A direct solver for elliptic PDEs in three dimensions based on hierarchical merging of Poincaré–Steklov operators

Hao, Sijia; Martinsson, Per-Gunnar

doi:10.1016/j.cam.2016.05.013

Cited by 19 publications

(17 citation statements)

References 23 publications

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“…Such a solver is currently under development and will be reported in future publications. Other extensions currently under way includes the development of adaptive refinement criteria (as opposed to the supervised adaptivity used in this work), and the extension to problems in three dimensions, analogous to the work in [6] for homogeneous equations.…”

Section: Discussionmentioning

confidence: 99%

An accelerated Poisson solver based on multidomain spectral discretization

et al. 2018

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This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The method works best for domains that can readily be mapped onto a rectangle, or a collection of nonoverlapping rectangles. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and higher have been tested and work well). Local mesh refinement results in highly accurate solutions even in the presence of local irregular behavior due to corner singularities, localized loads, etc. The system of linear equations attained upon discretization is solved using a direct (as opposed to iterative) solver with O(N 1.5 ) complexity for the factorization stage and O(N log N ) complexity for the solve. The scheme is ideally suited for executing the elliptic solve required when parabolic problems are discretized via time-implicit techniques. In situations where the geometry remains unchanged between time-steps, very fast execution speeds are obtained since the solution operator for each implicit solve can be pre-computed.

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Section: Discussionmentioning

confidence: 99%

An accelerated Poisson solver based on multidomain spectral discretization

et al. 2018

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“…We seek to compute the Schur complement of the 11 block. Computing Schur complements is a core computation in many factorization strategies, e.g., [18], and we plan to build on it in future work to develop complete solvers for GPUs. Given a factorization of A 11 = L 11 L t 11 on the GPU, the CUPARSE library can compute the product of the Schur complement S = A 22 − A 12 (L 11 L t 11 )\A 21 with vectors, via sparse matrix-vector products and sparse triangular forward and backward solves.…”

Section: Schur Complement Computationmentioning

confidence: 99%

Randomized GPU Algorithms for the Construction of Hierarchical Matrices from Matrix-Vector Operations

Boukaram¹,

Turkiyyah²,

Keyes³

2019

SIAM J. Sci. Comput.

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Randomized algorithms for the generation of low rank approximations of large dense matrices have become popular methods in scientific computing and machine learning. In this paper, we extend the scope of these methods and present batched GPU randomized algorithms for the efficient generation of low rank representations of large sets of small dense matrices, as well as their generalization to the construction of hierarchically low rank symmetric H 2 matrices with general partitioning structures. In both cases, the algorithms need to access the matrices only through matrix-vector multiplication operations which can be done in blocks to increase the arithmetic intensity and substantially boost the resulting performance. The batched GPU kernels are adaptive, allow nonuniform sizes in the matrices of the batch, and are more effective than SVD factorizations on matrices with fast decaying spectra. The hierarchical matrix generation consists of two phases, interleaved at every level of the matrix hierarchy. A first phase adaptively generates low rank approximations of matrix blocks through randomized matrix-vector sampling. A second phase accumulates and compresses these blocks into a hierarchical matrix that is incrementally constructed. The accumulation expresses the low rank blocks of a given level as a set of local low rank updates that are performed simultaneously on the whole matrix allowing high-performance batched kernels to be used in the compression operations. When the ranks of the blocks generated in the first phase are too large to be processed in a single operation, the low rank updates can be split into smaller-sized updates and applied in sequence. Assuming representative rank k, the resulting matrix has optimal O(kN) asymptotic storage complexity because of the nested bases it uses. The ability to generate an H 2 matrix from matrix-vector products allows us to support a general randomized matrix-matrix multiplication operation, an important kernel in hierarchical matrix computations. Numerical experiments demonstrate the high performance of the algorithms and their effectiveness in generating hierarchical matrices to a desired target accuracy.

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“…Therefore, the interpolative decomposition can be formed efficiently. Readers are referred to Section 2.3 for the explicit forms of each matrix in (21). Looping over all faces at level ℓ, we obtain…”

Section: Sequential Hifmentioning

confidence: 99%

“…For distributed-memory implementations, however, the former lacks parallel scalability [24,25] while the latter demonstrates scalability only for 1D and 2D problems [1]. For 3D problems, these methods typically suffer from large prefactors that make them less efficient for practical large-scale problems.A recent group of methods explore the idea of integrating the MF method with the hierarchical matrix [28,40,38,39,17,37,21] or block low-rank matrix [34,35,2] approach in order to leverage the efficiency of both methods. Instead of directly applying the hierarchical matrix structure to the 3D problems, these methods apply it to the representation of the frontal matrices (i.e., the interactions between the lower dimensional fronts).…”

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confidence: 99%

Distributed-memory hierarchical interpolative factorization

Li¹,

Ying²

2017

Res Math Sci

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The hierarchical interpolative factorization (HIF) offers an efficient way for solving or preconditioning elliptic partial differential equations. By exploiting locality and low-rank properties of the operators, the HIF achieves quasi-linear complexity for factorizing the discrete positive definite elliptic operator and linear complexity for solving the associated linear system. In this paper, the distributed-memory HIF (DHIF) is introduced as a parallel and distributed-memory implementation of the HIF. The DHIF organizes the processes in a hierarchical structure and keeps the communication as local as possible. The computation complexity is O( N log N P ) and O( N P ) for constructing and applying the DHIF, respectively, where N is the size of the problem and P is the number of processes. The communication complexity is O()β where α is the latency and β is the inverse bandwidth. Extensive numerical examples are performed on the NERSC Edison system with up to 8192 processes. The numerical results agree with the complexity analysis and demonstrate the efficiency and scalability of the DHIF.

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A direct solver for elliptic PDEs in three dimensions based on hierarchical merging of Poincaré–Steklov operators

Cited by 19 publications

References 23 publications

An accelerated Poisson solver based on multidomain spectral discretization

An accelerated Poisson solver based on multidomain spectral discretization

Randomized GPU Algorithms for the Construction of Hierarchical Matrices from Matrix-Vector Operations

Distributed-memory hierarchical interpolative factorization

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