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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>High Performance Statistical Computing with R</title>
<meta charset="utf-8" />
<meta name="author" content="George Ostrouchov" />
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<textarea id="source">
class: right, inverse, title-slide
.title[
# High Performance Statistical Computing <br> with R
]
.author[
### George Ostrouchov
]
.institute[
### Oak Ridge National Laboratory, University of Tennessee,<br> and Charles University
]
.date[
### <br> Robust22 <br><br> Volyně, CZ, June 12, 2022 <br><br><span style="font-size: 50%;"> Background Image: FRONTIER, First Top500 exascale system, announced June 2022</span>
]
---
# Get this presentation:
`https://github.com/RBigData/ROBUST2022/archive/refs/heads/main.zip`
* Unzip
* Open <br><br>`ROBUST2022-main/Robust2022.html` <br><br> in your web browser
* In this class, we will also show how to get it more directly with **git**
---
# Why HPSC?
* More memory
* Faster code
* Statisticians are needed in HPC
# What is Fast Enough?
* Methodology: seconds
* Optimize parameters: 100 to 1000 `\(\times\)` methodology
* Bootstrap uncertainty in optimization: 100 to 1000 `\(\times\)` optimization
### Faster code can do more
---
## Part I: Software Installatons on Laptop
## Part II: Workflow from Laptop to Cluster
## Part III: Parallel Hardware
## Part IV: Parallel Software
## Part V: Shared Memory Tools
## Part VI: Distributed Memory Tools
---
## Part I: <br> <br> Software Installatons on a Laptop for work with a Cluster
---
### For work with a cluster
* Mac
* R, RStudio
* terminal, git (in Xcode app)
* Windows
* R, RStudio
* putty
* git
* WinSCP
---
## Links for Software on Laptop and Access
* RStudio: [RStudio Desktop Free](https://www.rstudio.com/products/rstudio/download/)
* git: [RStudio with Git](https://support.rstudio.com/hc/en-us/articles/200532077-Version-Control-with-Git-and-SVN)
* ssh: [Wikipedia ssh](https://en.wikipedia.org/wiki/Secure_Shell)
* IT4Innovations:
* [Get Project](https://docs.it4i.cz/general/applying-for-resources/)
* [Example application](pics/IT4I/DD-form.pdf)
* [Access](https://docs.it4i.cz/general/shell-and-data-access/)
* [ssh](https://docs.it4i.cz/general/accessing-the-clusters/shell-access-and-data-transfer/ssh-key-management/)
* [Windows](https://docs.it4i.cz/general/accessing-the-clusters/shell-access-and-data-transfer/putty/)
* Karlova Universita MFF:
* [Sněhurka](https://cluster.karlin.mff.cuni.cz/pouziti-clusteru/)
* [Chiméra](https://www.mff.cuni.cz/cs/verejnost/aktuality/otevreni-fakultniho-hpc-clusteru)
---
## For faster R on your macOS laptop
```r
## Default BLAS from Netlib
> x = matrix(rnorm(1e7), nrow = 1e4)
> system.time(crossprod(x))
user system elapsed
6.752 0.023 6.801
```
* Apple vecLib (via Xcode) can be swapped in
`ln -sf /Library/Frameworks/R.framework/Resources/lib/libRblas.veclib.dylib /Library/Frameworks/R.framework/Resources/lib/libRblas.dylib`
* Or `brew install openblas`, and link OpenBLAS to R
`ln -sf /usr/local/opt/openblas/lib/libopenblas.dylib /Library/Frameworks/R.framework/Resources/lib/libRblas.dylib`
.pull-left[
```r
## vecLib
> system.time(crossprod(x))
user system elapsed
0.666 0.003 0.120
```
]
.pull-right[
```r
## OpenBLAS
> system.time(crossprod(x))
user system elapsed
0.822 0.042 0.121
```
]
---
## For faster R on your Windows laptop
Assessing R performance with optimized BLAS across three operating systems [link](https://thomasmcrow.com/blog/2021-08-optimized-blas-in-r/)
<br>
Building R 4+ for Windows with OpenBLAS [link](https://www.r-bloggers.com/2020/05/building-r-4-for-windows-with-openblas/)
---
# Part II: <br> <br> Workflow from Laptop to Cluster
---
## Working with a remote cluster using R
<img src="pics/01-intro/Workflow.jpg" height="500" />
---
background-image: url(pics/01-intro//Workflow.jpg)
