like, “what is the probability that x(T) > 1 at the final time T?”, can similarly be phrased as an expected value (probability statements are expected values of characteristic functions which are 1 if true 0 if false). So in general, the kinds of questions we want to ask and answer are expectations about the solutions of the differential equation.","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"The trivial to solve this problem is to sample 100,000 sets of parameters from our parameter distribution given by the Bayesian estimation, solve the ODE 100,000 times, and then take the average. But is 100,000 ODE solves enough? Well it's hard to tell, and even then, the convergence of this approach is slow. This is the Monte Carlo approach, and it converges to the correct answer by sqrt(N). Slow.","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"However, the Koopman expectation can converge with much fewer points, allowing the use of higher order quadrature methods to converge exponentially faster in many cases. To use the Koopman expectation, we first need to define our observable function g. This function designates the thing about the solution we wish to calculate the expectation of. Thus for our question “what is the expected value of xat time t=10?” we would simply use:","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"function g(sol, p)\n sol[1, end]\nend","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Now we need to use the expectation call, where we need to provide our initial condition and parameters as probability distributions. For this case, we will use the same constant u0 as before. But, let's turn our Bayesian MCMC chains into distributions through kernel density estimation (the plots of the distribution above are just KDE plots!).","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"p_kde = [kde(vec(Array(chain[:α]))), kde(vec(Array(chain[:β]))),\n kde(vec(Array(chain[:γ]))), kde(vec(Array(chain[:δ])))]","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Now that we have our observable and our uncertainty distributions, let's calculate the expected value. Using GenericDistribution, SciMLExpectation can calculate expectations of any distribution-like object that supports evaluating the probability density function, for the Koopman solver algorithm, and generating random numbers for the MonteCarlo solver algorithm.","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"pdf_func(x) = prod(pdf.(p_kde, x))\nrand_func() = nothing # not needed for Koopman\nlb = [p_kde_i.x[begin] for p_kde_i in p_kde] #lower bound for integration\nub = [p_kde_i.x[end] for p_kde_i in p_kde] #upper bound for integration\ngd = GenericDistribution(pdf_func, rand_func, lb, ub)\nh(x, u, p) = u, x\nsm = SystemMap(prob1, Tsit5())\nexprob = ExpectationProblem(sm, g, h, gd; nout = 1)\nsol = solve(exprob, Koopman(); reltol = 1e-2, abstol = 1e-2)\nsol.u","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Note how that gives the expectation and a residual for the error bound!","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"sol.resid","category":"page"},{"location":"tutorials/gpu_bayesian/#GPU-Accelerated-Expectations","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Expectations","text":"","category":"section"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"note: Note\nBatch functionality not yet fully implemented in version 2 of SciMLExpectations.","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Are we done? No, we need to add some GPUs! As mentioned earlier, probability calculations can take quite a bit of ODE solves, so let's parallelize across the parameters. DiffEqGPU.jl allows you to GPU-parallelize across parameters by using the Ensemble interface. Note that you do not have to do any of the heavy lifting: all the conversion to GPU kernels is done automatically by simply specifying EnsembleGPUArray as the ensembling method. For example:","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"function lotka_volterra(du, u, p, t)\n @inbounds begin\n x = u[1]\n y = u[2]\n α = p[1]\n β = p[2]\n γ = p[3]\n δ = p[4]\n du[1] = (α - β * y) * x\n du[2] = (δ * x - γ) * y\n end\nend\np = [1.5, 1.0, 3.0, 1.0]\nu0 = [1.0, 1.0]\nprob = ODEProblem(lotka_volterra, u0, (0.0, 10.0), p)\nprob_func = (prob, i, repeat) -> remake(prob, p = rand(Float64, 4) .* p)\nmonteprob = EnsembleProblem(prob, prob_func = prob_func, safetycopy = false)\n@time sol = solve(monteprob, Tsit5(), EnsembleGPUArray(), trajectories = 10_000,\n saveat = 1.0f0)","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"Let's now use this in the ensembling method. We need to specify a batch for the number of ODEs solved at the same time, and pass in our ensembling method. The following is a GPU-accelerated uncertainty quantified estimate of the expectation of the solution:","category":"page"},{"location":"tutorials/gpu_bayesian/","page":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","title":"GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators","text":"# batchmode = EnsembleGPUArray() #where to pass this?\nsol = solve(exprob, Koopman(), batch = 100, quadalg = CubaSUAVE())","category":"page"},{"location":"tutorials/process_noise/#Expectation-of-Process-Noise","page":"Expectation of Process Noise","title":"Expectation of Process Noise","text":"","category":"section"},{"location":"tutorials/process_noise/","page":"Expectation of Process Noise","title":"Expectation of Process Noise","text":"SciMLExpectations.jl is able to calculate the average trajectory of a stochastic differential equation. This done by representing the Wiener process using the Kosambi–Karhunen–Loève theorem.","category":"page"},{"location":"tutorials/process_noise/","page":"Expectation of Process Noise","title":"Expectation of Process Noise","text":"using SciMLExpectations\nusing IntegralsCuba\nusing StochasticDiffEq\nusing DiffEqNoiseProcess\nusing Distributions\n\nf(du, u, p, t) = (du .= u)\ng(du, u, p, t) = (du .= u)\nu0 = collect(1:4)\n\nW = WienerProcess(0.0, 0.0, 0.0)\nprob = SDEProblem(f, g, u0, (0.0, 1.0), noise = W)\nsm = ProcessNoiseSystemMap(prob, 8, LambaEM(), abstol = 1e-3, reltol = 1e-3)\ncov(x, u, p) = x, p\nobserved(sol, p) = sol[:, end]\nexprob = ExpectationProblem(sm, observed, cov; nout = length(u0))\nsol1 = solve(exprob, Koopman(), ireltol = 1e-3, iabstol = 1e-3, batch = 64,\n quadalg = CubaDivonne())\nsol1.u","category":"page"},{"location":"tutorials/process_noise/","page":"Expectation of Process Noise","title":"Expectation of Process Noise","text":"sol2 = solve(exprob, MonteCarlo(1_000_000))\nsol2.u","category":"page"},{"location":"tutorials/process_noise/","page":"Expectation of Process Noise","title":"Expectation of Process Noise","text":"In theory, any numerical integration method from Integrals.jl is supported, but in practice many of the techniques struggle with correctly calculating the expected value. We got the best results with the Divonne algorithm from Cuba.jl.","category":"page"},{"location":"manual/problem/#ExpectationProblem","page":"ExpectationProblem","title":"ExpectationProblem","text":"","category":"section"},{"location":"manual/problem/","page":"ExpectationProblem","title":"ExpectationProblem","text":"ExpectationProblem\nSystemMap\nProcessNoiseSystemMap\nGenericDistribution","category":"page"},{"location":"manual/problem/#SciMLExpectations.ExpectationProblem","page":"ExpectationProblem","title":"SciMLExpectations.ExpectationProblem","text":"ExpectationProblem(S, g, h, pdist, params, nout)\nExpectationProblem(g, pdist, params; nout = 1)\nExpectationProblem(sm::SystemMap, g, h, d; nout = 1)\n\nDefines ∫ g(S(h(x,u0,p)))*f(x)dx\n\nArguments\n\nLet 𝕏 = uncertainty space, 𝕌 = Initial condition space, ℙ = model parameter space\n\nS: 𝕌 × ℙ → 𝕌 also known as system map.\ng: 𝕌 × ℙ → ℝⁿᵒᵘᵗ also known as the observables or output function.\nh: 𝕏 × 𝕌 × ℙ → 𝕌 × ℙ also known as covariate function.\npdf(d,x): 𝕏 → ℝ the uncertainty distribution of the initial states.\nparams\nnout\n\n\n\n\n\n","category":"type"},{"location":"manual/problem/#SciMLExpectations.SystemMap","page":"ExpectationProblem","title":"SciMLExpectations.SystemMap","text":"SystemMap(prob, args...; kwargs...)\n\nRepresentation of a system solution map for a given prob::DEProblem. args and kwargs are forwarded to the equation solver.\n\n\n\n\n\n","category":"type"},{"location":"manual/problem/#SciMLExpectations.ProcessNoiseSystemMap","page":"ExpectationProblem","title":"SciMLExpectations.ProcessNoiseSystemMap","text":"ProcessNoiseSystemMap(prob, n, args...; kwargs...)\n\nRepresentation of a system solution map for a given prob::SDEProblem. args and kwargs are forwarded to the equation solver. n is the number of terms in the Kosambi–Karhunen–Loève representation of the process noise.\n\n\n\n\n\n","category":"type"},{"location":"manual/problem/#SciMLExpectations.GenericDistribution","page":"ExpectationProblem","title":"SciMLExpectations.GenericDistribution","text":"GenericDistribution(d, ds...)\n\nDefines a generic distribution that just wraps functions for pdf function, rand and bounds. User can use this for define any arbitrary joint pdf. Included b/c Distributions.jl Product method of mixed distributions are type instable.\n\n\n\n\n\n","category":"type"},{"location":"#SciMLExpectations.jl:-Expected-Values-of-Simulations-and-Uncertainty-Quantification","page":"Home","title":"SciMLExpectations.jl: Expected Values of Simulations and Uncertainty Quantification","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"SciMLExpectations.jl is a package for quantifying the uncertainties of simulations by calculating the expectations of observables with respect to input uncertainties. Its goal is to make it fast and easy to compute solution moments in a differentiable way, to enable fast optimization under uncertainty.","category":"page"},{"location":"","page":"Home","title":"Home","text":"warn: Warn\nThis package is still under heavy construction. Use at your own risk!","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install SciMLExpectations.jl, use the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(\"SciMLExpectations\")","category":"page"},{"location":"#Contributing","page":"Home","title":"Contributing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to SciML.\nSee the SciML Style Guide for common coding practices and other style decisions.\nThere are a few community forums:\nThe #diffeq-bridged and #sciml-bridged channels in the Julia Slack\nThe #diffeq-bridged and #sciml-bridged channels in the Julia Zulip\nOn the Julia Discourse forums\nSee also SciML Community page","category":"page"},{"location":"#Reproducibility","page":"Home","title":"Reproducibility","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"
The documentation of this SciML package was built using these direct dependencies,","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg # hide\nPkg.status() # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"
and using this machine and Julia version.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using InteractiveUtils # hide\nversioninfo() # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"
A more complete overview of all dependencies and their versions is also provided.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg # hide\nPkg.status(; mode = PKGMODE_MANIFEST) # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"using TOML\nusing Markdown\nversion = TOML.parse(read(\"../../Project.toml\", String))[\"version\"]\nname = TOML.parse(read(\"../../Project.toml\", String))[\"name\"]\nlink_manifest = \"https://github.com/SciML/\" * name * \".jl/tree/gh-pages/v\" * version *\n \"/assets/Manifest.toml\"\nlink_project = \"https://github.com/SciML/\" * name * \".jl/tree/gh-pages/v\" * version *\n \"/assets/Project.toml\"\nMarkdown.parse(\"\"\"You can also download the\n[manifest]($link_manifest)\nfile and the\n[project]($link_project)\nfile.\n\"\"\")","category":"page"},{"location":"manual/algorithms/#Expectation-Algorithms","page":"Expectation Algorithms","title":"Expectation Algorithms","text":"","category":"section"},{"location":"manual/algorithms/","page":"Expectation Algorithms","title":"Expectation Algorithms","text":"MonteCarlo\nKoopman","category":"page"},{"location":"manual/algorithms/#SciMLExpectations.MonteCarlo","page":"Expectation Algorithms","title":"SciMLExpectations.MonteCarlo","text":"MonteCarlo(trajectories::Int)\n\n\n\n\n\n","category":"type"},{"location":"manual/algorithms/#SciMLExpectations.Koopman","page":"Expectation Algorithms","title":"SciMLExpectations.Koopman","text":"Koopman()\n\n\n\n\n\n","category":"type"},{"location":"manual/solve/#Solving-Expectation-Problems","page":"Solving Expectation Problems","title":"Solving Expectation Problems","text":"","category":"section"},{"location":"manual/solve/","page":"Solving Expectation Problems","title":"Solving Expectation Problems","text":"solve(prob::ExpectationProblem, expalg::MonteCarlo)\nsolve(prob::ExpectationProblem, expalg::Koopman)","category":"page"},{"location":"manual/solve/#CommonSolve.solve-Tuple{ExpectationProblem, MonteCarlo}","page":"Solving Expectation Problems","title":"CommonSolve.solve","text":"solve(exprob::ExpectationProblem, expalg::MonteCarlo)\n\nSolve an ExpectationProblem using Monte Carlo integration.\n\n\n\n\n\n","category":"method"},{"location":"manual/solve/#CommonSolve.solve-Tuple{ExpectationProblem, Koopman}","page":"Solving Expectation Problems","title":"CommonSolve.solve","text":"solve(exprob::ExpectationProblem, expalg::Koopman;\n maxiters = 1000000, batch = 0, quadalg = HCubatureJL(),\n ireltol = 1e-2, iabstol = 1e-2, kwargs...)\n\nSolve an ExpectationProblem using Koopman integration.\n\n\n\n\n\n","category":"method"}] +} diff --git a/v2.1.0/siteinfo.js b/v2.1.0/siteinfo.js new file mode 100644 index 0000000..35f509d --- /dev/null +++ b/v2.1.0/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v2.1.0"; diff --git a/v2.1.0/tutorials/gpu_bayesian/658ef11d.svg b/v2.1.0/tutorials/gpu_bayesian/658ef11d.svg new file mode 100644 index 0000000..8cc44ce --- /dev/null +++ b/v2.1.0/tutorials/gpu_bayesian/658ef11d.svg @@ -0,0 +1,54 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v2.1.0/tutorials/gpu_bayesian/a69a6699.svg b/v2.1.0/tutorials/gpu_bayesian/a69a6699.svg new file mode 100644 index 0000000..08e1b39 --- /dev/null +++ b/v2.1.0/tutorials/gpu_bayesian/a69a6699.svg @@ -0,0 +1,248 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v2.1.0/tutorials/gpu_bayesian/bf88ecc4.svg b/v2.1.0/tutorials/gpu_bayesian/bf88ecc4.svg new file mode 100644 index 0000000..9aa3100 --- /dev/null +++ b/v2.1.0/tutorials/gpu_bayesian/bf88ecc4.svg @@ -0,0 +1,350 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v2.1.0/tutorials/gpu_bayesian/index.html b/v2.1.0/tutorials/gpu_bayesian/index.html new file mode 100644 index 0000000..0ea2266 --- /dev/null +++ b/v2.1.0/tutorials/gpu_bayesian/index.html @@ -0,0 +1,156 @@ + +GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators · SciMLExpectations.jl
GitHub

