The inverse fast multipole method: using a fast approximate direct solver as a preconditioner for dense linear systems

Coulier, Pieter; Pouransari, Hadi; Darve, Eric

doi:10.48550/arxiv.1508.01835

Cited by 11 publications

(30 citation statements)

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“…The sparse factorization is constructed of parameters of the H 2 matrix. In the numerical experiments, we show the consistency of this approach in comparison to the other approximate block low-rank hierarchical solvers, such as HODLR[3], H2Lib[5], and IFMM [11].…”

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confidence: 79%

“…, where r i ∈ R 3 is the position of the i-th element. This example is used for testing of IFMM (Inverse Fast Multipole Method) method as a preconditioning in [11], so we have chosen this example for the convenient comparison.…”

Section: Comparison To H2lib Liberarymentioning

confidence: 99%

“…The nested basis property leads to the additional improvement in terms of storage and complexity of different operations such as matrix-vector products. Approximate solution and preconditioning of systems with H 2 matrices is a rapidly developed area [9,3,11,19], however, construction of the accurate, time and memory efficient factorization that leads to approximate solution is still a challenging problem. In this paper, we propose a new representation of H 2 matrices.…”

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confidence: 99%

“…• The algorithm that allows to construct factors U, S and V from parameters of H 2 matrix. • Numerical comparison of the proposed method with HODLR [1], IFMM [11] and H2Lib [5] packages. The main idea of the sparsification algorithm is the compression of the low-rank blocks (the very close idea of the compression of the fill-in blocks during the block Cholesky factorization of a sparse matrix is presented in works [25,29]).…”

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confidence: 99%

“…The main idea of the sparsification algorithm is the compression of the low-rank blocks (the very close idea of the compression of the fill-in blocks during the block Cholesky factorization of a sparse matrix is presented in works [25,29]). The main difference between the presented sparse factorization and the other methods of sparsification [2,11,26] is a size of factors. In the other methods, the hierarchical matrix is factorized into the sparse matrices of larger sizes.…”

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confidence: 99%

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Simple non-extensive sparsification of the hierarchical matrices

Sushnikova¹,

Oseledets²

2017

Preprint

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In this paper, we consider the matrices approximated in H 2 format. The direct solution, as well as the preconditioning, of systems with such matrices is a challenging problem. We propose a non-extensive sparse factorization of the H 2 matrix that allows to substitute direct H 2 solution with the solution of the system with an equivalent sparse matrix of the same size. The sparse factorization is constructed of parameters of the H 2 matrix. In the numerical experiments, we show the consistency of this approach in comparison to the other approximate block low-rank hierarchical solvers, such as HODLR[3], H2Lib[5], and IFMM [11].

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confidence: 79%

Section: Comparison To H2lib Liberarymentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Simple non-extensive sparsification of the hierarchical matrices

Sushnikova¹,

Oseledets²

2017

Preprint

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Sparse hierarchical solvers with guaranteed convergence

Yang

Pouransari

Darve

2019

Numerical Meth Engineering

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Summary Solving sparse linear systems from discretized partial differential equations is challenging. Direct solvers have, in many cases, quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and efficient. Approximate factorization preconditioners such as incomplete LU factorization provide cheap approximations to the system matrix. However, even a highly accurate preconditioner may have deteriorating performance when the condition number of the system matrix increases. By increasing the accuracy on low‐frequency errors, we propose a novel hierarchical solver with improved robustness with respect to the condition number of the linear system. This solver retains the linear computational cost and memory footprint of the original algorithm.

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An integral equation formulation for rigid bodies in Stokes flow in three dimensions

Corona

Greengard

Rachh

et al. 2017

Journal of Computational Physics

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We present a new derivation of a boundary integral equation (BIE) for simulating the threedimensional dynamics of arbitrarily-shaped rigid particles of genus zero immersed in a Stokes fluid, on which are prescribed forces and torques. Our method is based on a single-layer representation and leads to a simple second-kind integral equation. It avoids the use of auxiliary sources within each particle that play a role in some classical formulations. We use a spectrally accurate quadrature scheme to evaluate the corresponding layer potentials, so that only a small number of spatial discretization points per particle are required. The resulting discrete sums are computed in O(n) time, where n denotes the number of particles, using the fast multipole method (FMM). The particle positions and orientations are updated by a high-order time-stepping scheme. We illustrate the accuracy, conditioning and scaling of our solvers with several numerical examples.

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The inverse fast multipole method: using a fast approximate direct solver as a preconditioner for dense linear systems

Cited by 11 publications

References 0 publications

Simple non-extensive sparsification of the hierarchical matrices

Simple non-extensive sparsification of the hierarchical matrices

Sparse hierarchical solvers with guaranteed convergence

An integral equation formulation for rigid bodies in Stokes flow in three dimensions

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