background-position: top right
background-size: 20%
## Why?
### Laptop RStudio
* Familiar custom editing environment (Windows, Mac, Unix)
* Interactive Syntax checking
### GitHub/GitLab
* Portability to remote computing
* Version control
* Collaboration
### Cluster unix
* Same environment for all
* Batch job submission
<br><br>
#### Advanced: interactive multinode development and debugging in RStudio
* Available now, but unstable (launchr, pbdCS, pbdRPC, remoter)
* Needs further development and standardization
---
background-image: url(pics/01-intro//WorkflowRunning.jpg)
background-position: top right
background-size: 20%
## Running Distributed on a Cluster
<img src="pics/01-intro/BatchRonCluster.jpg" height="500" />
---
background-image: url(pics/01-intro//WorkflowCluster.jpg)
background-position: top right
background-size: 20%
# Clusters are Linux systems
* Linux is one of many descendants of original Unix. MacOS is another.
* Like all file systems, Linux files are organized as a tree.
* When in a terminal, you are talking to a *shell* program (*bash* is most common)
* Each command can have a list of *options* and a list of *arguments*
* *Standard input* and *standard output* of a command is the terminal but can be redirected
* **<**, **<<**, **>**, **>>** redirect standard input and output
* *command1* **|** *command2* pipes standard output1 to standard input2
* Commands are looked up in directories listed in your PATH variable (try "echo $PATH")
* $ means substitute variable value
* *export* lists (or sets) all your variables and their values
* There are many resources on the web to learn Linux basics
---
background-image: url(pics/01-intro//WorkflowCluster.jpg)
background-position: top right
background-size: 20%
# Some useful Linux commands
* <mark>pwd</mark> Show curent directory
* <mark>ls</mark> List files in current directory
* <mark>ls -a</mark> Include files that start with <mark>.<mark>
* <mark>ls -l</mark> Long listing with *permissions*, *owners*, and *last change time*
* <mark>cd *dir_name* </mark> Change directory to dir_name
* <mark>mkdir *dir_name*</mark> Creates directory dir_name
* <mark>rmdir *dir_name*</mark> Deletes directory (must be empty)
* <mark>rm *file_name*</mark> Deletes file_name
* <mark>cat *file_name*</mark> Displays content of entire file_name
* <mark>less *file_name*</mark> Displays content of file_name with paging
* <mark>man *command*</mark> Displays the manual page for *command* with paging
* <mark>which *command*</mark> Returns location of command
* <mark>exit</mark> Quit shell and logout
* Much advanced power comes from learning *regular expressions*
---
background-image: url(pics/01-intro/WorkflowCluster.jpg)
background-position: top right
background-size: 20%
## Job Submission on Cluster
* Command line submission
* Write a shell script to request resources and submit a batch job (preferred)
.pull-left[
#### PBS (Karolina, Barbora)
<mark>qsub *script.sh*
<mark>qstat -u *uid*
<mark>qdel *jobname*</mark>
]
.pull-right[
#### Slurm (Sněhurka, Chimera)
<mark>sbatch *script.sh* </mark>
<mark>squeue -u *uid*</mark>
<mark>scancel *jobnumber*</mark>
]
<br>
* **module** to set software environment (PATH)
* <mark>*module load R*</mark>
* <mark>*module load r*</mark>
* <mark>*module list*</mark>
* <mark>*module avail*</mark>
---
background-image: url(pics/01-intro/WorkflowGit.jpg)
background-position: top left
background-size: 20%
.right[
# GitHub and git (laptop to cluster)
]
.w80.pull-left[
<img src="pics/01-intro/Git_operations.svg" height="450" />
<font size="-4">*By Daniel Kinzler - Own work, CC BY 3.0, https://commons.wikimedia.org/w/index.php?curid=25223536</font>
]
.w20.pull-right[
<br>
<img src="pics/01-intro/WorkflowCluster.jpg" height="80" />
<br><br><br><br><br><br><br><br><br><br>
<img src="pics/01-intro/WorkflowLaptop.jpg" height="80" />
]
---
# Making it easy: set ssh keys
* A message encrypted by public key can be decrypted by private key
* Works like a single-use password generator and authenticator
.w80.pull-left[
<img src="pics/01-intro/ssh-key-based-authentication.png" width="450" height="400" />
<font size="-4">Graphic from </font>
<img src="pics/01-intro/ssh-credit.png" width="300" height="10" />
]
.w20.pull-right[
Your private key is protected on your local resource
<br><br><br><br>
Put your public key on the remote resource to enable easy access
]
---
## Demo ... `\(\qquad\)` Workflow
* Fork ROBUST2022 to your GitHub account
* Login to GitHub
* Navigate to RBigData/ROBUST2022
* Click Fork button near top-right
* clone to New Project in RStudio
* Open Terminal window (ssh or putty)
* Login to cluster
* clone your ROBUST2022 (git clone ...)