GPU-Accelerated Data-Driven Bayesian Uncertainty Quantification with Koopman Operators

What if you have data and a general model and would like to evaluate the probability that the fitted model outcomes would have had a given behavior? The purpose of this tutorial is to demonstrate a fast workflow for doing exactly this. It composes together a few different pieces of the SciML ecosystem:

  1. Parameter estimation with uncertainty with Bayesian differential equations by integrating the differentiable differential equation solvers with the Turing.jl library.
  2. Fast calculation of probabilistic estimates of differential equation solutions with parametric uncertainty using the Koopman expectation.
  3. GPU-acceleration of batched differential equation solves.

Let's dive right in.

Bayesian Parameter Estimation with Uncertainty

Let's start by importing all the necessary libraries:

using OrdinaryDiffEq
+using Turing, MCMCChains, Distributions
+using KernelDensity
+using SciMLExpectations, IntegralsCuba
+using DiffEqGPU
+using Plots, StatsPlots
+using Random;
+Random.seed!(1);
Random.TaskLocalRNG()

For this tutorial, we will use the Lotka-Volterra equation:

function lotka_volterra(du, u, p, t)
+    @inbounds begin
+        x = u[1]
+        y = u[2]
+        α = p[1]
+        β = p[2]
+        γ = p[3]
+        δ = p[4]
+        du[1] = (α - β * y) * x
+        du[2] = (δ * x - γ) * y
+    end
+end
+p = [1.5, 1.0, 3.0, 1.0]
+u0 = [1.0, 1.0]
+prob1 = ODEProblem(lotka_volterra, u0, (0.0, 10.0), p)
+sol = solve(prob1, Tsit5())
+plot(sol)
Example block output

