Introduction to MATLAB parallel computing toolbox

1. Introduction to MATLAB parallel computing toolbox Kadin Tseng Boston University Scientific Computing and Visualization

2. MATLAB Parallel Computing Toolbox Log On To PC For Hands-on Practice • Log on with your BU userid and Kerboros password • If you don’t have BU userid, then use this: userid: tuta1 . . . tuta18 password: SCVsummer12 The number after tuta should match the number affixed on the front of your PC tower • Start MATLAB M-Files on T drive There are some files that you can copy over to your local folder for hands-on practices: >> copyfile(‘T:\kadin\pct\*’ , ’.\’)

3. MATLAB Parallel Computing Toolbox What is the PCT ? • The Parallel Computing Toolbox is a MATLAB tool box. • This tool box provides parallel utility functions to enable users to run MATLAB operations or procedures in parallel to speed up processing time.

4. MATLAB Parallel Computing Toolbox Where To Run The PCT ? • Run on a desktop or laptop • MATLAB must be installedon local machine • Starting with R2011b, up to 12 processors can be used; up to 8 processors for older versions • Must have multi-core to gain speedup. • The thin client you are using has a dual-core processor • Requires BU userid (to access MATLAB, PCT licenses) • Run on a Katana Cluster node (as if a multi-cored desktop) • Requires SCV userid • Run on multiple Katana nodes ( for up to 32 processors) • Requires SCV userid

5. MATLAB Parallel Computing Toolbox Types of Parallel Jobs ? There are two types of parallel applications. • Distributed Jobs – task parallel • Parallel Jobs – data parallel

6. MATLAB Parallel Computing Toolbox Distributed Jobs This type of parallel processing is classified as: Multiple tasks running independently on multiple workers with no information passed among them. On Katana, a distributed job is a series of single-processor batch jobs. This is also known as task-parallel, or “embarrassingly parallel” , jobs. Examples of distributed jobs: Monte Carlo simulations, image processing Parallel utility function: dfeval

7. MATLAB Parallel Computing Toolbox Parallel Jobs A parallel job is: A single task running concurrently on multiple workers that may communicate with each other. On Katana, this results in one batch job with multiple processors. This is also known as a data-parallel job. Examples of a parallel job include many linear algebra applications : matrix multiply; linear algebraic system of equations solvers; Eigen solvers. Some may run efficiently in parallel and others may not. It depends on the underlining algorithms and operations. This also include jobs that mix serial and parallel processing. Parallel utility functions: spmd, drange, parfor, . . .

8. MATLAB Parallel Computing Toolbox How much work to parallelize my code ? • Code dependent • Many MATLAB functions are overloaded to handle vector or parallel operations based on the variables’ data type, i.e., scalars, vectors, or distributed arrays. For some application, may just have to turn on parallel capability. [No effort to a few lines of code modification] • If arrays have been declared distributed (i.e., parallel), subsequent operations of these arrays using built-in or user functions will be processed in parallel without additional instructions. Arrays not declared as parallel will also be promoted to parallel if they are used with distributed arrays. [moderate effort: code, algorithm change] • For custom parallelization, MPI-like utilities are available. [More extensive effort: code, algorithm]

9. MATLAB Parallel Computing Toolbox Distributed Jobs — dfeval Example: Run a dfeval job “interactively” Computes 1x4, 3x2 random arrays locally or on Katana (with pre-defined batch configuration ‘SGE’) input to rand application m-file or ‘SGE’ >> y = dfeval(@rand,{1 3},{4 2},'Configuration',‘local‘) Submitting task 1 Job output will be written to: /usr1/scv/kadin/Job6_Task1.out QSUB output: Your job 211908 ("Job6.1") has been submitted Submitting task 2 Job output will be written to: /usr1/scv/kadin/Job6_Task2.out QSUB output: Your job 211909 ("Job6.2") has been submitted y = [1x4 double] [3x2 double] Job ran in batch or background, output returns to client workspace.

10. MATLAB Parallel Computing Toolbox Distributed Jobs – SCV script For task-parallel applications on the Katana Cluster, we strongly recommend the use of an SCV script instead of dfeval. This script does not use the PCT. For details, see http://www.bu.edu/tech/research/computation/linux-cluster/katana-cluster/runningjobs/multiple_matlab_tasks/ If your application fits the description of a distributed job, you don’t need to know any more beyond this point . . .

