add Trapezoidal rule

Project.toml

@@ -13,29 +13,36 @@ Reexport = "189a3867-3050-52da-a836-e630ba90ab69"
 Requires = "ae029012-a4dd-5104-9daa-d747884805df"
 SciMLBase = "0bca4576-84f4-4d90-8ffe-ffa030f20462"
 
+[weakdeps]
+ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838"
+ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
+Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
+
+[extensions]
+IntegralsFastGaussQuadratureExt = "FastGaussQuadrature"
+IntegralsForwardDiffExt = "ForwardDiff"
+IntegralsZygoteExt = ["Zygote", "ChainRulesCore"]
+
 [compat]
 ChainRulesCore = "0.10.7, 1"
 CommonSolve = "0.2"
 Distributions = "0.23, 0.24, 0.25"
+FastGaussQuadrature = "0.5"
 ForwardDiff = "0.10"
 HCubature = "1.4"
 MonteCarloIntegration = "0.0.1, 0.0.2, 0.0.3"
 QuadGK = "2.5"
 Reexport = "0.2, 1.0"
 Requires = "1"
-SciMLBase = "1.70"
+SciMLBase = "1.98"
 Zygote = "0.4.22, 0.5, 0.6"
 julia = "1.6"
-FastGaussQuadrature = "0.5"
-
-[extensions]
-IntegralsForwardDiffExt = "ForwardDiff"
-IntegralsZygoteExt = ["Zygote", "ChainRulesCore"]
-IntegralsFastGaussQuadratureExt = "FastGaussQuadrature"
 
 [extras]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
 Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
+FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838"
 FiniteDiff = "6a86dc24-6348-571c-b903-95158fe2bd41"
 ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 Pkg = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
@@ -44,13 +51,6 @@ SciMLSensitivity = "1ed8b502-d754-442c-8d5d-10ac956f44a1"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
-FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838"
 
 [targets]
 test = ["SciMLSensitivity", "StaticArrays", "FiniteDiff", "Pkg", "SafeTestsets", "Test", "Distributions", "ForwardDiff", "Zygote", "ChainRulesCore", "FastGaussQuadrature"]
-
-[weakdeps]
-ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
-ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
-Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
-FastGaussQuadrature = "442a2c76-b920-505d-bb47-c5924d526838"

docs/pages.jl

@@ -2,6 +2,7 @@ pages = ["index.md",
     "Tutorials" => Any["tutorials/numerical_integrals.md",
         "tutorials/differentiating_integrals.md"],
     "Basics" => Any["basics/IntegralProblem.md",
+        "basics/SampledIntegralProblem.md",
         "basics/solve.md",
         "basics/FAQ.md"],
     "Solvers" => Any["solvers/IntegralSolvers.md"],

docs/src/basics/SampledIntegralProblem.md

@@ -0,0 +1,73 @@
+# Integrating pre-sampled data
+
+In some cases, instead of a function that acts as integrand, 
+one only possesses a list of data points `y` at a set of sampling 
+locations `x`, that must be integrated. This package contains functionality
+for doing that. 
+
+## Example
+
+Say, by some means we have generated a dataset `x` and `y`:
+```example 1
+using Integrals # hide
+f = x -> x^2
+x = range(0, 1, length=20)
+y = f.(x)
+```
+
+Now, we can integrate this data set as follows:
+
+```example 1
+problem = SampledIntegralProblem(y, x)
+method = TrapezoidalRule()
+solve(problem, method)
+```
+
+The exact aswer is of course \$ 1/3 \$.
+
+## Details
+
+### Non-equidistant grids
+
+If the sampling points `x` are provided as an `AbstractRange` 
+(constructed with the `range` function for example), faster methods are used that take advantage of
+the fact that the points are equidistantly spaced. Otherwise, general methods are used for 
+non-uniform grids.
+
+Example:
+
+```example 2
+using Integrals # hide
+f = x -> x^7
+x = [0.0; sort(rand(1000)); 1.0]
+y = f.(x)
+problem = SampledIntegralProblem(y, x)
+method = TrapezoidalRule()
+solve(problem, method)
+```
+
+### Evaluating multiple integrals at once
+
+If the provided data set `y` is a multidimensional array, the integrals are evaluated across only one
+of its axes. For performance reasons, the last axis of the array `y` is chosen by default, but this can be modified with the `dim`
+keyword argument to the problem definition.
+
+```example 3
+using Integrals # hide
+f1 = x -> x^2
+f2 = x -> x^3
+f3 = x -> x^4
+x = range(0, 1, length=20)
+y = [f1.(x) f2.(x) f3.(x)]
+problem = SampledIntegralProblem(y, x; dim=1)
+method = TrapezoidalRule()
+solve(problem, method)
+```
+
+### Supported methods
+
+Right now, only the `TrapezoidalRule` is supported, [see wikipedia](https://en.wikipedia.org/wiki/Trapezoidal_rule).
+
+```@docs
+TrapezoidalRule
+```