* You are ready for development loop:
* edit -> commit -> push -> pull -> run -> examine output
.pull-left[
<img src="pics/01-intro/Workflow.jpg" height="250" />
]
.pull-right[
<img src="pics/01-intro/BatchRonCluster.jpg" height="250" />
]
---
# Part III: <br> <br> Parallel Hardware
---
background-image: url(pics/Mangalore/ParallelHardware/Slide7.png)
background-position: bottom
background-size: 90%
# Three Basic Concepts in Hardware
???
# GPU - NVIDIA
# MIC - Intel KNL - ARM
* Manycore chip with memory on the chip instead of separate memory boards: https://www.youtube.com/watch?v=eXhlDt2SD8o
* A manycore that can act as a GPU
* Multocores are MIMD and can run anything
* GPU are SIMD and more limited
---
background-image: url(pics/Mangalore/ParallelHardware/Slide1.png)
background-position: bottom
background-size: 90%
# Three Basic Concepts in Hardware
???
# GPU - NVIDIA
# MIC - Intel KNL - ARM
* Manycore chip with memory on the chip instead of separate memory boards: https://www.youtube.com/watch?v=eXhlDt2SD8o
* A manycore that can act as a GPU
* Multocores are MIMD and can run anything
* GPU are SIMD and more limited
---
background-image: url(pics/Mangalore/ParallelHardware/Slide2.png)
background-position: bottom
background-size: 90%
???
Graphics
---
background-image: url(pics/Mangalore/ParallelHardware/Slide3.png)
background-position: bottom
background-size: 90%
???
GPU - NVIDIA
MIC - Intel KNL
---
background-image: url(pics/Mangalore/ParallelHardware/Slide4.png)
background-position: bottom
background-size: 90%
---
background-image: url(pics/Mangalore/ParallelHardware/Slide5.png)
background-position: bottom
background-size: 90%
---
background-image: url(pics/Mangalore/ParallelHardware/Slide6.png)
background-position: bottom
background-size: 90%
---
# Part IV: <br> <br> Parallel Software
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide2.png)
background-position: bottom
background-size: 90%
# Native Programming Mindset
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide3.png)
background-position: bottom
background-size: 90%
# Native Programming Models and Tools
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide4.png)
background-position: bottom
background-size: 90%
# 35+ Years of Parallel Computing Research
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide5.png)
background-position: bottom
background-size: 90%
# Last 15+ years of Advances
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide6.png)
background-position: bottom
background-size: 90%
## Distributed Programming Works in Shared Memory
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7.png)
background-position: bottom
background-size: 90%
# R Interfaces to Low-Level Native Tools
---
# Part V: <br> <br> Shared Memory Tools
## Working with a single node
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7shared.jpg)
background-position: bottom
background-size: 90%
# Working with a single node
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7fork.jpg)
background-position: bottom
background-size: 90%
# fork via mclapply
???
* we begin with `paralel`'s multicore parts
* continue with Foreign language via libraries (OpenBLAS, nvBLAS)
* go to SPMD MPI with collectives
* reverse of history - because we are used to a laptop
* Distributed - some things are recomputed rather than communicated
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7fork.jpg)
background-position: top right
background-size: 20%
# Unix `fork`
* A memory efficient parallelism on shared memory devices
* Copy-on-write: copy page if forked process tries to write
* R: **parallel** package `mclapply` and friends
* Use for numerical sections only
* Avoid GUI, I/O, and graphics sections
* Convenient for data (not modified)
* Convenient for functional languages like R
* Careful with nested parallelism
* OpenBLAS takes all cores by default
* data.table switches to single threaded mode upon fork
.footnote[A deeper discussion of `fork` memory (if you have interest) on [YouTube](https://www.youtube.com/watch?v=8hVLcyBkSXY) by Chris Kanich (UIC)]
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7fork.jpg)
background-position: top right
background-size: 20%
# Copy-on-write
<img src="pics/MC/Fork/Slide1.png" height="500" style="display: block; margin: auto;" />
???