From the Lotka-Volterra equation, we will generate a dataset with known parameters:

sol1 = solve(prob1, Tsit5(), saveat = 0.1)
retcode: Success
+Interpolation: 1st order linear
+t: 101-element Vector{Float64}:
+  0.0
+  0.1
+  0.2
+  0.3
+  0.4
+  0.5
+  0.6
+  0.7
+  0.8
+  0.9
+  ⋮
+  9.2
+  9.3
+  9.4
+  9.5
+  9.6
+  9.7
+  9.8
+  9.9
+ 10.0
+u: 101-element Vector{Vector{Float64}}:
+ [1.0, 1.0]
+ [1.0610780673356455, 0.8210842775886171]
+ [1.1440276717257598, 0.6790526689784505]
+ [1.2491712125724483, 0.5668931465841184]
+ [1.3776445705636382, 0.4788129513795158]
+ [1.531230817748013, 0.41015646708661463]
+ [1.7122697558187638, 0.3572654487994841]
+ [1.9235782758301574, 0.31734720616177187]
+ [2.168391089699407, 0.28838884378732427]
+ [2.450250667140253, 0.2690537093960073]
+ ⋮
+ [1.8172823219555445, 4.064946595043299]
+ [1.4427612988383434, 3.5397375780465032]
+ [1.208908107884442, 2.9914550030313953]
+ [1.068592596962787, 2.482072920162587]
+ [0.9910229623276097, 2.0372445701968505]
+ [0.957421348475831, 1.6632055724973926]
+ [0.9569793912886637, 1.3555870283301144]
+ [0.98356090632007, 1.1062868199419809]
+ [1.0337581256020674, 0.9063703842886032]

Now let's assume our dataset should have noise. We can add this noise in and plot the noisy data against the generating set:

odedata = Array(sol1) + 0.8 * randn(size(Array(sol1)))
+plot(sol1, alpha = 0.3, legend = false);
+scatter!(sol1.t, odedata');
Example block output

Now let's assume that all we know is the data odedata and the model form. What we want to do is use the data to inform us of the parameters, but also get a probabilistic sense of the uncertainty around our parameter estimate. This is done via Bayesian estimation. For a full look at Bayesian estimation of differential equations, look at the Bayesian differential equation tutorial from Turing.jl.

Following that tutorial, we choose a set of priors and perform NUTS sampling to arrive at the MCMC chain:

Turing.setadbackend(:forwarddiff)
+
+@model function fitlv(data, prob1)
+    σ ~ InverseGamma(2, 3) # ~ is the tilde character
+    α ~ truncated(Normal(1.5, 0.5), 1.0, 2.0)
+    β ~ truncated(Normal(1.2, 0.5), 0.5, 1.5)
+    γ ~ truncated(Normal(3.0, 0.5), 2, 4)
+    δ ~ truncated(Normal(1.0, 0.5), 0.5, 1.5)
+
+    p = [α, β, γ, δ]
+    prob = remake(prob1, p = p)
+    predicted = solve(prob, Tsit5(), saveat = 0.1)
+
+    for i in 1:length(predicted)
+        data[:, i] ~ MvNormal(predicted[i], σ)
+    end
+end
+
+model = fitlv(odedata, prob1)
+
+# This next command runs 3 independent chains without using multithreading.
+chain = mapreduce(c -> sample(model, NUTS(0.45), 1000), chainscat, 1:3)
Chains MCMC chain (1000×17×3 Array{Float64, 3}):
+
+Iterations        = 501:1:1500
+Number of chains  = 3
+Samples per chain = 1000
+Wall duration     = 79.88 seconds
+Compute duration  = 72.61 seconds
+parameters        = σ, α, β, γ, δ
+internals         = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
+
+Summary Statistics
+  parameters      mean       std      mcse   ess_bulk   ess_tail      rhat   e ⋯
+      Symbol   Float64   Float64   Float64    Float64    Float64   Float64     ⋯
+
+           σ    0.8417    0.0431    0.0078    27.7432   109.8148    1.1156     ⋯
+           α    1.5174    0.0491    0.0107    21.8564    73.3739    1.1784     ⋯
+           β    1.0732    0.0495    0.0094    28.9358   127.7724    1.1281     ⋯
+           γ    3.0287    0.1443    0.0298    23.5354    93.4607    1.1650     ⋯
+           δ    0.9742    0.0505    0.0104    23.6050    76.1056    1.1623     ⋯
+                                                                1 column omitted
+
+Quantiles
+  parameters      2.5%     25.0%     50.0%     75.0%     97.5%
+      Symbol   Float64   Float64   Float64   Float64   Float64
+
+           σ    0.7666    0.8073    0.8397    0.8728    0.9333
+           α    1.4264    1.4813    1.5155    1.5504    1.6238
+           β    0.9894    1.0369    1.0640    1.1084    1.1829
+           γ    2.7360    2.9281    3.0304    3.1281    3.3134
+           δ    0.8724    0.9384    0.9747    1.0111    1.0733
+

This chain gives a discrete approximation to the probability distribution of our desired quantities. We can plot the chains to see these distributions in action:

plot(chain)
Example block output

Great! From our data, we have arrived at a probability distribution for our parameter values.

Evaluating Model Hypotheses with the Koopman Expectation

Now, let's try to ask a question: what is the expected value of x (the first term in the differential equation) at time t=10 given the known uncertainties in our parameters? This is a good tutorial question because all other probabilistic statements can be phrased similarly. Asking a question like, “what is the probability that x(T) > 1 at the final time T?”, can similarly be phrased as an expected value (probability statements are expected values of characteristic functions which are 1 if true 0 if false). So in general, the kinds of questions we want to ask and answer are expectations about the solutions of the differential equation.

The trivial to solve this problem is to sample 100,000 sets of parameters from our parameter distribution given by the Bayesian estimation, solve the ODE 100,000 times, and then take the average. But is 100,000 ODE solves enough? Well it's hard to tell, and even then, the convergence of this approach is slow. This is the Monte Carlo approach, and it converges to the correct answer by sqrt(N). Slow.

However, the Koopman expectation can converge with much fewer points, allowing the use of higher order quadrature methods to converge exponentially faster in many cases. To use the Koopman expectation, we first need to define our observable function g. This function designates the thing about the solution we wish to calculate the expectation of. Thus for our question “what is the expected value of xat time t=10?” we would simply use:

function g(sol, p)
+    sol[1, end]
+end
g (generic function with 1 method)

Now we need to use the expectation call, where we need to provide our initial condition and parameters as probability distributions. For this case, we will use the same constant u0 as before. But, let's turn our Bayesian MCMC chains into distributions through kernel density estimation (the plots of the distribution above are just KDE plots!).

p_kde = [kde(vec(Array(chain[:α]))), kde(vec(Array(chain[:β]))),
+    kde(vec(Array(chain[:γ]))), kde(vec(Array(chain[:δ])))]
4-element Vector{KernelDensity.UnivariateKDE{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}:
+ KernelDensity.UnivariateKDE{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}(1.3324718591150952:0.00018870493018994144:1.7187508512139054, [9.907373544226772e-6, 1.000033177378512e-5, 1.0161812437736728e-5, 1.0392729763442077e-5, 1.069434817413395e-5, 1.1068286942084171e-5, 1.1516525906696273e-5, 1.2041412213115166e-5, 1.2645668139743549e-5, 1.3332399973364772e-5  …  1.2791587909927182e-5, 1.217384336985039e-5, 1.1634445038793961e-5, 1.117083746526859e-5, 1.0780824484579732e-5, 1.0462563863805618e-5, 1.0214562933680282e-5, 1.0035675159170143e-5, 9.92509772643202e-6, 9.88237009072468e-6])
+ KernelDensity.UnivariateKDE{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}(0.883394267330297:0.00019301142994834292:1.278488664434555, [1.2031528841216499e-5, 1.2287766372409692e-5, 1.2631969006804411e-5, 1.306611224372034e-5, 1.3592645361715228e-5, 1.4214499684861437e-5, 1.493509831052009e-5, 1.5758367306162313e-5, 1.6688748381010754e-5, 1.7731213028504555e-5  …  1.4202249829481373e-5, 1.3586199029386181e-5, 1.3063657949619056e-5, 1.2631992557299299e-5, 1.2289041628588127e-5, 1.2033110103865097e-5, 1.1862963970848961e-5, 1.177782664235627e-5, 1.1777376830723885e-5, 1.1861747925140387e-5])
+ KernelDensity.UnivariateKDE{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}(2.4802871887330267:0.0005682177709970292:3.6434289659639454, [3.831747204863234e-6, 3.885507411593991e-6, 3.967659508941068e-6, 4.0786874420106756e-6, 4.219230549606934e-6, 4.390085879713013e-6, 4.592210987919287e-6, 4.826727233231409e-6, 5.094923564596954e-6, 5.398260803257671e-6  …  4.858149579484161e-6, 4.621099093715664e-6, 4.415986904572622e-6, 4.2417633488375794e-6, 4.097537666246376e-6, 3.98257549735076e-6, 3.896296855904202e-6, 3.838274573668421e-6, 3.808233222206414e-6, 3.8060485192115134e-6])
+ KernelDensity.UnivariateKDE{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}(0.7948065916692116:0.00020431348469314916:1.213036294836088, [1.0414353169174717e-5, 1.0602032376905157e-5, 1.0872104300840846e-5, 1.1226502340644373e-5, 1.1667649407520742e-5, 1.2198470054958221e-5, 1.2822404415580024e-5, 1.354342392811958e-5, 1.4366048913139196e-5, 1.5295368051670977e-5  …  1.2977961284604822e-5, 1.2345670017947785e-5, 1.1801885953538482e-5, 1.1343950152606364e-5, 1.0969661706938805e-5, 1.0677274001003045e-5, 1.0465492434108548e-5, 1.0333473612025856e-5, 1.0280826042152746e-5, 1.0307612366630692e-5])