docs/src/tutorials/caching_interface.md

@@ -50,3 +50,50 @@ Note that the types of these variables is not allowed to change.
 If it is necessary to change the integrand `f` instead of defining a new
 `IntegralProblem`, consider using
 [FunctionWrappers.jl](https://github.com/yuyichao/FunctionWrappers.jl).
+
+## Caching for sampled integral problems
+
+For sampled integral problems, it is possible to cache the weights and reuse
+them for multiple data sets.
+```@example cache2
+using Integrals
+
+x = 0.0:0.1:1.0
+y = sin.(x)
+
+prob = SampledIntegralProblem(y, x)
+alg = TrapezoidalRule()
+
+cache = init(prob, alg)
+sol1 = solve!(cache)
+```
+
+```@example cache2
+cache.y = cos.(x)   # use .= to update in-place
+sol2 = solve!(cache)
+```
+If the grid is modified, the weights are recomputed.
+```@example cache2
+cache.x = 0.0:0.2:2.0
+cache.y = sin.(cache.x)
+sol3 = solve!(cache)
+```
+
+For multi-dimensional datasets, the integration dimension can also be changed
+```@example cache3
+using Integrals
+
+x = 0.0:0.1:1.0
+y = sin.(x) .* cos.(x')
+
+prob = SampledIntegralProblem(y, x)
+alg = TrapezoidalRule()
+
+cache = init(prob, alg)
+sol1 = solve!(cache)
+```
+
+```@example cache3
+cache.dim = 1
+sol2 = solve!(cache)
+```

src/Integrals.jl

@@ -13,6 +13,8 @@ include("init.jl")
 include("algorithms.jl")
 include("infinity_handling.jl")
 include("quadrules.jl")
+include("sampled.jl")
+include("trapezoidal.jl")
 
 abstract type QuadSensitivityAlg end
 struct ReCallVJP{V}
@@ -148,5 +150,5 @@ function __solvebp_call(prob::IntegralProblem, alg::VEGAS, sensealg, lb, ub, p;
     SciMLBase.build_solution(prob, alg, val, err, chi = chi, retcode = ReturnCode.Success)
 end
 
-export QuadGKJL, HCubatureJL, VEGAS, GaussLegendre, QuadratureRule
+export QuadGKJL, HCubatureJL, VEGAS, GaussLegendre, QuadratureRule, TrapezoidalRule
 end # module

src/algorithms.jl

@@ -124,7 +124,28 @@ function GaussLegendre(; n = 250, subintervals = 1, nodes = nothing, weights = n
 end
 
 """
-    QuadratureRule(q; n=250)
+    TrapezoidalRule
+
+Struct for evaluating an integral via the trapezoidal rule.
+
+
+Example with sampled data:
+
+```
+using Integrals 
+f = x -> x^2
+x = range(0, 1, length=20)
+y = f.(x)
+problem = SampledIntegralProblem(y, x)
+method = TrapezoidalRul()
+solve(problem, method)
+```
+"""
+struct TrapezoidalRule <: SciMLBase.AbstractIntegralAlgorithm
+end
+
+"""
+  QuadratureRule(q; n=250)
 
 Algorithm to construct and evaluate a quadrature rule `q` of `n` points computed from the
 inputs as `x, w = q(n)`. It assumes the nodes and weights are for the standard interval