11. MATLAB Parallel Computing Toolbox How Do I Run Parallel Jobs ? Two ways to run parallel jobs: pmode or matlabpool. This procedure turns on (off) parallelism and allocate (deallocate) resources. pmode is a special mode of application; useful for learning the PCT and parallel program prototyping interactively. matlabpoolis the general mode of application; it can be used for interactive and batch processing.

12. MATLAB Parallel Computing Toolbox pmode >> pmode start local % assuming 4workers are available Above is a MATLAB window on a Katana node. A separate Parallel Command Window is spawned. (Similar for Windows) In Room B27, each thin client has 2 cores (workers).

13. MATLAB Parallel Computing Toolbox pmode >> pmode start local % use 4 workers in parallel mode PCT terminologies: worker = processor labindex: processor number numlabs: Number of processors Worker number Command entered Enter command here • Any command issued at the “P>>” prompt is executed on all • workers. Enter “labindex” to query for workers’ ID. • Use if conditional with labindex to issue instructions to • specific workers, likethis:if labindex==1, numlabs, end

14. Replicate array • P>> A = magic(3); % A is replicated on every worker • Variant array • P>> A = magic(3) + labindex – 1; % labindex=1,2,3,4 • LAB 1 LAB 2 LAB 3 LAB 4 • | | | | • |8 1 6|9 2 7|10 3 8|11 4 9 • |3 5 7|4 6 8| 5 7 9| 6 8 10 • |4 9 2|5 10 3| 6 11 4| 7 12 5 • Private array • P>> if labindex==2, A = magic(3) + labindex – 1; end • LAB 1 LAB 2 LAB 3 LAB 4 • | | | | • | |9 2 7| | • |undefined|4 6 8|undefined|undefined • | |5 10 3| | pmode

15. If you are running MATLAB pmode, exit it. • P>> exit • MATLAB allows only one parallel environment at a time. • pmode may be started with the keyword open or start. • matlabpool can only be started with the keyword open. • You can also close pmode from the MATLAB window: • >> pmode close • Now, open matlabpool • >> matlabpool open local % rely on default worker size Switch from pmode to matlabpool

16. matlabpool is the general paradigm for MATLAB parallel computing; it can be used for interactive and batch jobs. >> matlabpool open local % open 2 workers on thin client >> % >> % serial or parallel applications ... >> % >> matlabpool close % ends matlabpool properly matlabpool

17. parfor is a parallel for-loop. Work load is distributed according • to loop index. Computation among loop indices must be • independent. Operates in matlabpool. • matlabpool open % use default number of workers ( 2 ) • s = 0; • parfor i=1:10 • x(i) = sin(2*pi*i/10); % each slice of 5 computed by a worker • s = s + i; % summation; a reduction operation • end • matlabpool close • Does parfor work for all kinds of operations? • No. As an example, a reduction operation (such as addition) • must satisfy x ◊ (y ◊ z) = (x ◊ y) ◊ z associative rule • Plus (+) and multiply (*) operators satisfy rule. • Subtract (-) and divide (/) operators fail ruleindeterministic Parallel Methods —parfor

18. Try this: s = 0; parfor i=1:10, s = s + i, end, s % s = 55 (s = 1 + 2 + 3 + . . . + n; s = n(n+1)/2) s=1000; parfor i=1:5, s=s*i,end, s % do a few times s=1000; parfor i=1:5, s=s/i, end, s % do a few times Above + , * operators satisfy associativerule. The - , / operators fail rule. They may or may not produce same (correct) result each time. Some other operational rules for parfor: Loop index must be consecutive integers. Loop index must not be altered within loop. Loop count need not be divisible by number of workers. No need to distribute arrays. Solution remains in the MATLAB workspace. Does not work in spmd; can’t use numlabs, labindex. No graphics can be displayed within parfor. More rules depending on operations; consult the PCT doc. Parallel Methods —parfor

19. An integration of the cosine function between 0 and π/2 • Integration scheme is mid-point rule for simplicity. • Several parallel methods will be demonstrated. Integration Example Worker 1 mid-point of increment Worker 2 a = 0; b = pi/2; % range m = 8; % # of increments h = (b-a)/m; % increment p = numlabs; n = m/p; % inc. / worker ai = a + (i-1)*n*h; aij = ai + (j-1)*h; cos(x) h Worker 3 Worker 4 x=a x=b

