GMRES with embedded ensemble propagation for the efficient solution of parametric linear systems in uncertainty quantification of computational models

Liegeois, Kim; Boman, Romain; Phipps, Eric Todd; Wiesner, Tobias A.; Arnst, Maarten

doi:10.1016/j.cma.2020.113188

Cited by 3 publications

(6 citation statements)

References 20 publications

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“…Mathematically, the coupled problem can be seen as the gathering of the matrices associated to the current subset into a block diagonal matrix followed by the application of the Krylov method on the gathered block diagonal system. In the field of ensemble propagation, this approach is known as ensemble reduction [13]. The convergence of this approach depends on the union of the spectra of all the matrices; in the best-case scenario, the union of the spectra is not worse than the worst spectra among the current subset.…”

Section: Batched Krylov Methodsmentioning

confidence: 99%

“…For a given subset, the Krylov method will iterate up to the convergence of the last system of the subset. In the field of ensemble propagation, this approach is known as ensemble-typed dot product or without ensemble reduction [13]. This approach requires every kernel to be a team level kernel (e.g., SPMV, dot product) which support subsets of values instead of one value so that the Krylov solver can be implemented using them.…”

Section: Batched Krylov Methodsmentioning

confidence: 99%

“…The Krylov methods might require different numbers of iterations for different systems to converge within the same subset. This can lead to issues such as overflow if not treated carefully as discussed in [13].…”

Section: Batched Krylov Methodsmentioning

confidence: 99%

“…This batched Krylov method can be called from a parallel-for so multiple teams can solve independent subsets as shown in Algorithm 2. For more details on the code divergence in the GMRES, we refer to [13]. Moreover, in the rest of this paper, if not stated otherwise, we will only discuss the implementation at the team level and therefore, to reduce the verbosity of the formula, we use the following notation shortcut: 𝓐 := 𝓐 (ℓ 1 :ℓ 2 ):: where ℓ 1 and ℓ 2 are as in Algorithm 2.…”

Section: Batched Krylov Methodsmentioning

confidence: 99%

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Performance Portable Batched Sparse Linear Solvers

Liegeois

Rajamanickam

Berger-Vergiat

2023

IEEE Trans. Parallel Distrib. Syst.

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Solving large number of small linear systems is increasingly becoming a bottleneck in computational science applications. While dense linear solvers for such systems have been studied before, batched sparse linear solvers are just starting to emerge. In this paper, we discuss algorithms for solving batched sparse linear systems and their implementation in the Kokkos Kernels library. The new algorithms are performance portable and map well to the hierarchical parallelism available in modern accelerator architectures. The sparse matrix vector product (SPMV) kernel is the main performance bottleneck of the Krylov solvers we implement in this work. The implementation of the batched SPMV and its performance are therefore discussed thoroughly in this paper. The implemented kernels are tested on different Central Processing Unit (CPU) and Graphic Processing Unit (GPU) architectures. We also develop batched Conjugate Gradient (CG) and batched Generalized Minimum Residual (GMRES) solvers using the batched SPMV. Our proposed solver was able to solve 20,000 sparse linear systems on V100 GPUs with a mean speedup of 76x and 924x compared to using a parallel sparse solver with a block diagonal system with all the small linear systems, and compared to solving the small systems one at a time, respectively. We see mean speedup of 0.51 compared to dense batched solver of cuSOLVER on V100, while using lot less memory. Thorough performance evaluation on three different architectures and analysis of the performance are presented.

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Section: Batched Krylov Methodsmentioning

confidence: 99%

Section: Batched Krylov Methodsmentioning

confidence: 99%

Section: Batched Krylov Methodsmentioning

confidence: 99%

Section: Batched Krylov Methodsmentioning

confidence: 99%

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Performance Portable Batched Sparse Linear Solvers

Liegeois

Rajamanickam

Berger-Vergiat

2023

IEEE Trans. Parallel Distrib. Syst.

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“…Two parts need to be computed for x, the domain integrals and the boundary integrals. During the computed procedure, the domain integral is computed with the help of the FMM-accelerated LIM, the boundary integrals are computed by the FMM FM-LIBEM for 3D heat conduction analysis accelerated BEM, and the result of the domain integral is added into the right vector, then the unknowns are computed by the generalized minimal residual method (Kpadonou et al, 2020;Liegeois et al, 2020). In FM-LIBEM, all the integrals are divided into two kinds for a node x, the near integrals (integral points in cubes that are the neighborhood of C P and itself) and the far integrals (integral points in cubes that are in the interaction list of C P and those wellseparated from itself), the previous one is computed by the tradition LIBEM directly, while the computation of the latter one is done by the FMM.…”

Section: 2mentioning

confidence: 99%

The fast multipole method–accelerated line integration boundary element method for 3D heat conduction analysis with heat source

Liu,

Wang,

Feng

et al. 2023

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Purpose3D steady heat conduction analysis considering heat source is conducted on the fundamental of the fast multipole method (FMM)-accelerated line integration boundary element method (LIBEM).Design/methodology/approachDue to considering the heat source, domain integral is generated in the traditional heat conduction boundary integral equation (BIE), which will counteract the well-known merit of the BEM, namely, boundary-only discretization. To avoid volume discretization, the enhanced BEM, the LIBEM with dimension reduction property is introduced to transfer the domain integral into line integrals. Besides, owing to the unsatisfactory performance of the LIBEM when it comes to large-scale structures requiring massive computation, the FMM-accelerated LIBEM (FM-LIBEM) is proposed to improve the computation efficiency further.FindingsAssuming N and M are the numbers of nodes and integral lines, respectively, the FM-LIBEM can reduce the time complexity from O(NM) to about O(N+ M), and a full discussion and verification of the advantage are done based on numerical examples under heat conduction.Originality/value(1) The LIBEM is applied to 3D heat conduction analysis with heat source. (2) The domain integrals can be transformed into boundary integrals with straight line integrals by the LIM. (3) A FM-LIBEM is proposed and can reduce the time complexity from O(NM) to O(N+ M). (4) The FM-LIBEM with high computational efficiency is exerted to solve 3D heat conduction analysis with heat source in massive computation successfully.

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A high-performance boundary element method and its applications in engineering

Liu¹

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GMRES with embedded ensemble propagation for the efficient solution of parametric linear systems in uncertainty quantification of computational models

Cited by 3 publications

References 20 publications

Performance Portable Batched Sparse Linear Solvers

Performance Portable Batched Sparse Linear Solvers

The fast multipole method–accelerated line integration boundary element method for 3D heat conduction analysis with heat source

A high-performance boundary element method and its applications in engineering

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