* All done with pointers
* Memory is in pages
* Processes not aware of each other or other's memory use
* OS is aware of memory use
* 16 forks write = 16 copies of memory
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7fork.jpg)
background-position: top right
background-size: 20%
# Mapping Threads to Cores
### Theory and Reality
* Operating system manages core affinity
* Operating system tasks can compete
* Core switching occurs frequently
.pull-left[
<img src="pics/WSC/iDVTstR_theory.jpg" height="350" style="display: block; margin: auto;" />
]
.pull-right[
<img src="pics/WSC/iDVTstR_reality.jpg" height="350" style="display: block; margin: auto;" />
]
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7fork.jpg)
background-position: top right
background-size: 20%
### `R`: Drop-in replacements (almost) <br> for `lapply`, `mapply`, and `Map`
`mclapply(X, FUN, ...,`
` mc.preschedule = TRUE, mc.set.seed = TRUE,`
` mc.silent = FALSE, mc.cores = getOption("mc.cores", 2L),`
` mc.cleanup = TRUE, mc.allow.recursive = TRUE, affinity.list = NULL)`
`mcmapply(FUN, ...,`
` MoreArgs = NULL, SIMPLIFY = TRUE, USE.NAMES = TRUE,`
` mc.preschedule = TRUE, mc.set.seed = TRUE,`
` mc.silent = FALSE, mc.cores = getOption("mc.cores", 2L),`
` mc.cleanup = TRUE, affinity.list = NULL)`
`mcMap(f, ...)`
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7fork.jpg)
background-position: top right
background-size: 20%
# Example Random forest Code
#### Letter recognition data ( `\(20\,000 \times 17\)` )
<img src="pics/MC/ML_FreySlate1991.png" height="350" style="display: block; margin: auto;" />
.footnote[*Parallel Statistical Computing with R: An Illustration on Two Architectures [ arXiv:1709.01195](https://arxiv.org/abs/1709.01195)]
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7fork.jpg)
background-position: top right
background-size: 20%
# Random Forest Classification
### Build many decision trees. Each tree from
* random subset of variables
* resampled (with replacement) data
### Use their majority votes to classify
---
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background-position: bottom
background-size: 100%
# Pseudo Random Number Generators (RNG)
.pull-left[
* Guaranteed reproducibility
* Possibly overlapping streams
]
.pull-right[
* Reproducibility for same number of streams
* Guaranteed independent streams
]
---
background-image: url(pics/MC/benchmark_mc.png)
background-position: bottom
background-size: 80%
# Example Random Forest Classification Code
#### Letter recognition data ( `\(20\,000 \times 17\)` ) - Scaling graph example
---
### `ROBUST2022/code/rf_serial.r`
```r
library(randomForest)
data(LetterRecognition, package = "mlbench")
set.seed(seed = 123)
n = nrow(LetterRecognition)
n_test = floor(0.2 * n)
i_test = sample.int(n, n_test)
train = LetterRecognition[-i_test, ]
test = LetterRecognition[i_test, ]
rf.all = randomForest(lettr ~ ., train, ntree = 500, norm.votes = FALSE)
pred = predict(rf.all, test)
correct = sum(pred == test$lettr)
cat("Proportion Correct:", correct/(n_test), "\n")
```
---
### `ROBUST2022/code/rf_mc.r`
```r
*library(parallel)
library(randomForest)
data(LetterRecognition, package = "mlbench")
*set.seed(seed = 123, "L'Ecuyer-CMRG")
n = nrow(LetterRecognition)
n_test = floor(0.2 * n)
i_test = sample.int(n, n_test)
train = LetterRecognition[-i_test, ]
test = LetterRecognition[i_test, ]
*nc = as.numeric(commandArgs(TRUE)[2])
*ntree = lapply(splitIndices(500, nc), length)
*rf = function(x, train) randomForest(lettr ~ ., train, ntree=x,
* norm.votes = FALSE)
*rf.out = mclapply(ntree, rf, train = train, mc.cores = nc)
*rf.all = do.call(combine, rf.out)
*crows = splitIndices(nrow(test), nc)