20. % serial integration (with for-loop) tic m = 10000; a = 0; % lower limit of integration b = pi/2; % upper limit of integration dx = (b – a)/m; % increment length intSerial = 0; % initialize intSerial for i=1:m x = a+(i-0.5)*dx; % mid-point of increment i intSerial = intSerial + cos(x)*dx; end toc Integration Example — Serial Integration dx X(1) = a + dx/2 X(m) = b - dx/2

21. % serial integration (with vector form) tic m = 10000; a = 0; % lower limit of integration b = pi/2; % upper limit of integration dx = (b – a)/m; % increment length x = a+dx/2:dx:b-dx/2; % mid-points of m increments intSerial = sum(cos(x)*dx); toc Integration Example — Serial Integration dx X(1) = a + dx/2 X(m) = b - dx/2

22. This example performs parallel integration in spmd. It uses labindex and numlabs to enable all workers to run the same command (“sp”) with different data (“md”). matlabpool open local tic % includes the overhead cost of spmd spmd m = 10000; a = 0; b = pi/2; n = m/numlabs; % # of increments per lab deltax = (b - a)/numlabs; % length per lab ai = a + (labindex - 1)*deltax; % local integration range bi = a + labindex*deltax; dx = deltax/n; % increment length for lab x = ai+dx/2:dx:bi-dx/2; % mid-points of n increments per worker intSPMD = sum(cos(x)*dx); % integral sum per worker intSPMD = gplus(intSPMD,1); % global sum over all workers end % spmd toc matlabpool close Integration Example — spmd

23. This example performs parallel integration with parfor. matlabpool open 4 tic m = 10000; nworkers = matlabpool(‘size’); % returns # of workers assigned n = m/nworkers; % number of increments per worker a = 0; b = pi/2; deltax = (b – a)/nworkers; % increment length per worker dx = deltax/n; intParfor1 = 0; parfor i=1:nworkers ai = a + (i- 1)*deltax; bi = a + i*deltax; x = ai+dx/2:dx:bi-dx/2; % mid-points of n increments per worker intParfor1 = intParfor1 + sum(cos(x)*dx); end toc matlabpool close Integration Example — parfor

24. This example performs parallel integration with parfor. matlabpool open 4 tic m = 10000; a = 0; b = pi/2; dx = (b – a)/m; % increment length intParfor2 = 0; parfor i=1:m intParfor2 = intParfor2 + cos(a+(i-0.5)*dx)*dx; end toc matlabpool close Integration Example — parfor

25. Similar to parforbut used within spmd. matlabpool open 4 tic m = 10000; a = 0; b = pi/2; spmd n = m/numlabs; % number of increments per lab (worker) deltax = (b - a)/numlabs; % increment length per worker for i=drange(1:numlabs) ai = a + (i - 1)*deltax; bi = a + i*deltax; dx = deltax/n; x = ai+dx/2:dx:bi-dx/2; % mid-points of n increments intDrange1 = sum(cos(x)*dx); end intDrange1 = gplus(intDrange1, 1); % send global sum to lab 1 end % spmd intDrange1 = intDrange1{1}; % send integral to client toc matlabpool close Integration Example — drange

26. Similar to parforbut used within spmd. matlabpool open 4 tic m = 10000; a = 0; b = pi/2; spmd dx = (b - a)/m; % increment length per worker intDrange2 = 0; for i=drange(1:m) intDrange2 = intDrange2 + cos(a+(i-0.5)*dx)*dx; end intDrange2 = gplus(intDrange2, 1); % send global sum to lab 1 end % spmd intDrange2 = intDrange2{1}; % send integral to client toc matlabpool close Integration Example — drange

27. Integration Example Benchmarks • Timings (seconds) obtained on a quad-core Xeon X5570 • Computation linearly proportional to # of increments. • serial and serialv are times by loop and vector, respectively • parfor1 and drange1 distribute work over workers; local work • performed in vector form. • parfor2and drange2 distribute m over workers in chunk. • FORTRAN and C timings are an order of magnitude faster.