Now that we have our observable and our uncertainty distributions, let's calculate the expected value. Using GenericDistribution, SciMLExpectation can calculate expectations of any distribution-like object that supports evaluating the probability density function, for the Koopman solver algorithm, and generating random numbers for the MonteCarlo solver algorithm.

pdf_func(x) = prod(pdf.(p_kde, x))
+rand_func() = nothing # not needed for Koopman
+lb = [p_kde_i.x[begin] for p_kde_i in p_kde] #lower bound for integration
+ub = [p_kde_i.x[end] for p_kde_i in p_kde] #upper bound for integration
+gd = GenericDistribution(pdf_func, rand_func, lb, ub)
+h(x, u, p) = u, x
+sm = SystemMap(prob1, Tsit5())
+exprob = ExpectationProblem(sm, g, h, gd; nout = 1)
+sol = solve(exprob, Koopman(); reltol = 1e-2, abstol = 1e-2)
+sol.u

Note how that gives the expectation and a residual for the error bound!

sol.resid

GPU-Accelerated Expectations

Note

Batch functionality not yet fully implemented in version 2 of SciMLExpectations.

Are we done? No, we need to add some GPUs! As mentioned earlier, probability calculations can take quite a bit of ODE solves, so let's parallelize across the parameters. DiffEqGPU.jl allows you to GPU-parallelize across parameters by using the Ensemble interface. Note that you do not have to do any of the heavy lifting: all the conversion to GPU kernels is done automatically by simply specifying EnsembleGPUArray as the ensembling method. For example:

function lotka_volterra(du, u, p, t)
+    @inbounds begin
+        x = u[1]
+        y = u[2]
+        α = p[1]
+        β = p[2]
+        γ = p[3]
+        δ = p[4]
+        du[1] = (α - β * y) * x
+        du[2] = (δ * x - γ) * y
+    end
+end
+p = [1.5, 1.0, 3.0, 1.0]
+u0 = [1.0, 1.0]
+prob = ODEProblem(lotka_volterra, u0, (0.0, 10.0), p)
+prob_func = (prob, i, repeat) -> remake(prob, p = rand(Float64, 4) .* p)
+monteprob = EnsembleProblem(prob, prob_func = prob_func, safetycopy = false)
+@time sol = solve(monteprob, Tsit5(), EnsembleGPUArray(), trajectories = 10_000,
+                  saveat = 1.0f0)

Let's now use this in the ensembling method. We need to specify a batch for the number of ODEs solved at the same time, and pass in our ensembling method. The following is a GPU-accelerated uncertainty quantified estimate of the expectation of the solution:

# batchmode = EnsembleGPUArray() #where to pass this?
+sol = solve(exprob, Koopman(), batch = 100, quadalg = CubaSUAVE())
diff --git a/v2.1.0/tutorials/introduction/2b224976.svg b/v2.1.0/tutorials/introduction/2b224976.svg new file mode 100644 index 0000000..be427a4 --- /dev/null +++ b/v2.1.0/tutorials/introduction/2b224976.svg @@ -0,0 +1,48 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v2.1.0/tutorials/introduction/312d27cf.svg b/v2.1.0/tutorials/introduction/312d27cf.svg new file mode 100644 index 0000000..0a0fb5f --- /dev/null +++ b/v2.1.0/tutorials/introduction/312d27cf.svg @@ -0,0 +1,69 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v2.1.0/tutorials/introduction/7728cebb.svg b/v2.1.0/tutorials/introduction/7728cebb.svg new file mode 100644 index 0000000..02a7502 --- /dev/null +++ b/v2.1.0/tutorials/introduction/7728cebb.svg @@ -0,0 +1,48 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v2.1.0/tutorials/introduction/9bf537ab.svg b/v2.1.0/tutorials/introduction/9bf537ab.svg new file mode 100644 index 0000000..10bdfc3 --- /dev/null +++ b/v2.1.0/tutorials/introduction/9bf537ab.svg @@ -0,0 +1,200 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/v2.1.0/tutorials/introduction/index.html b/v2.1.0/tutorials/introduction/index.html new file mode 100644 index 0000000..0bb4e89 --- /dev/null +++ b/v2.1.0/tutorials/introduction/index.html @@ -0,0 +1,187 @@ + +An Introduction to Expectations via SciMLExpectations.jl · SciMLExpectations.jl
GitHub

An Introduction to Expectations via SciMLExpectations.jl

System Model

First, let's consider the following linear model.

\[u' = p u\]

f(u, p, t) = p .* u
f (generic function with 1 method)

We then wish to solve this model on the timespan t=0.0 to t=10.0, with an initial condition u0=10.0 and parameter p=-0.3. We can then set up the differential equations, solve, and plot as follows

using OrdinaryDiffEq, Plots
+u0 = [10.0]
+p = [-0.3]
+tspan = (0.0, 10.0)
+prob = ODEProblem(f, u0, tspan, p)
+sol = solve(prob, Tsit5())
+plot(sol)
Example block output

However, what if we wish to consider a random initial condition? Assume u0 is distributed uniformly from -10.0 to 10.0, i.e.,

using Distributions
+u0_dist = [Uniform(-10.0, 10.0)]
1-element Vector{Distributions.Uniform{Float64}}:
+ Distributions.Uniform{Float64}(a=-10.0, b=10.0)

We can then run a Monte Carlo simulation of 100,000 trajectories by solving an EnsembleProblem.

prob_func(prob, i, repeat) = remake(prob, u0 = rand.(u0_dist))
+ensemble_prob = EnsembleProblem(prob, prob_func = prob_func)
+
+ensemblesol = solve(ensemble_prob, Tsit5(), EnsembleThreads(), trajectories = 100000)
EnsembleSolution Solution of length 100000 with uType:
+ODESolution{Float64, 2, Vector{Vector{Float64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Vector{Float64}}}, ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, false, Vector{Float64}, ODEFunction{false, SciMLBase.AutoSpecialize, typeof(Main.f), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, OrdinaryDiffEq.InterpolationData{ODEFunction{false, SciMLBase.AutoSpecialize, typeof(Main.f), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Vector{Vector{Float64}}, Vector{Float64}, Vector{Vector{Vector{Float64}}}, OrdinaryDiffEq.Tsit5ConstantCache}, DiffEqBase.Stats, Nothing}

Plotting a summary of the trajectories produces

summ = EnsembleSummary(ensemblesol)
+plot(summ)
Example block output

Given the ensemble solution, we can then compute the expectation of a function $g\left(\text{sol},p\right)$ of the system state u at any time in the timespan, e.g., the state itself at t=4.0 as

g(sol, p) = sol(4.0)
+mean([g(sol, prob.p) for sol in ensemblesol])
1-element Vector{Float64}:
+ 0.006729399779693562

Alternatively, SciMLExpectations.jl offers a convenient interface for this type of calculation, using ExpectationProblem.

using SciMLExpectations
+gd = GenericDistribution(u0_dist...)
+h(x, u, p) = x, p
+sm = SystemMap(prob, Tsit5())
+exprob = ExpectationProblem(sm, g, h, gd; nout = 1)
+sol = solve(exprob, MonteCarlo(100000))
+sol.u
1-element Vector{Float64}:
+ 0.004849252422331459

ExpectationProblem takes a SystemMap, which contains the ODEProblem that we are working with, and how to solve it. The function of interest $g$, which maps the solution of the ODEProblem to the quantities you want to take the expectation of, in this case the state after 4 seconds. A GenericDistribution containing all the aspects of the dynamical system you are uncertain about, in this example this is the initial condition. The function $h$, which maps a realization of the uncertainty space to the initial conditions and parameters of the ODEProblem. Here, only the initial conditions are uncertain, while the parameters are deterministic. See further down for an example with both uncertain initial conditions and parameters.

