I am Computer Scientist and now I am interested in matrix multiplication on GPUs. My research are focused on matrix free finite elements method where I multiply sparse matrix. Sparse matrix could multiply regular or matrix free. In general based on special coordinate function. I have a few general question: How popular is this method? Does any another name exist for this method? I am also looking for books and article concentrate on finite element method especially matrix free multiplication and I consider about general books and article. Because many article based on rather complicated examples like conjugate gradient method or something different.

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    $\begingroup$ I'm voting to close this question as off-topic because it is about a discipline in academia, not about Academia itself. $\endgroup$ – scaaahu Oct 22 '15 at 9:14
  • $\begingroup$ What service on stack exchange you recommended for this question? $\endgroup$ – Konrad Oct 22 '15 at 9:16
  • $\begingroup$ You may want to try Mathematics SE, Computer Science SE, Computational Science SE. Please do not cross-post i.e. please delete this one before you ask the same question on another SE. $\endgroup$ – scaaahu Oct 22 '15 at 9:19
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    $\begingroup$ Scicomp.se is definitely the right place for a question on this topic. $\endgroup$ – Federico Poloni Oct 22 '15 at 9:37
  • $\begingroup$ Ok, thanks for reply. I dont' know why I cannot delete this question by myself. I don't have delete button. $\endgroup$ – Konrad Oct 22 '15 at 9:49

Matrix-free finite elements are relatively well-known. For explicit methods for transient problems, this involves applying the finite element matrix using small reference matrices and geometry-specific transformations.

For implicit problems, this is usually done in conjunction with an iterative solver such as CG, GMRES, etc. Note that this typically also requires a preconditioner, many of which may not map well directly to GPUs. For nonlinear implicit problems, this may also be paired with a matrix-free approximation of the Jacobian in a linearization (see for example Ben Kirk's thesis for an application of matrix-free implicit methods in CFD).

For GPU-specific implementations of explicit methods, there is ample literature on matrix-free implementations of finite element (specifically Discontinuous Galerkin) methods; see, for example, Hesthaven/Warburton (not GPU specific, but implementation is similar) and Klockner et al.. For some literature on the GPU implementation of matrix-free implicit solvers, see Remacle, Gandham, Warburton, where they solve the heat equation using a two-grid overlapping additive Schwarz preconditioner.

  • $\begingroup$ This question also appears to be a duplicate. Flagging. $\endgroup$ – Jesse Chan Oct 22 '15 at 14:47
  • $\begingroup$ Since this question will be deleted, would you copy your answer to the other version? It complements the other answer. $\endgroup$ – Christian Clason Oct 22 '15 at 15:42

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