首页|Algebraic temporal blocking for sparse iterative solvers on multi-core CPUs

Algebraic temporal blocking for sparse iterative solvers on multi-core CPUs

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  • NETL
  • NSTL
  • Sage
Sparse linear iterative solvers are essential for many large-scale simulations. Much of the runtime of these solvers is often spent in the implicit evaluation of matrix polynomials via a sequence of sparse matrix-vector products. A variety of approaches has been proposed to make these polynomial evaluations explicit (i.e., fix the coefficients), e.g., polynomial preconditioners or s-step Krylov methods. Furthermore, it is nowadays a popular practice to approximate triangular solves by a matrix polynomial to increase parallelism. Such algorithms allow to evaluate the polynomial using a so-called matrix power kernel (MPK), which computes the product between a power of a sparse matrix A and a dense vector x, i.e., A~px, or a related operation. Recently we have shown that using the level-based formulation of sparse matrix-vector multiplications in the Recursive Algebraic Coloring Engine (RACE) framework we can perform temporal cache blocking of MPK to increase its performance. In this work, we demonstrate the application of this cache-blocking optimization in sparse iterative solvers. By integrating the RACE library into the Trilinos framework, we demonstrate the speedups achieved in (preconditioned) s-step GMRES, polynomial preconditioners, and algebraic multigrid (AMG). For MPK-dominated algorithms we achieve speedups of up to 3× on modem multi-core compute nodes. For algorithms with moderate contributions from subspace orthogonalization, the gain reduces significantly, which is often caused by the insufficient quality of the or-thogonalization routines. Finally, we showcase the application of RACE-accelerated solvers in a real-world wind turbine simulation (Nalu-Wind) and highlight the new opportunities and perspectives opened up by RACE as a cache-blocking technique for MPK-enabled sparse solvers.

Sparse matricesiterative solversmatrix polynomialcache blockingperformance

Christie Alappat、Jonas Thies、Georg Hager、Holger Fehske、Gerhard Wellein

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Erlangen National High Performance Computing Center, Friedrich- Alexander-Universitaet Erlangen-Nuernberg, Erlangen, Germany

Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

Erlangen National High Performance Computing Center, Friedrich- Alexander-Universitaet Erlangen-Nuernberg, Erlangen, Germany||Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands||Department of Computer Science, Friedrich-Alexander-Universitaet Erlangen-Nuernberg, Erlangen, Germany

2025

10.1177/10943420241283828

International journal of high performance computing applications

International journal of high performance computing applications

ISSN：1094-3420
年,卷(期)：2025.39(2)
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