The ExpectationProblem can then be solved using an AbstractExpectationAlgorithm. Here we use MonteCarlo() to use the Monte Carlo algorithm. Recall, that u0_dist = [Uniform(-10.0,10.0)], while p = [-0.3]. From this specification, the expectation is solved as

\[\mathbb{E}\left[g\left(S\left(h\left(x,u_0,p\right)\right)\right)\vert x\sim \text{gd}\right]\]

where $Pf$ is the “push-forward” density of the initial joint pdf $f$ on initial conditions and parameters.

Alternatively, we could solve the same problem using the Koopman() algorithm.

sol = solve(exprob, Koopman())
+sol.u
1-element Vector{Float64}:
+ 0.0

Being that this system is linear, we can analytically compute the solution as a deterministic ODE with its initial condition set to the expectation of the initial condition, i.e.,

\[e^{pt}\mathbb{E}\left[u_0\right]\]

exp(p[1] * 4.0) * mean(u0_dist[1])
0.0

We see that for this case, the Koopman() algorithm produces a more accurate solution than MonteCarlo(). Not only is it more accurate, it is also much faster

@time solve(exprob, MonteCarlo(100000))
SciMLExpectations.ExpectationSolution{Vector{Float64}, Nothing, Nothing}([-0.002384377270005703], nothing, nothing)
@time solve(exprob, Koopman())
SciMLExpectations.ExpectationSolution{Vector{Float64}, Float64, SciMLBase.IntegralSolution{Float64, 1, Vector{Float64}, Float64, IntegralProblem{false, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, SciMLExpectations.var"#23#24"{GenericDistribution{SciMLExpectations.var"#pdf_func#10"{Tuple{Distributions.Uniform{Float64}}}, SciMLExpectations.var"#rand_func#12"{Tuple{Distributions.Uniform{Float64}}}, StaticArraysCore.SVector{1, Float64}, StaticArraysCore.SVector{1, Float64}}, typeof(Main.h), typeof(Main.g), SystemMap{ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, false, Vector{Float64}, ODEFunction{false, SciMLBase.AutoSpecialize, typeof(Main.f), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, EnsembleThreads, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}}}, StaticArraysCore.SVector{1, Float64}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}}, HCubatureJL{typeof(LinearAlgebra.norm)}, Nothing, Nothing}}([0.0], 0.0, [0.0])

Changing the distribution, we arrive at

u0_dist = [Uniform(0.0, 10.0)]
+gd = GenericDistribution(u0_dist...)
+exprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))
+sol = solve(exprob, MonteCarlo(100000))
+sol.u
1-element Vector{Float64}:
+ 1.5099604868513306

and

u0_dist = [Uniform(0.0, 10.0)]
+gd = GenericDistribution(u0_dist...)
+exprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))
+sol = solve(exprob, Koopman())
+sol.u
1-element Vector{Float64}:
+ 1.5059722132990783

where the analytical solution is

exp(p[1] * 4.0) * mean(u0_dist[1])
1.5059710595610107

Note that the Koopman() algorithm doesn't currently support infinite or semi-infinite integration domains, where the integration domain is determined by the extrema of the given distributions. So, trying to use a Normal distribution will produce NaN

u0_dist = [Normal(3.0, 2.0)]
+gd = GenericDistribution(u0_dist...)
+exprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))
+sol = solve(exprob, Koopman())
+sol.u
1-element Vector{Float64}:
+ 0.9035426476099572

Here, the analytical solution is

exp(p[1] * 4.0) * mean(u0_dist[1])
0.9035826357366064

Using a truncated distribution will alleviate this problem. However, there is another gotcha. If a large majority of the probability mass of the distribution exists in a small region in the support, then the adaptive methods used to solve the expectation can “miss” the non-zero portions of the distribution and erroneously return 0.0.

u0_dist = [truncated(Normal(3.0, 2.0), -1000, 1000)]
+gd = GenericDistribution(u0_dist...)
+exprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))
+sol = solve(exprob, Koopman())
+sol.u
1-element Vector{Float64}:
+ 0.0

whereas truncating at $\pm 4\sigma$ produces the correct result

u0_dist = [truncated(Normal(3.0, 2.0), -5, 11)]
+gd = GenericDistribution(u0_dist...)
+exprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))
+sol = solve(exprob, Koopman())
+sol.u
1-element Vector{Float64}:
+ 0.9035833577703978

If a large truncation is required, it is best practice to center the distribution on the truncated interval. This is because many of the underlying quadrature algorithms use the center of the interval as an evaluation point.

u0_dist = [truncated(Normal(3.0, 2.0), 3 - 1000, 3 + 1000)]
+gd = GenericDistribution(u0_dist...)
+exprob = ExpectationProblem(sm, g, h, gd; nout = length(u0))
+sol = solve(exprob, Koopman())
+sol.u
1-element Vector{Float64}:
+ 0.9035843608115925

Vector-Valued Functions

ExpectationProblem can also handle vector-valued functions. Simply pass the vector-valued function and set the nout kwarg to the length of the vector the function returns.

Here, we demonstrate this by computing the expectation of u at t=4.0s and t=6.0s

g(sol, p) = [sol(4.0)[1], sol(6.0)[1]]
+exprob = ExpectationProblem(sm, g, h, gd; nout = 2)
+sol = solve(exprob, Koopman())
+sol.u
2-element Vector{Float64}:
+ 0.9035843608115925
+ 0.4958955682159018

with analytical solution

exp.(p .* [4.0, 6.0]) * mean(u0_dist[1])
2-element Vector{Float64}:
+ 0.9035826357366064
+ 0.4958966646647597

this can be used to compute the expectation at a range of times simultaneously

saveat = tspan[1]:0.5:tspan[2]
+g(sol, p) = sol[1, :]
+prob = ODEProblem(f, u0, tspan, p, saveat = saveat)
+sm = SystemMap(prob, Tsit5())
+exprob = ExpectationProblem(sm, g, h, gd; nout = length(saveat))
+sol = solve(exprob, Koopman())
+sol.u
21-element Vector{Float64}:
+ 3.00000342997363
+ 2.5821268324641955
+ 2.222456270936201
+ 1.9128866158620428
+ 1.6464354737189055
+ 1.4171010439585803
+ 1.2197090633820724
+ 1.0498111825687724
+ 0.903584360811593
+ 0.7777191272838139
+ ⋮
+ 0.49589556821590175
+ 0.42681769377636924
+ 0.36736961463153206
+ 0.31620019646595693
+ 0.27215081053485346
+ 0.23424232180420576
+ 0.2016186364835746
+ 0.17353590778751105
+ 0.1493637534733821

We can then plot these values along with the analytical solution

plot(t -> exp(p[1] * t) * mean(u0_dist[1]), tspan..., xlabel = "t", label = "analytical")
+scatter!(collect(saveat), sol.u, marker = :o, label = nothing)
Example block output

Benefits of Using Vector-Valued Functions

In the above examples, we used vector-valued expectation calculations to compute the various expectations required. Alternatively, one could simply compute multiple scalar-valued expectations. However, in most cases it is more efficient to use the vector-valued form. This is especially true when the ODE to be solved is computationally expensive.

To demonstrate this, let's compute the expectation of $x$, $x^2$, and $x^3$ using both approaches while counting the number of times g() is evaluated. This is the same as the number of simulation runs required to arrive at the solution. First, consider the scalar-valued approach. Here, we follow the same method as before, but we add a counter to our function evaluation that stores the number of function calls for each expectation calculation to an array.

function g(sol, p, power, counter)
+    counter[power] += 1
+    sol(4.0)[1]^power
+end
+counter = [0, 0, 0]
+g(sol, p, power) = g(sol, p, power, counter)
+g(sol, p) = g(sol, p, 1)
+exprob = ExpectationProblem(sm, g, h, gd; nout = 1)
+sol = solve(exprob, Koopman())
+sol.u
0.903584360811593
g(sol, p) = g(sol, p, 2)
+exprob = ExpectationProblem(sm, g, h, gd; nout = 1)
+sol = solve(exprob, Koopman())
+sol.u
1.1793351995541244
g(sol, p) = g(sol, p, 3)
+exprob = ExpectationProblem(sm, g, h, gd; nout = 1)
+sol = solve(exprob, Koopman())
+sol.u
1.7213917763291955
counter
3-element Vector{Int64}:
+ 375
+ 405
+ 375

Leading to a total of j sum(counters) function evaluations.