*rfp = function(x) as.vector(predict(rf.all, test[x, ]))
*cpred = mclapply(crows, rfp, mc.cores = nc)
*pred = do.call(c, cpred)
correct <- sum(pred == test$lettr)
cat("Proportion Correct:", correct/(n_test), "\n")
```
---
# Demo ... <br><br> Random Forest via mclapply
---
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background-size: 90%
# Libraries via compiled language interfaces
---
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background-position: top right
background-size: 20%
# R-LAPACK-BLAS
* BLAS: Basic Linear Algebra Subroutines - A matrix multiplication library
* `%*%`, `crossprod()`, `sweep()`, `scale()`, and many more
* LAPACK: dense and banded matrix decomposition and more
* `svd()`, `La.svd()`, `prcomp()`, `princomp()`, `qr()`, `solve()`, `chol()`, `norm()`, and many more
* But not `lm()`, careful with `qr(x, LAPACK = TRUE)`: column pivoting
* Implementations: OpenBLAS, Intel MKL, Nvidia nvBLAS, Apple vecLib, AMD BLIS, Arm Performance Libraries
* **FlexiBLAS**: A BLAS and LAPACK wrapper library with runtime exchangeable backends
* Great for benchmarking implementations
* Great for dynamic core assignment
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7libs.jpg)
background-position: top right
background-size: 20%
# OpenBLAS
OpenBLAS is an optimized BLAS library based on GotoBLAS2 (2010, Kazushige Goto).
* [openblas.net](https://www.openblas.net)
* Optimizes algorithm to chip microarchitecture details of memory hierarchies (L1 cache, L2 cache, etc.) and register vector length
* IT4I FlexiBLAS: "OPENBLAS" backend
.footnote[
Wang Qian, Zhang Xianyi, Zhang Yunquan, Qing Yi, AUGEM: Automatically Generate High Performance Dense Linear Algebra Kernels on x86 CPUs, In the International Conference for High Performance Computing, Networking, Storage and Analysis (SC'13), Denver CO, November 2013.
]
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7libs.jpg)
background-position: top right
background-size: 20%
### FlexiBLAS
`flexiblas_setup.r`
```r
library(flexiblas)
flexiblas_avail()
flexiblas_version()
flexiblas_current_backend()
flexiblas_list()
flexiblas_list_loaded()
getthreads = function() {
flexiblas_get_num_threads()
}
setthreads = function(thr, label = "") {
cat(label, "Setting", thr, "threads\n")
flexiblas_set_num_threads(thr)
}
setback = function(backend, label = "") {
cat(label, "Setting", backend, "backend\n")
flexiblas_switch(flexiblas_load_backend(backend))
}
```
.footnote[
[https://github.com/Enchufa2/r-flexiblas](https://github.com/Enchufa2/r-flexiblas)
[https://cran.r-project.org/package=flexiblas](https://cran.r-project.org/package=flexiblas)
]
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7libs.jpg)
background-position: top right
background-size: 20%
# Demo ... <br><br> FlexiBLAS
---
# The Singular Value Decomposition
`$$X = UDV^T$$`
For `\(n \geq p\)`,
`\(U\)` is an `\(n\times p\)` orthogonal matrix of left singular vectors
`\(V\)` is a `\(p\times p\)` orthogonal matrix of right singular vectors
`\(D\)` is a `\(p\times p\)` diagonal matrix of singular values
---
## Truncated SVD as Regression Basis Vectors
Suppose we have `\(n\)` images, each with `\(p\)` pixel values. The well known MNIST data set of digit images is an example with `\(n = 60\, 000\)` and `\(p = 784\,\, (28\times 28)\)`.
<img src="mnist/RplotsDigits.png" height="300" />
Let `\(A\)` be the matrix of `\(n_A\)` images of a single digit, say digit zero, the pixel values of each image as a column.
---
## Truncated SVD as Regression Basis Vectors
Let `\(A\)` be the matrix of `\(n_A\)` images of a single digit, the pixel values of each image as a column.
The SVD of `\(A = UDV^T\)`.
