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Mat2Stencil: A Modular Matrix-Based DSL for Explicit and Implicit Matrix-Free PDE Solvers on Structured Grid

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Wenguang ChenAuthors Info & Claims

Proceedings of the ACM on Programming Languages, Volume 7, Issue OOPSLA2

Article No.: 246, Pages 686 - 715

https://doi.org/10.1145/3622822

Published: 16 October 2023 Publication History

Abstract

Partial differential equation (PDE) solvers are extensively utilized across numerous scientific and engineering fields. However, achieving high performance and scalability often necessitates intricate and low-level programming, particularly when leveraging deterministic sparsity patterns in structured grids.

In this paper, we propose an innovative domain-specific language (DSL), Mat2Stencil, with its compiler, for PDE solvers on structured grids. Mat2Stencil introduces a structured sparse matrix abstraction, facilitating modular, flexible, and easy-to-use expression of solvers across a broad spectrum, encompassing components such as Jacobi or Gauss-Seidel preconditioners, incomplete LU or Cholesky decompositions, and multigrid methods built upon them. Our DSL compiler subsequently generates matrix-free code consisting of generalized stencils through multi-stage programming. The code allows spatial loop-carried dependence in the form of quasi-affine loops, in addition to the Jacobi-style stencil’s embarrassingly parallel on spatial dimensions. We further propose a novel automatic parallelization technique for the spatially dependent loops, which offers a compile-time deterministic task partitioning for threading, calculates necessary inter-thread synchronization automatically, and generates an efficient multi-threaded implementation with fine-grained synchronization.

Implementing 4 benchmarking programs, 3 of them being the pseudo-applications in NAS Parallel Benchmarks with 6.3% lines of code and 1 being matrix-free High Performance Conjugate Gradients with 16.4% lines of code, we achieve up to 1.67× and on average 1.03× performance compared to manual implementations.

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Cited By

Aydoğdu MAydoğdu ÖPehlivan H(2024)MxPL: A Programming Language for Matrix-Related OperationsSymmetry10.3390/sym1602018116:2(181)Online publication date: 2-Feb-2024
https://doi.org/10.3390/sym16020181
Zhu Q(2024)FreeStencil: A Fine-Grained Solver Compiler with Graph and Kernel Optimizations on Structured Meshes for Modern GPUsProceedings of the 53rd International Conference on Parallel Processing10.1145/3673038.3673076(1022-1031)Online publication date: 12-Aug-2024
https://dl.acm.org/doi/10.1145/3673038.3673076

Index Terms

Mat2Stencil: A Modular Matrix-Based DSL for Explicit and Implicit Matrix-Free PDE Solvers on Structured Grid
1. Computing methodologies
  1. Parallel computing methodologies
    1. Parallel algorithms
      1. Shared memory algorithms
    2. Parallel programming languages
2. Software and its engineering
  1. Software notations and tools
    1. Compilers
      1. Source code generation
    2. Context specific languages
      1. Domain specific languages

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cover image Proceedings of the ACM on Programming Languages

Proceedings of the ACM on Programming Languages Volume 7, Issue OOPSLA2

October 2023

2250 pages

EISSN:2475-1421

DOI:10.1145/3554312

Editor:
Michael Hicks
Amazon, USA

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Association for Computing Machinery

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Published: 16 October 2023

Published in PACMPL Volume 7, Issue OOPSLA2

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Cited By

Aydoğdu MAydoğdu ÖPehlivan H(2024)MxPL: A Programming Language for Matrix-Related OperationsSymmetry10.3390/sym1602018116:2(181)Online publication date: 2-Feb-2024
https://doi.org/10.3390/sym16020181
Zhu Q(2024)FreeStencil: A Fine-Grained Solver Compiler with Graph and Kernel Optimizations on Structured Meshes for Modern GPUsProceedings of the 53rd International Conference on Parallel Processing10.1145/3673038.3673076(1022-1031)Online publication date: 12-Aug-2024
https://dl.acm.org/doi/10.1145/3673038.3673076

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