Now, let's compare this to the vector-valued approach

function g(sol, p, counter)
+    counter[1] += 1
+    v = sol(4.0)[1]
+    [v, v^2, v^3]
+end
+counter = [0]
+g(sol, p) = g(sol, p, counter)
+exprob = ExpectationProblem(sm, g, h, gd; nout = 3)
+sol = solve(exprob, Koopman())
+sol.u
3-element Vector{Float64}:
+ 0.9035833264285407
+ 1.1793351995541244
+ 1.721398472894846
counter
1-element Vector{Int64}:
+ 405

This is j round(counter[1]/sum(counters)*100,digits=2)% the number of simulations required when using scalar-valued expectations. Note how the number of evaluations used in the vector-valued form is equivalent to the maximum number of evaluations for the 3 scalar-valued expectation calls.

Higher-Order Moments

Leveraging this vector-valued capability, we can also efficiently compute higher-order central moments.

Variance

The variance, or 2nd central moment, of a random variable $X$ is defined as

\[\mathrm{Var}\left(X\right)=\mathbb{E}\left[\left(X-\mu\right)^2\right]\]

where

\[\mu = \mathbb{E}\left[X\right]\]

The expression for the variance can be expanded to

\[\mathrm{Var}\left(X\right)=\mathbb{E}\left[X^2\right]-\mathbb{E}\left[X\right]^2\]

Using this, we define a function that returns the expectations of $X$ and $X^2$ as a vector-valued function and then compute the variance from these

function g(sol, p)
+    x = sol(4.0)[1]
+    [x, x^2]
+end
+exprob = ExpectationProblem(sm, g, h, gd; nout = 2)
+sol = solve(exprob, Koopman())
+sol.u
+mean_g = sol.u[1]
+var_g = sol.u[2] - mean_g^2
0.36287237175445763

For a linear system, we can propagate the variance analytically as

\[e^{2pt}\mathrm{Var}\left(u_0\right)\]

exp(2 * p[1] * 4.0) * var(u0_dist[1])
0.36287181315765005

This can be computed at multiple time instances as well

saveat = tspan[1]:0.5:tspan[2]
+g(sol, p) = [sol[1, :]; sol[1, :] .^ 2]
+prob = ODEProblem(f, u0, tspan, p, saveat = saveat)
+sm = SystemMap(prob, Tsit5())
+exprob = ExpectationProblem(sm, g, h, gd; nout = length(saveat) * 2)
+sol = solve(exprob, Koopman())
+sol.u
+
+μ = sol.u[1:length(saveat)]
+σ = sqrt.(sol.u[(length(saveat) + 1):end] - μ .^ 2)
+
+plot(t -> exp(p[1] * t) * mean(u0_dist[1]), tspan...,
+     ribbon = t -> -sqrt(exp(2 * p[1] * t) * var(u0_dist[1])),
+     label = "Analytical Mean, 1 std bounds")
+scatter!(collect(saveat), μ, marker = :x, yerror = σ, c = :black,
+         label = "Koopman Mean, 1 std bounds")
Example block output

Skewness

A similar approach can be used to compute skewness

function g(sol, p)
+    x = sol(4.0)[1]
+    [x, x^2, x^3]
+end
+exprob = ExpectationProblem(sm, g, h, gd; nout = 3)
+sol = solve(exprob, Koopman())
+sol.u
+mean_g = sol.u[1]
+var_g = sol.u[2] - mean_g^2
+skew_g = (sol.u[3] - 3.0 * mean_g * var_g - mean_g^3) / var_g^(3 / 2)
3.795455433349072e-9

As the system is linear, we expect the skewness to be unchanged from the initial distribution. Because the distribution is a truncated Normal distribution centered on the mean, the true skewness is 0.0.

Batch-Mode

It is also possible to solve the various simulations in parallel by using the batch kwarg and a batch-mode supported quadrature algorithm via the quadalg kwarg. To view the list of batch compatible quadrature algorithms, refer to Integrals.jl. Note: Batch-mode operation is built on top of DifferentialEquation.jl's EnsembleProblem. See the EnsembleProblem documentation for additional options.

The default quadrature algorithm to solve ExpectationProblem does not support batch-mode evaluation. So, we first load dependencies for additional quadrature algorithms

using IntegralsCuba

We then solve our expectation as before, using a batch=10 multi-thread parallelization via EnsembleThreads() of Cuba's SUAVE algorithm. We also introduce additional uncertainty in the model parameter.

u0_dist = truncated(Normal(3.0, 2.0), -5, 11)
+p_dist = truncated(Normal(-0.7, 0.1), -1, 0)
+gd = GenericDistribution(u0_dist, p_dist)
+g(sol, p) = sol(6.0)[1]
+h(x, u, p) = [x[1]], [x[2]]
+sm = SystemMap(prob, Tsit5(), EnsembleThreads())
+exprob = ExpectationProblem(sm, g, h, gd; nout = 1)
+# batchmode = EnsembleThreads() #where to pass this?
+sol = solve(exprob, Koopman(), batch = 10, quadalg = CubaSUAVE())
+sol.u
1-element Vector{Float64}:
+ 0.053978607864753524

Now, let's compare the performance of the batch and non-batch modes

@time solve(exprob, Koopman(), batch = 10, quadalg = CubaSUAVE())
SciMLExpectations.ExpectationSolution{Vector{Float64}, Vector{Float64}, SciMLBase.IntegralSolution{Float64, 1, Vector{Float64}, Vector{Float64}, IntegralProblem{true, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, SciMLExpectations.var"#28#35"{SciMLExpectations.var"#output_func#34"{GenericDistribution{SciMLExpectations.var"#pdf_func#10"{Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Float64}}}, SciMLExpectations.var"#rand_func#12"{Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Float64}}}, StaticArraysCore.SVector{2, Float64}, StaticArraysCore.SVector{2, Float64}}, typeof(Main.g)}, SciMLExpectations.var"#27#33"{typeof(Main.h)}, SystemMap{ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, false, Vector{Float64}, ODEFunction{false, SciMLBase.AutoSpecialize, typeof(Main.f), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, Tuple{Symbol}, NamedTuple{(:saveat,), Tuple{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, EnsembleThreads, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}}}, StaticArraysCore.SVector{2, Float64}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}}, IntegralsCuba.CubaSUAVE{Float64}, Vector{Float64}, Nothing}}([0.053978607864753524], [0.004462032598640491], [0.053978607864753524])
solve(exprob, Koopman(), batch = 0, quadalg = CubaSUAVE())
+@time solve(exprob, Koopman(), batch = 0, quadalg = CubaSUAVE())
SciMLExpectations.ExpectationSolution{Float64, Vector{Float64}, SciMLBase.IntegralSolution{Float64, 0, Float64, Vector{Float64}, IntegralProblem{false, RecursiveArrayTools.ArrayPartition{Float64, Tuple{Vector{Float64}, Vector{Float64}}}, SciMLExpectations.var"#23#24"{GenericDistribution{SciMLExpectations.var"#pdf_func#10"{Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Float64}}}, SciMLExpectations.var"#rand_func#12"{Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Float64}}}, StaticArraysCore.SVector{2, Float64}, StaticArraysCore.SVector{2, Float64}}, typeof(Main.h), typeof(Main.g), SystemMap{ODEProblem{Vector{Float64}, Tuple{Float64, Float64}, false, Vector{Float64}, ODEFunction{false, SciMLBase.AutoSpecialize, typeof(Main.f), LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing}, Base.Pairs{Symbol, StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}, Tuple{Symbol}, NamedTuple{(:saveat,), Tuple{StepRangeLen{Float64, Base.TwicePrecision{Float64}, Base.TwicePrecision{Float64}, Int64}}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEq.Tsit5{typeof(OrdinaryDiffEq.trivial_limiter!), typeof(OrdinaryDiffEq.trivial_limiter!), Static.False}, EnsembleThreads, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}}}, StaticArraysCore.SVector{2, Float64}, Base.Pairs{Symbol, Union{}, Tuple{}, NamedTuple{(), Tuple{}}}}, IntegralsCuba.CubaSUAVE{Float64}, Vector{Float64}, Nothing}}(0.053978607864753524, [0.004462032598640491], fill(0.053978607864753524))

It is also possible to parallelize across the GPU. However, one must be careful of the limitations of ensemble solutions with the GPU. Please refer to DiffEqGPU.jl for details.

