A domain-specific language and matrix-free stencil code for investigating electronic properties of Dirac and topological materials

Pieper, Andreas; Hager, Georg; Fehske, Holger

doi:10.1177/1094342020959423

Cited by 3 publications

(1 citation statement)

References 60 publications

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“…Addressing the issue, researchers have come up with many stencil libraries, and DSLs, including [Huang et al 2019;Lengauer et al 2020;Louboutin et al 2019;Pieper et al 2021;Tang et al 2011;Zhang et al 2017], providing easier interfaces for programming stencils. Among those, Devito [Louboutin et al 2019] and ExaStencil [Lengauer et al 2020] presents a set of program representations that is pretty much close to the math formula of the differential equation, being the easiest ones in programming.…”

Section: Related Workmentioning

confidence: 99%

Programming Matrices as Staged Sparse Rows to Generate Efficient Matrix-free Differential Equation Solver

Cao¹,

Tang²,

Yu³

et al. 2022

Preprint

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Solving differential equations is a critical task in scientific computing. Domain-specific languages (DSLs) have been a promising direction in achieving performance and productivity, but the current state of the art only supports stencil computation, leaving solvers requiring loop-carried dependencies aside. Alternatively, sparse matrices can represent such equation solvers and are more general than existing DSLs, but the performance is sacrificed.This paper points out that sparse matrices can be represented as programs instead of data, having both the generality from the matrix-based representation and the performance from program optimizations. Based on the idea, we propose the Staged Sparse Row (SSR) sparse matrix representation that can efficiently cover applications on structured grids. With SSR representation, users can intuitively define SSR matrices using generator functions and use SSR matrices through a concise object-oriented interface. SSR matrices can then be chained and applied to construct the algorithm, including those with loop-carried dependences. We then apply a set of dedicated optimizations, and ultimately simplify the SSR matrix-based codes into straightforward matrix-free ones, which are efficient and friendly for further analysis.Implementing BT pseudo application in the NAS Parallel Benchmark, with less than 10% lines of code compared with the matrix-free reference FORTRAN implementation, we achieved up to 92.8% performance. Implementing a matrix-free variant for the High-Performance Conjugate Gradient benchmark, we achieve 3.29× performance compared with the reference implementation, while our implementation shares the same algorithm on the same programming abstraction, which is sparse matrices.

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Section: Related Workmentioning

confidence: 99%

Programming Matrices as Staged Sparse Rows to Generate Efficient Matrix-free Differential Equation Solver

Cao¹,

Tang²,

Yu³

et al. 2022

Preprint

View full text Add to dashboard Cite

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Orthogonal Layers of Parallelism in Large-Scale Eigenvalue Computations

Alvermann

Hager

Fehske

2023

ACM Trans. Parallel Comput.

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We address the communication overhead of distributed sparse matrix-(multiple)-vector multiplication in the context of large-scale eigensolvers, using filter diagonalization as an example. The basis of our study is a performance model which includes a communication metric that is computed directly from the matrix sparsity pattern without running any code. The performance model quantifies to which extent scalability and parallel efficiency are lost due to communication overhead. To restore scalability, we identify two orthogonal layers of parallelism in the filter diagonalization technique. In the horizontal layer the rows of the sparse matrix are distributed across individual processes. In the vertical layer bundles of multiple vectors are distributed across separate process groups. An analysis in terms of the communication metric predicts that scalability can be restored if, and only if, one implements the two orthogonal layers of parallelism via different distributed vector layouts. Our theoretical analysis is corroborated by benchmarks for application matrices from quantum and solid state physics, road networks, and nonlinear programming. We finally demonstrate the benefits of using orthogonal layers of parallelism with two exemplary application cases—an exciton and a strongly correlated electron system—which incur either small or large communication overhead.

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

Cao,

Tang,

Zhu

et al. 2023

Proc. ACM Program. Lang.

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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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A domain-specific language and matrix-free stencil code for investigating electronic properties of Dirac and topological materials

Cited by 3 publications

References 60 publications

Programming Matrices as Staged Sparse Rows to Generate Efficient Matrix-free Differential Equation Solver

Programming Matrices as Staged Sparse Rows to Generate Efficient Matrix-free Differential Equation Solver

Orthogonal Layers of Parallelism in Large-Scale Eigenvalue Computations

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

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