GPU speed up for pcg() is disappointing
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I am using the pcg() to solve x =A\b. b is about 2e6 long. I am using R2021b on Ubuntu 20.04 with a Ryzen 9 5950x CPU and an nVidia A4000 GPU.
Running this code...
tol = 1e-4;
%solve on the CPU
L = ichol(A, struct('michol','on'));
x = pcg(A, b, tol, 5000, L, L');
fprintf('solve time: %0.4g s\n', toc);
%solve on the GPU
x = pcg(gpuArray(A), b, tol, 50000);
fprintf('solve time: %0.4g s\n', toc);
pcg converged at iteration 347 to a solution with relative residual 9.6e-05.
solve time: 16.24 s
pcg converged at iteration 7281 to a solution with relative residual 9.9e-05.
solve time: 13.91 s
So, I am seeing a small GPU speed up (nice, but not very exciting). The fact that the GPU takes 20x the iterations of the CPU makes me think there could be a >20x speed up possible, which would be much more exciting.
The L arguments make a big difference to the CPU speed, but I can't get the GPU version to take it. Doing
x = pcg(gpuArray(A), b, tol, 5000, L, L');
throws "Error using gpuArray/pcg (line 58) When the first input argument is a sparse matrix, the second preconditioner cannot be a matrix. Use functions for both preconditioners, or multiply the precondition matrices". Doing
x = pcg(gpuArray(A), b, tol, 50000, L*L');
seems to hang MATLAB (after waiting 5 minutes I gave up and had to terminate by restarting MATLAB; ctrl-C did nothing).
Can anyone tell me what is going on here? Is it simply that the GPU version is using a different algorithm and so it makes no sense to compare number of iterations (and so the 15% speed up I see is all I should hope for)? Or is it that I need a different preconditioning approach?
I see there is a possible bug related to this: https://uk.mathworks.com/support/bugreports/details/2534618.
Joss Knight on 11 Sep 2022
I'm guessing LL' is extremely dense, which will explain why the solver stalls. On the GPU the preconditioning is (currently) performed using ILU, which, like most sparse operations should be passed a satisfactorily sparse matrix. Try just passing A as the preconditioner and you may get a better result. Also, try the other solvers (CGS, GMRES, LSQR, BICG etc).
The reason why the solvers on the GPU work differently is because a sparse direct solve does not parallelize well, which is why sparse backslash (\) is generally slow - mostly because of the amount of memory needed. That doesn't explain why the solvers do not accept two triangular sparse matrices as preconditioner input - that is something that should be rectified. But the ILU should have much the same effect as ICHOL does.
I thought I'd be telling you that the GPU is slow because your card isn't very fast for double precision (only 599 GFLOPS). But actually you're doing 20 times the iterations in less time, so it seems you're right, if you hit the right combination of solver and preconditioner there's a good chance you'll get to the result much faster.
More Answers (2)
Yair Altman on 8 Sep 2022
The gpuArray version of pcg has not been updated since 2018, so it is somewhat lagging compared to the CPU version. Preconditioner input for sparse input is only supported for a single preconditioner matrix, not two as in the CPU case. Refer to the help of the GPU version (type help('gpuArray/pcg') or edit(...) for details).
Perhaps you could try to use the GPU version with a dense (non-sparse) input - it's wasteful in memory but perhaps possibly faster than the sparse-CPU version:
x = pcg(gpuArray(double(A)), b, tol, 5000, L, L');
Christine Tobler on 12 Sep 2022
It looks like you can simply replace your current call to pcg with
x = pcg(A, b, tol, 5000, @(y) L\y, @(y)L'\y);
as the error message just says that it only supports function handle input here (though it's not clear to me why that restriction is there).
Perhaps it makes sense to also cast the matrix L to a gpuArray?