Here we load DiffEqGPU and modify our problem to use Float32 and to put the ODE in the required GPU form

Note

Switch EnsembleCPUArray() to EnsembleGPUArray() to make this example work on your GPU. Currently, these docs are built on a machine without a GPU.

using DiffEqGPU
+
+function f(du, u, p, t)
+    @inbounds begin du[1] = p[1] * u[1] end
+    nothing
+end
+
+u0 = Float32[10.0]
+p = Float32[-0.3]
+tspan = (0.0f0, 10.0f0)
+prob = ODEProblem(f, u0, tspan, p)
+
+g(sol, p) = sol(6.0)[1]
+
+u0_dist = truncated(Normal(3.0f0, 2.0f0), -5.0f0, 11.0f0)
+p_dist = truncated(Normal(-0.7f0, 0.1f0), -1.0f0, 0.0f0)
+gd = GenericDistribution(u0_dist, p_dist)
+g(sol, p) = sol(6.0f0)[1]
+h(x, u, p) = [x[1]], [x[2]]
+prob = ODEProblem(f, u0, tspan, p)
+sm = SystemMap(prob, Tsit5(), EnsembleCPUArray())
+exprob = ExpectationProblem(sm, g, h, gd; nout = 1)
+sol = solve(exprob, Koopman(), batch = 10, quadalg = CubaSUAVE())
+sol.u

The performance gains realized by leveraging batch GPU processing is problem-dependent. In this case, the number of batch evaluations required to overcome the overhead of using the GPU exceeds the number of simulations required to converge to the quadrature solution.

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GitHub

Optimization Under Uncertainty

This tutorial showcases how to leverage the efficient Koopman expectation method from SciMLExpectations to perform optimization under uncertainty. We demonstrate this by using a bouncing ball model with an uncertain model parameter. We also demonstrate its application to problems with probabilistic constraints, in particular a special class of constraints called chance constraints.

System Model

First let's consider a 2D bouncing ball, where the states are the horizontal position $x$, horizontal velocity $\dot{x}$, vertical position $y$, and vertical velocity $\dot{y}$. This model has two system parameters, acceleration due to gravity and coefficient of restitution (models energy loss when the ball impacts the ground). We can simulate such a system using ContinuousCallback as

using DifferentialEquations, Plots
+
+function ball!(du, u, p, t)
+    du[1] = u[2]
+    du[2] = 0.0
+    du[3] = u[4]
+    du[4] = -p[1]
+end
+
+ground_condition(u, t, integrator) = u[3]
+ground_affect!(integrator) = integrator.u[4] = -integrator.p[2] * integrator.u[4]
+ground_cb = ContinuousCallback(ground_condition, ground_affect!)
+
+u0 = [0.0, 2.0, 50.0, 0.0]
+tspan = (0.0, 50.0)
+p = [9.807, 0.9]
+
+prob = ODEProblem(ball!, u0, tspan, p)
+sol = solve(prob, Tsit5(), callback = ground_cb)
+plot(sol, vars = (1, 3), label = nothing, xlabel = "x", ylabel = "y")
Example block output

For this particular problem, we wish to measure the impact distance from a point $y=25$ on a wall at $x=25$. So, we introduce an additional callback that terminates the simulation on wall impact.

stop_condition(u, t, integrator) = u[1] - 25.0
+stop_cb = ContinuousCallback(stop_condition, terminate!)
+cbs = CallbackSet(ground_cb, stop_cb)
+
+tspan = (0.0, 1500.0)
+prob = ODEProblem(ball!, u0, tspan, p)
+sol = solve(prob, Tsit5(), callback = cbs)
+plot(sol, vars = (1, 3), label = nothing, xlabel = "x", ylabel = "y")
Example block output

To help visualize this problem, we plot as follows, where the star indicates a desired impact location

rectangle(xc, yc, w, h) = Shape(xc .+ [-w, w, w, -w] ./ 2.0, yc .+ [-h, -h, h, h] ./ 2.0)
+
+begin
+    plot(sol, vars = (1, 3), label = nothing, lw = 3, c = :black)
+    xlabel!("x [m]")
+    ylabel!("y [m]")
+    plot!(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)
+    scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)
+    ylims!(0.0, 50.0)
+end
Example block output

Considering Uncertainty

We now wish to introduce uncertainty in p[2], the coefficient of restitution. This is defined via a continuous univariate distribution from Distributions.jl. We can then run a Monte Carlo simulation of 100 trajectories via the EnsembleProblem interface.

using Distributions
+
+cor_dist = truncated(Normal(0.9, 0.02), 0.9 - 3 * 0.02, 1.0)
+trajectories = 100
+
+prob_func(prob, i, repeat) = remake(prob, p = [p[1], rand(cor_dist)])
+ensemble_prob = EnsembleProblem(prob, prob_func = prob_func)
+ensemblesol = solve(ensemble_prob, Tsit5(), EnsembleThreads(), trajectories = trajectories,
+    callback = cbs)
+
+begin # plot
+    plot(ensemblesol, vars = (1, 3), lw = 1)
+    xlabel!("x [m]")
+    ylabel!("y [m]")
+    plot!(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)
+    scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)
+    plot!(sol, vars = (1, 3), label = nothing, lw = 3, c = :black, ls = :dash)
+    xlims!(0.0, 27.5)
+end
Example block output

Here, we plot the first 350 Monte Carlo simulations along with the trajectory corresponding to the mean of the distribution (dashed line).

We now wish to compute the expected squared impact distance from the star. This is called an “observation” of our system or an “observable” of interest.

We define this observable as

obs(sol, p) = abs2(sol[3, end] - 25)
obs (generic function with 1 method)

With the observable defined, we can compute the expected squared miss distance from our Monte Carlo simulation results as

mean_ensemble = mean([obs(sol, p) for sol in ensemblesol])
35.90267990627214

Alternatively, we can use the Koopman() algorithm in SciMLExpectations.jl to compute this expectation much more efficiently as

using SciMLExpectations
+gd = GenericDistribution(cor_dist)
+h(x, u, p) = u, [p[1]; x[1]]
+sm = SystemMap(prob, Tsit5(), callback = cbs)
+exprob = ExpectationProblem(sm, obs, h, gd; nout = 1)
+sol = solve(exprob, Koopman(), ireltol = 1e-5)
+sol.u
36.00862821417056

Optimization Under Uncertainty

We now wish to optimize the initial position ($x_0,y_0$) and horizontal velocity ($\dot{x}_0$) of the system to minimize the expected squared miss distance from the star, where $x_0\in\left[-100,0\right]$, $y_0\in\left[1,3\right]$, and $\dot{x}_0\in\left[10,50\right]$. We will demonstrate this using a gradient-based optimization approach from NLopt.jl using ForwardDiff.jl AD through the expectation calculation.

using Optimization, OptimizationNLopt, OptimizationMOI
+make_u0(θ) = [θ[1], θ[2], θ[3], 0.0]
+function 𝔼_loss(θ, pars)
+    prob = ODEProblem(ball!, make_u0(θ), tspan, p)
+    sm = SystemMap(prob, Tsit5(), callback = cbs)
+    exprob = ExpectationProblem(sm, obs, h, gd; nout = 1)
+    sol = solve(exprob, Koopman(), ireltol = 1e-5)
+    sol.u
+end
+opt_f = OptimizationFunction(𝔼_loss, Optimization.AutoForwardDiff())
+opt_ini = [-1.0, 2.0, 50.0]
+opt_lb = [-100.0, 1.0, 10.0]
+opt_ub = [0.0, 3.0, 50.0]
+opt_prob = OptimizationProblem(opt_f, opt_ini; lb = opt_lb, ub = opt_ub)
+optimizer = OptimizationMOI.MOI.OptimizerWithAttributes(NLopt.Optimizer,
+    "algorithm" => :LD_MMA)
+opt_sol = solve(opt_prob, optimizer)
+minx = opt_sol.u
3-element Vector{Float64}:
+  0.0
+  2.4428947026478425
+ 49.20927899180528

Let's now visualize 100 Monte Carlo simulations

ensembleprob = EnsembleProblem(remake(prob, u0 = make_u0(minx)), prob_func = prob_func)
+ensemblesol = solve(ensembleprob, Tsit5(), EnsembleThreads(), trajectories = 100,
+    callback = cbs)
+
+begin
+    plot(ensemblesol, vars = (1, 3), lw = 1, alpha = 0.1)
+    plot!(solve(remake(prob, u0 = make_u0(minx)), Tsit5(), callback = cbs),
+        vars = (1, 3), label = nothing, c = :black, lw = 3, ls = :dash)
+    xlabel!("x [m]")
+    ylabel!("y [m]")
+    plot!(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)
+    scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)
+    ylims!(0.0, 50.0)
+    xlims!(minx[1], 27.5)
+end
Example block output

Looks pretty good! But, how long did it take? Let's benchmark.