28. The purpose is to distribute data (e.g., an array) among workers to reduce the memory usage and workload on each worker in return for smaller wall clock time. For some parallel applications, creating a distributed array often is the only thing you need to do to make your application to run in parallel (e.g., due to function overloading). Some operations distribute data automatically while others require manual distribution. Array Distributions

29. Utilities to distribute arrays: • distributed – distribute data from client; convenient but • with restrictions • codistributed– used in spmd to distribute data on backend • Composite– distribute data on backend; access onclient • Methods to distribute data: • Partitioning a larger array. • Building from smaller arrays. • Created with MATLAB constructor function (rand, zeros, . . .). How To Distribute Arrays

30. >> matlabpool open 4 >> n = 3000; A = rand(n); B = rand(n); >> C = A * B; % run with 4 threads >> maxNumCompThreads(1);% set threads to 1 >> C1 = A * B; % run on single thread >> a = distributed(A); % distributes A, B from client >> b = distributed(B); % a, b on workers; accessible from client >> c = a * b; % run on workers; c is distributed >> matlabpool close Wall clock time, in seconds, for the above operations Data Parallel Example – Matrix Multiply • The cost for distributing the matrices is recorded separately as this is incurred only once over the life of the distributed matrices. • Time required to distribute matrices is not significantly affected by matrix size.

31. There are alternative ways to distribute matrices. >> matlabpool open 4 >> A = rand(3000); B = rand(3000); >> spmd p = rand(n, codistributor1d(1)); % 2 ways to directly create q = codistributed.rand(n); % distributed random array s = p* q; % run on workers; s is distributed % distribute matrix after it is created u = codistributed(A, codistributor1d(1)); % by row v = codistributed(B, codistributor1d(2)); % by column w = u* v; % run on workers; w is distributed end >> matlabpool close Additional Ways to Distribute Matrices

32. For matrix-matrix multiply, there are 4 combinations on how to distribute the 2 matrices (by row or column). While all 4 ways lead to the correct solution, some perform better than others. n = 3000; A = rand(n); B = rand(n); spmd ar = codistributed(A, codistributor1d(1)) % distributed by row ac = codistributed(A, codistributor1d(2)) % distributed by column br= codistributed(B, codistributor1d(1)) % distributed by row bc= codistributed(B, codistributor1d(2)) % distributed by column crr = ar* br; crc = ar* bc; ccr= ac* br; ccc = ac * bc; end Wall clock times of the four ways to distribute A and B Preferred Way to Distribute Matrices ? The above is true for on-node or across-nodes runs. Across-nodes array distribution and runs are more likely to be slower than on-node ones due to slower communications. Specifically for MATLAB, Katana uses 100-Mbits Ethernet for across-nodes communications instead of Gigabit InfiniBand.

33. matlabpool open 4 % serial operations n = 3000; M = rand(n); x = ones(n,1); [A, b] = linearSystem(M, x); u = A\b; % solves Au = b; u should equal x clear A b % parallel operations in spmd spmd m = codistributed(M, codistributor('1d',2)); % by column y = codistributed(x, codistributor(‘1d’,1)); % by row [A, b] = linearSystem(m, y); v = A\b; end clear A b m y % parallel operations from client m = distributed(M); y = distributed(x); [A, b] = linearSystem(m, y); W = A\b; matlabpool close Linear algebraic system Example: Ax = b function [A, b] = linearSystem(M, x) % Returns A and b of linear system Ax = b A = M + M'; % A is real and symmetric b = A * x; % b is the RHS of linear system

34. matlabpool open 4 n = 3000; M = rand(n); x = ones(n,1); % Solves 4 cases of Ax=b sequentially, each with 4 workers for i=1:4 m = distributed(M); y = distributed(x*i); [A, b] = linearSystem(m, y); % computes with 4 workers u = A\b; % solves each case with 4 workers end clear A b m y % solves 4 cases of Ax=b concurrently (with parfor) parfor i=1:4 [A, b] = linearSystem(M, x*i); % computes on 1 worker v = A\b; % solves with 1 worker end % solves 4 cases of Ax=b concurrently (with drange) spmd for i=drange(1:4) [A, b] = linearSystem(M, x*i); % computes on 1 worker w = A\b; % 1 worker end end matlabpool close Task Parallel vs. Data Parallel function [A, b] = linearSystem(M, x) • % Returns A and b of linear system Ax = b A = M + M'; % A is real and symmetric b = A * x; % b is the RHS of linear system

35. Profile serial code with profile. Identify section of code or function within code using the most CPU time. Look for ways to improve code section performance. See if section is parallelizable and worth parallelizing. If warranted, research and choose a suitable parallel algorithm and parallel paradigm for code section. Parallelize code section with chosen parallel algorithm. To improve performance further, work on the next most CPU-time-intensive section. Repeats steps 2 – 7. Analyze the performance efficiency to know what the sweet-spot is, i.e., given the implementation and platforms on which code is intended, what is the minimum number of workers for speediest turn-around (see the Amdahl’s Law page). 7. How Do I Parallelize My Code ?