src/common.jl

@@ -94,3 +94,65 @@ end
 function __solvebp_call(cache::IntegralCache, args...; kwargs...)
     __solvebp_call(build_problem(cache), args...; kwargs...)
 end
+
+
+mutable struct SampledIntegralCache{Y, X, D, PK, A, K, Tc}
+    y::Y
+    x::X
+    dim::D
+    prob_kwargs::PK
+    alg::A
+    kwargs::K
+    isfresh::Bool   # state of whether weights have been calculated
+    cacheval::Tc    # store alg weights here
+end
+
+function Base.setproperty!(cache::SampledIntegralCache, name::Symbol, x)
+    if name === :x
+        setfield!(cache, :isfresh, true)
+    end
+    setfield!(cache, name, x)
+end
+
+function SciMLBase.init(prob::SampledIntegralProblem,
+    alg::SciMLBase.AbstractIntegralAlgorithm;
+    kwargs...)
+    NamedTuple(kwargs) == NamedTuple() || throw(ArgumentError("There are no keyword arguments allowed to `solve`"))
+
+    cacheval = init_cacheval(alg, prob)
+    isfresh = true
+
+    SampledIntegralCache(
+        prob.y,
+        prob.x,
+        prob.dim,
+        prob.kwargs,
+        alg,
+        kwargs,
+        isfresh,
+        cacheval)
+end
+
+
+"""
+```julia
+solve(prob::SampledIntegralProblem, alg::SciMLBase.AbstractIntegralAlgorithm; kwargs...)
+```
+
+## Keyword Arguments
+
+There are no keyword arguments used to solve `SampledIntegralProblem`s
+"""
+function SciMLBase.solve(prob::SampledIntegralProblem,
+    alg::SciMLBase.AbstractIntegralAlgorithm;
+    kwargs...)
+    solve!(init(prob, alg; kwargs...))
+end
+
+function SciMLBase.solve!(cache::SampledIntegralCache)
+    __solvebp(cache, cache.alg; cache.kwargs...)
+end
+
+function build_problem(cache::SampledIntegralCache)
+    SampledIntegralProblem(cache.y, cache.x; dim = dimension(cache.dim), cache.prob_kwargs...)
+end

src/sampled.jl

@@ -0,0 +1,71 @@
+abstract type AbstractWeights end
+
+# must have field `n` for length, and a field `h` for stepsize
+abstract type UniformWeights <: AbstractWeights end
+@inline Base.iterate(w::UniformWeights) = (0 == w.n) ? nothing : (w[1], 1)
+@inline Base.iterate(w::UniformWeights, i) = (i == w.n) ? nothing : (w[i+1], i+1)
+Base.length(w::UniformWeights) = w.n
+Base.eltype(w::UniformWeights) = typeof(w.h)
+Base.size(w::UniformWeights) = (length(w), )
+
+# must contain field `x` which are the sampling points
+abstract type NonuniformWeights <: AbstractWeights end
+@inline Base.iterate(w::NonuniformWeights) = (0 == length(w.x)) ? nothing : (w[firstindex(w.x)], firstindex(w.x))
+@inline Base.iterate(w::NonuniformWeights, i) = (i == lastindex(w.x)) ? nothing : (w[i+1], i+1)
+Base.length(w::NonuniformWeights) = length(w.x)
+Base.eltype(w::NonuniformWeights) = eltype(w.x)
+Base.size(w::NonuniformWeights) = (length(w), )
+
+_eachslice(data::AbstractArray; dims=ndims(data)) = eachslice(data; dims=dims)
+_eachslice(data::AbstractArray{T, 1}; dims=ndims(data)) where T = data
+
+
+# these can be removed when the Val(dim) is removed from SciMLBase
+dimension(::Val{D}) where {D} = D
+dimension(D::Int) = D
+
+
+function evalrule(data::AbstractArray, weights, dim)
+    fw = zip(_eachslice(data, dims=dim), weights)
+    next = iterate(fw)
+    next === nothing && throw(ArgumentError("No points to integrate"))
+    (f1, w1), state = next
+    out = w1 * f1
+    next = iterate(fw, state)
+    if isbits(out)
+        while next !== nothing
+            (fi, wi), state = next
+            out += wi * fi
+            next = iterate(fw, state)
+        end
+    else
+        while next !== nothing
+            (fi, wi), state = next
+            out .+= wi .* fi
+            next = iterate(fw, state)
+        end
+    end
+    return out
+end
+
+
+# can be reused for other sampled rules, which should implement find_weights(x, alg)
+
+function init_cacheval(alg::SciMLBase.AbstractIntegralAlgorithm, prob::SampledIntegralProblem)
+    find_weights(prob.x, alg)
+end
+
+function __solvebp_call(cache::SampledIntegralCache, alg::SciMLBase.AbstractIntegralAlgorithm; kwargs...)
+    dim = dimension(cache.dim)
+    err = nothing
+    data = cache.y
+    grid = cache.x
+    if cache.isfresh
+        cache.cacheval = find_weights(grid, alg)
+        cache.isfresh = false
+    end
+    weights = cache.cacheval
+    I = evalrule(data, weights, dim)
+    prob = build_problem(cache)
+    return SciMLBase.build_solution(prob, alg, I, err, retcode = ReturnCode.Success)
+end