@time solve(opt_prob, optimizer)
u: 3-element Vector{Float64}:
+  0.0
+  2.4428947026478425
+ 49.20927899180528

Not bad for bound constrained optimization under uncertainty of a hybrid system!

Probabilistic Constraints

With this approach, we can also consider probabilistic constraints. Let us now consider a wall at $x=20$ with height 25.

constraint = [20.0, 25.0]
+begin
+    plot(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)
+    xlabel!("x [m]")
+    ylabel!("y [m]")
+    plot!([constraint[1], constraint[1]], [0.0, constraint[2]], lw = 5, c = :black,
+        label = nothing)
+    scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)
+    ylims!(0.0, 50.0)
+    xlims!(minx[1], 27.5)
+end
Example block output

We now wish to minimize the same loss function as before, but introduce an inequality constraint such that the solution must have less than a 1% chance of colliding with the wall at $x=20$. This class of probabilistic constraints is called a chance constraint.

To do this, we first introduce a new callback and solve the system using the previous optimal solution

constraint_condition(u, t, integrator) = u[1] - constraint[1]
+function constraint_affect!(integrator)
+    integrator.u[3] < constraint[2] ? terminate!(integrator) : nothing
+end
+constraint_cb = ContinuousCallback(constraint_condition, constraint_affect!,
+    save_positions = (true, false));
+constraint_cbs = CallbackSet(ground_cb, stop_cb, constraint_cb)
+
+ensemblesol = solve(ensembleprob, Tsit5(), EnsembleThreads(), trajectories = 500,
+    callback = constraint_cbs)
+
+begin
+    plot(ensemblesol, vars = (1, 3), lw = 1, alpha = 0.1)
+    plot!(solve(remake(prob, u0 = make_u0(minx)), Tsit5(), callback = constraint_cbs),
+        vars = (1, 3), label = nothing, c = :black, lw = 3, ls = :dash)
+
+    xlabel!("x [m]")
+    ylabel!("y [m]")
+    plot!(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)
+    plot!([constraint[1], constraint[1]], [0.0, constraint[2]], lw = 5, c = :black)
+    scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)
+    ylims!(0.0, 50.0)
+    xlims!(minx[1], 27.5)
+end
Example block output

That doesn't look good!

We now need a second observable for the system. To compute a probability of impact, we use an indicator function for if a trajectory impacts the wall. In other words, this functions returns 1 if the trajectory hits the wall and 0 otherwise.

function constraint_obs(sol, p)
+    sol((constraint[1] - sol[1, 1]) / sol[2, 1])[3] <= constraint[2] ? one(sol[1, end]) :
+    zero(sol[1, end])
+end
constraint_obs (generic function with 1 method)

Using the previously computed optimal initial conditions, let's compute the probability of hitting this wall

sm = SystemMap(remake(prob, u0 = make_u0(minx)), Tsit5(), callback = cbs)
+exprob = ExpectationProblem(sm, constraint_obs, h, gd; nout = 1)
+sol = solve(exprob, Koopman(), ireltol = 1e-5)
+sol.u
0.9988981583360147

We then set up the constraint function for NLopt just as before.

function 𝔼_constraint(res, θ, pars)
+    prob = ODEProblem(ball!, make_u0(θ), tspan, p)
+    sm = SystemMap(prob, Tsit5(), callback = cbs)
+    exprob = ExpectationProblem(sm, constraint_obs, h, gd; nout = 1)
+    sol = solve(exprob, Koopman(), ireltol = 1e-5)
+    res .= sol.u
+end
+opt_lcons = [-Inf]
+opt_ucons = [0.01]
+optimizer = OptimizationMOI.MOI.OptimizerWithAttributes(NLopt.Optimizer,
+    "algorithm" => :LD_MMA)
+opt_f = OptimizationFunction(𝔼_loss, Optimization.AutoForwardDiff(), cons = 𝔼_constraint)
+opt_prob = OptimizationProblem(opt_f, opt_ini; lb = opt_lb, ub = opt_ub, lcons = opt_lcons,
+    ucons = opt_ucons)
+opt_sol = solve(opt_prob, optimizer)
+minx2 = opt_sol.u
3-element Vector{Float64}:
+ -0.8933087495240182
+  2.0005561486807566
+ 50.0

The probability of impacting the wall is now

container = zeros(1)
+𝔼_constraint(container, minx2, nothing)
+λ = container[1]
0.0024804394521740487

We can check if this is within tolerance by

λ <= 0.01 + 1e-5
true

Again, we plot some Monte Carlo simulations from this result as follows

ensembleprob = EnsembleProblem(remake(prob, u0 = make_u0(minx2)), prob_func = prob_func)
+ensemblesol = solve(ensembleprob, Tsit5(), EnsembleThreads(),
+    trajectories = 500, callback = constraint_cbs)
+
+begin
+    plot(ensemblesol, vars = (1, 3), lw = 1, alpha = 0.1)
+    plot!(solve(remake(prob, u0 = make_u0(minx2)), Tsit5(), callback = constraint_cbs),
+        vars = (1, 3), label = nothing, c = :black, lw = 3, ls = :dash)
+    plot!([constraint[1], constraint[1]], [0.0, constraint[2]], lw = 5, c = :black)
+
+    xlabel!("x [m]")
+    ylabel!("y [m]")
+    plot!(rectangle(27.5, 25, 5, 50), c = :red, label = nothing)
+    scatter!([25], [25], marker = :star, ms = 10, label = nothing, c = :green)
+    ylims!(0.0, 50.0)
+    xlims!(minx[1], 27.5)
+end
Example block output
diff --git a/v2.1.0/tutorials/process_noise/index.html b/v2.1.0/tutorials/process_noise/index.html new file mode 100644 index 0000000..f03f9f5 --- /dev/null +++ b/v2.1.0/tutorials/process_noise/index.html @@ -0,0 +1,29 @@ + +Expectation of Process Noise · SciMLExpectations.jl
GitHub

Expectation of Process Noise

SciMLExpectations.jl is able to calculate the average trajectory of a stochastic differential equation. This done by representing the Wiener process using the Kosambi–Karhunen–Loève theorem.

using SciMLExpectations
+using IntegralsCuba
+using StochasticDiffEq
+using DiffEqNoiseProcess
+using Distributions
+
+f(du, u, p, t) = (du .= u)
+g(du, u, p, t) = (du .= u)
+u0 = collect(1:4)
+
+W = WienerProcess(0.0, 0.0, 0.0)
+prob = SDEProblem(f, g, u0, (0.0, 1.0), noise = W)
+sm = ProcessNoiseSystemMap(prob, 8, LambaEM(), abstol = 1e-3, reltol = 1e-3)
+cov(x, u, p) = x, p
+observed(sol, p) = sol[:, end]
+exprob = ExpectationProblem(sm, observed, cov; nout = length(u0))
+sol1 = solve(exprob, Koopman(), ireltol = 1e-3, iabstol = 1e-3, batch = 64,
+             quadalg = CubaDivonne())
+sol1.u
4-element Vector{Float64}:
+  4.371665736791912
+  8.743331473583824
+ 13.114997210375737
+ 17.48666294716765
sol2 = solve(exprob, MonteCarlo(1_000_000))
+sol2.u
4-element Vector{Float64}:
+  4.375566158316454
+  8.751132316632908
+ 13.126698474949364
+ 17.502264633265817

In theory, any numerical integration method from Integrals.jl is supported, but in practice many of the techniques struggle with correctly calculating the expected value. We got the best results with the Divonne algorithm from Cuba.jl.

diff --git a/versions.js b/versions.js index 19c0ce3..0cc0eac 100644 --- a/versions.js +++ b/versions.js @@ -1,7 +1,8 @@ var DOC_VERSIONS = [ "stable", + "v2.1", "v2.0", "dev", ]; -var DOCUMENTER_NEWEST = "v2.0.1"; +var DOCUMENTER_NEWEST = "v2.1.0"; var DOCUMENTER_STABLE = "stable";