MATLAB: Preconditioning algorithm on GPU for solution of sparse matrices

gpupreconditioner

I solve large sparse Ax=b equations with conjugate gradient algorithms with a preconditioner. Since Matlab 2016a, Matlab started to support some conjugate gradient algorithms like bicgstab, pcg, gmres on GPU with a preconditioner for sparse matrices. Those functions only accept M sparse matrix (M=M1*M2 for M1 lower M2 upper triangular sparse matrix) not M1 and M2.

I'm wondering how Matlab apply preconditioner? I know that sparse triangular matrix solving on GPU is notoriously slow. So I think it might use some kind of iterative method. Maybe preconditioner applying might be done on CPU instead. So what exactly is done on the background while applying the preconditioner?

Best Answer

MATLAB's preconditioning for sparse iterative solvers on the GPU is currently implemented using ILU and sparse triangular solves. If you have a solution more appropriate to your problem then you can use the functional form - this diverts to a different implementation but can be faster and/or converge better depending on your problem.

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MATLAB: Iterative solver with gpuArray

Even for much larger problem sizes (n=10240) and a not so new graphics card (GTX 580), I see negligible overhead in time to swap between CPU and GPU,

   n = 1024*10; 
    Acpu = rand(n)+100*eye(n);
    bcpu = rand(n,1); 
    Agpu = gpuArray(Acpu); 
    bgpu= gpuArray(bcpu);
    gputimeit(@() Agpu*bgpu) %all data on gpu
    %0.0052sec
    gputimeit(@() gather( Agpu*bcpu )) %requires data transfer
    %0.0054sec

Speed-up in GMRES also seems pretty good (factor of 4)

   tic;
    x = gmres(@(x) Acpu*x,bcpu,[]);
   toc
   %Elapsed time is 0.391786 seconds.
   tic;
    x = gmres(@(x)gather(Agpu*x),bcpu,[]);
   toc
   %Elapsed time is 0.097924 seconds.

MATLAB: How `gpuArray` save sparse matrix when running Preconditioned conjugated gradient

gpuArray currently stores sparse matrices internally in CSR format. This matches the NVIDIA cusparse routines that are used for basic algebra.

I don't know quite what you mean by dynamic allocation. All MATLAB variables are allocated dynamically in some sense, because they are not defined before the application is run. However, MATLAB uses a variety of pooling techniques to ensure actual dynamic allocations (such as calls to cudaMalloc) happen as infrequently as possible. If you are noticing some performance delays when data is copied to the device then sometimes the conversion between CSC (the CPU storage format) and CSR is responsible.

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MATLAB: Iterative solver with gpuArray

MATLAB: How `gpuArray` save sparse matrix when running Preconditioned conjugated gradient

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