36. Task parallel applications generally scale linearly. • Data parallel applications’ parallel efficiency depend • on individual code and algorithm used. • My personal experience is that the runtime of a reasonably • well tuned MATLAB program running on single or multi- • processor is at least an order of magnitude slower than an • equivalent C/FORTRAN code. • Additionally, the PCT’s communication is based on the • Ethernet. MPI on Katana uses Infiniband and is faster than • Ethernet. This further disadvantaged PCT if your code is • communication-bound. 8. How Well Does PCT Scales ?

37. S is ratio of T1 over TN , elapsed times of 1 and N workers. f is fraction of T1 due to code sections not parallelizable. Amdahl’s Law above states that a code with its parallelizable component comprising 90% of total computation time can at best achieve a 10X speedup with lots of workers. A code that is 50% parallelizable speeds up two-fold with lots of workers. The parallel efficiency is E = S / N Program that scales linearly (S = N) has parallel efficiency 1. A task-parallel program is usually more efficient than a data- parallel program. Parallel codes can sometimes achieve super-linear behavior due to efficient cache usage per worker. Speedup Ratio and Parallel Efficiency

38. Example of Speedup Ratio & Parallel Efficiency

39. MATLAB provides various ways to submit and run jobs in the background or in batch. SCV recommends a simple and effective method for all PCT batch processing on Katana. Cut-and-paste the below into a file, say, mbatch Batch Processing On Katana #!/bin/csh -f # MATLAB script for running serial or parallel background jobs # ForMATLAB applications that contain MATLAB Parallel Computing Toolbox # parallel operations request (matlabpool or dfeval), a batch job # will be queued and run in batch (using the SGE configuration) # Script name: mbatch (you can change it) # Usage: katana% mbatch <m-file> <output-file> # <m-file>: name of m-file to be executed, DONOT include .m ($1) # <output-file>: output file name; may include path ($2) nohup matlab –nodisplay -r “$1 exit” >! $2 & katana% chmod +x mbatchgive mbatch execute attribute Katana% mbatch my_mfile myOutput Do not start MATLAB with -nojvm. PCT requires Java.

40. You can use the Katana Cluster as if it is ‘local’ with 4 workers • for interactive usages: • Request 4 processors on the same node • (You need x-win32 on your client computer) • Katana% qsh –pe omp 4 • 2. In the new X-window, run matlab • Katana% matlab • 3. Request workers in MATLAB window • >> matlabpool open local 4 • or • >> pmode start local 4 • Generally, ‘SGE’ is the default configuration. If ‘local’ is not • specified, matlabpool would request 4 workers using the ‘SGE’ • configuration and hence the processors allocated with the qsh • interactive batch shell would not be used. Use ‘local’ Config. on Katana Request a node with 4 processors. Only need a PCT license; no worker licenses needed.

41. Communications Among Workers, Client • Communications between workers and MATLAB client • pmode:lab2client, client2lab • matlabpool:A = a{1} % from lab 1 to client • Collective communications among workers • gather, gop, gplus, gcat

42. MPI Point-to-Point Communications • MPI point-to-point communication among workers • labSend and labReceive • % Example: each lab sends its lab # to lab 1 and sum data on lab 1 • matlabpool open 4 • spmd % MPI requires spmd • a = labindex; % define a on workers • If labindex == 1 % on worker 1 . . . • % lab 1 is designated to accumulate sum on a • s = a; % sum s starts with a on worker 1 • for k = 2:numlabs % loop over remaining workers • s = s + labReceive(k); % receive a from worker k, then add to s • end • else % for all other workers . . . • labSend(a,1); % send a on workers 2 to 4 to worker 1 • end • end % spmd • indexSum = s{1}; % copy s on lab 1 to client • matlabpool close • It is illegal to communicate with itself.

43. MATLAB Parallel Computing Toolbox Useful SCV Info Please help us do better in the future by participating in aquick survey: http://scv.bu.edu/survey/tutorial_evaluation.html • SCV home page (www.bu.edu/tech/research) • Resource Applicationswww.bu.edu/tech/accounts/special/research/accounts • Help • System • help@katana.bu.edu, bu.service-now.com • Web-based tutorials (www.bu.edu/tech/research/training/tutorials) (MPI, OpenMP, MATLAB, IDL, Graphics tools) • HPC consultations by appointment • Kadin Tseng (kadin@bu.edu)