Finite Element Algorithms and Data Structures on Graphical Processing Units

The finite element method (FEM) is one of the most commonly used techniques for the solution of partial differential equations on unstructured meshes. This paper discusses both the assembly and the solution phases of the FEM with special attention to the balance of computation and data movement. We...

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Vydáno v:International journal of parallel programming Ročník 43; číslo 2; s. 203 - 239
Hlavní autoři: Reguly, I. Z., Giles, M. B.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Boston Springer US 01.04.2015
Springer Nature B.V
Témata:
Algorithms
Approximation
Assembly
Computation
Computer architecture
Computer Science
Finite element analysis
Finite element method
Indexing
Linear algebra
Mathematical analysis
Mathematical models
Methods
Partial differential equations
Phases
Processor Architectures
Software Engineering/Programming and Operating Systems
Sparsity
Studies
Theory of Computation
Finite element method
Preconditioned conjugate gradient method
Sparse matrix-vector multiplication
Performance analysis
Graphical processing unit
ISSN:0885-7458, 1573-7640
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Shrnutí:The finite element method (FEM) is one of the most commonly used techniques for the solution of partial differential equations on unstructured meshes. This paper discusses both the assembly and the solution phases of the FEM with special attention to the balance of computation and data movement. We present a GPU assembly algorithm that scales to arbitrary degree polynomials used as basis functions, at the expense of redundant computations. We show how the storage of the stiffness matrix affects the performance of both the assembly and the solution. We investigate two approaches: global assembly into the CSR and ELLPACK matrix formats and matrix-free algorithms, and show the trade-off between the amount of indexing data and stiffness data. We discuss the performance of different approaches in light of the implicit caches on Fermi GPUs and show a speedup over a two-socket 12-core CPU of up to 10 times in the assembly and up to 6 times in the solution phase. We present our sparse matrix-vector multiplication algorithms that are part of a conjugate gradient iteration and show that a matrix-free approach may be up to two times faster than global assembly approaches and up to 4 times faster than NVIDIA’s cuSPARSE library, depending on the preconditioner used.
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ISSN:0885-7458
1573-7640
DOI:10.1007/s10766-013-0301-6