If `\(u_i\)` and `\(v_i\)` are the columns of `\(U\)` and `\(V\)`, respectively,
then
`$$A = \sum_{i=1}^p d_i u_iv_i^T.$$`
and image `\(j\)` in column `\(a_j = \sum_{i=1}^p (d_i v_{ij})u_i.\)`
From matrix approximation, we know that this SVD can be truncated to some `\(k \ll p\)` components and still represent each image well.
---
## Truncated SVD as Regression Basis Vectors
For some `\(k \ll p\)`, we have `\(a_j = \sum_{i=1}^k (d_iv_{ij})u_i.\)`
The `\(u_i\)` are basis functions constructed from data, a set of orthogonalized "images", which are the regressors and the `\(d_iv_{ij}\)` are the regression coefficients.
We can now look at classification of a new image of a digit by regressing it onto each of the 10 digit bases and classifying it into the category that fits best.
The tuning parameter `\(k\)` can be optimized with crossvalidation.
---
# Demo ... MNIST svd regression
.pull-left[
<img src="mnist/one_basis95.png" height="400" style="display: block; margin: auto auto auto 0;" />
]
.pull-right[
<img src="mnist/five_basis95.png" height="400" style="display: block; margin: auto auto auto 0;" />
]
---
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background-position: bottom
background-size: 90%
# Part VI: `\(\qquad\)` Distributed Memory Tools
---
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# Message Passing Interface (MPI)
---
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background-size: 20%
# pbdR - MPI
<img src="pics/01-intro/pbdRlib.jpg" height="100" style="display: block; margin: auto auto auto 0;" />
* MPI: Message Passing Interface - *de facto* standard for distributed communication in supercomputing
* Used for data mostly via collective communication - high level
* `pbdMPI`, `kazaam`, and `cop` R packages
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7mpi.jpg)
background-position: top right
background-size: 20%
# Single Program Multiple Data (SPMD)
#### Hello world!
```r
suppressMessages(library(pbdMPI))
my_rank = comm.rank()
ranks = comm.size()
msg = paste0("Hello World! My name is Empi", my_rank,
". We are ", ranks, " identical siblings.")
cat(msg, "\n")
finalize()
```
#### One code and a parallel mindset
#### A generalization of a serial code
#### Many rank-aware operations are automated
#### No manager, it is all cooperation
#### Explicit point-to-point communications are an advanced topic
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7mpi.jpg)
background-position: top right
background-size: 20%
# High-level Collective Communications
`$$\bf A = \sum_{i=1}^nX_i$$`
#### `pbdMPI`: `\(\qquad\)` **reduce(X)** `\(\qquad\)` `\(\qquad\)` `\(\qquad\)` **allreduce(X**)
`$$\bf A = \left[ X_1 | X_2 | \cdots | X_n \right]$$`
#### `pbdMPI`: `\(\qquad\)` **gather(X)** `\(\qquad\)` `\(\qquad\)` `\(\qquad\)` **allgather(X**)
<img src="pics/01-intro/RHistory3sub.png" height="250" style="display: block; margin: auto auto auto 0;" />
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7mpi.jpg)
background-position: top right
background-size: 20%
# Demo ... <br><br> MPI communication in SPMD programming
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7mpi.jpg)
background-position: top right
background-size: 20%
# pbdR - ScaLAPACK - MPI
.pull-left[
<img src="pics/01-intro/pbdRlib.jpg" height="100" style="display: block; margin: auto auto auto 0;" />
]
.pull-right[
**pbdr.org**
]
* ScaLAPACK: Scalable LAPACK - Distributed version of LAPACK (uses PBLAS/BLAS but not LAPACK)
* 2d Block-Cyclic data layout - mostly automated in `pbdDMAT` package
* BLACS: Communication collectives for distributed matrix computation
* PBLAS: BLAS - distributed BLAS (uses shared memory BLAS within blocks)
* `pbdDMAT` and `pbdML` R packages - most matrix operations identical to serial through overloading operators and `ddmatrix` class
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7mpi.jpg)
background-position: top right
background-size: 20%
## Constructing a Distributed Matrix from Data
* Each MPI rank reads different data in a contiguous block
.pull-left[
`$$\large\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\[1ex]a_{21}&a_{22}&a_{23}\\[1ex] a_{31}&a_{32}&a_{33}\end{array}\right]$$`
`$$\large\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\[1ex]a_{21}&a_{22}&a_{23}\\[1ex] a_{31}&a_{32}&a_{33}\end{array}\right]$$`
]
.pull-right[
`$$\qquad$$`
`\(\large a_{11}\;a_{12}\;a_{13}\;a_{21}\;a_{22}\;a_{23}\;a_{31}\;a_{32}\;a_{33}\)`
C, C++, NumPy `\(\quad\)` **Row-Block**
`$$\qquad$$`
`$$\qquad$$`
`\(\large a_{11}\;a_{21}\;a_{31}\;a_{12}\;a_{22}\;a_{32}\;a_{13}\;a_{23}\;a_{33}\)`
Fortran, R, Matlab `\(\quad\)` **Column-Block**
]
* Each MPI rank adds attributes for global context
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7mpi.jpg)
background-position: top right
background-size: 20%
# Demo ... <br><br> ScaLAPACK via pbdDMAT
---
## Randomized sketching algorithms
<br><br>
Fast new alternatives to classical numerical linear algebra computations.
<br>
Guarantees are given with probability statements instead of classical error analysis.
<br> <br>
Martinsson, P., & Tropp, J. (2020). Randomized numerical linear algebra: Foundations and algorithms. Acta Numerica, 29, 403-572. [https://doi.org/10.48550/arXiv.2002.01387](https://doi.org/10.48550/arXiv.2002.01387)
---
`mnist_rsvd.R`
```r
source("mnist_read_mpi.R") # reads blocks of rows
suppressMessages(library(pbdDMAT))
suppressMessages(library(pbdML))
init.grid()
## construct block-cyclic ddmatrix
bldim = c(allreduce(nrow(my_train), op = "max"), ncol(my_train))
gdim = c(allreduce(nrow(my_train), op = "sum"), ncol(my_train))
dmat_train = new("ddmatrix", Data = my_train, dim = gdim,
ldim = dim(my_train), bldim = bldim, ICTXT = 2)
cyclic_train = as.blockcyclic(dmat_train)
comm.print(comm.size())
t1 = as.numeric(Sys.time())
rsvd_train = rsvd(cyclic_train, k = 10, q = 3, retu = FALSE, retv = FALSE)
t2 = as.numeric(Sys.time())
t1 = allreduce(t1, op = "min")
t2 = allreduce(t2, op = "max")
comm.cat("Time:", t2 - t1, "seconds\n")
comm.cat("rsvd top 10 singular values:", rsvd_train$d, "\n")
finalize()
```
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7mpi.jpg)
background-position: top right
background-size: 20%
## Randomized SVD via subspace embedding
Given an `\(n\times p\)` matrix `\(X\)` and `\(k = r + 10\)`, where `\(r\)` is the *effective rank* of `\(X\)`:
1. Construct a `\(p\times k\)` random matrix `\(\Omega\)`
2. Form `\(Y = X \Omega\)`
3. Decompose `\(Y = QR\)`
`\(Q\)` is an orthogonal basis for the columnspace of `\(Y\)`, which with high probability is the columnspace of `\(X\)`. To get the SVD of `\(X\)`:
1. Compute `\(C= Q^TX\)`
2. Decompose `\(C = \hat{U}\Sigma V^T\)`
3. Compute `\(U = Q\hat{U}\)`
4. Truncate factorization to `\(r\)` columns
---
background-image: url(pics/Mangalore/ParallelSoftware/Slide7mpi.jpg)
background-position: top right
background-size: 20%
## Randomized SVD via subspace embedding
Given an `\(n\times p\)` matrix `\(X\)` and `\(k = r + 10\)`, where `\(r\)` is the *effective rank* of `\(X\)`:
1. Construct a `\(p\times k\)` random matrix `\(\Omega\)`
2. Let `\(Y_0 = \Omega\)`
3. For `\(i\)` in `\(1:q\)`
2. Decompose `\(Y_{i-1} = Q_{i}R_{i}\)`
1. `\(Y_i = X(X^TQ_i)\)`
4. Decompose `\(Y_q = QR\)`
`\(Q\)` is an orthogonal basis for the columnspace of `\(Y\)`, which with high probability is the columnspace of `\(X\)`. To get the SVD of `\(X\)`:
1. Compute `\(C= Q^TX\)`
2. Decompose `\(C = \hat{U}\Sigma V^T\)`
3. Compute `\(U = Q\hat{U}\)`
4. Truncate factorization to `\(r\)` columns
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