An Adaptive Fast Direct Solver for Boundary Integral Equations in Two Dimensions

Kong, W. Y.; Bremer, J.; Rokhlin, V.

doi:10.21236/ada555115

Cited by 13 publications

(18 citation statements)

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“…To name few references: [9,10,18,30,25]. As a subclass of such solvers, HODLR direct solvers [1,3,21] have been introduced. These solvers compute approximate sparse LU-like factorization of HSS-matrices.…”

Section: Final Comparisonmentioning

confidence: 99%

``Compress and Eliminate” Solver for Symmetric Positive Definite Sparse Matrices

Sushnikova¹,

Oseledets²

2018

SIAM J. Sci. Comput.

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We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in. The systems that have efficient compression of the fill-in mostly arise from discretization of partial differential equations. We show that the resulting factorization can be used as an efficient preconditioner and compare the proposed approach with the state-of-art direct and iterative solvers.

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Section: Final Comparisonmentioning

confidence: 99%

``Compress and Eliminate” Solver for Symmetric Positive Definite Sparse Matrices

Sushnikova¹,

Oseledets²

2018

SIAM J. Sci. Comput.

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“…These matrices are formalized as, with an increasing level of complexity, hierarchically off-diagonal low-rank (HODLR) matrices, 55 hierarchically semiseparable (HSS) matrices, 56,57 -matrices, 46,58 and  2 -matrices. 59,60 Fast methods for efficiently constructing and storing approximate inverses of these matrices have been developed in recent years, [61][62][63] mainly for the aforementioned HODLR and HSS matrices. Moreover, once the approximate inverse, which only depends on the geometry, is computed and stored, it can be applied with a minimal amount of CPU time to multiple right-hand sides.…”

Section: Figurementioning

confidence: 99%

An efficient preconditioner for the fast simulation of a 2D stokes flow in porous media

Quaife

Coulier

Darve

2017

Numerical Meth Engineering

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Summary We consider an efficient preconditioner for a boundary integral equation (BIE) formulation of the two‐dimensional Stokes equations in porous media. While BIEs are well‐suited for resolving the complex porous geometry, they lead to a dense linear system of equations that is computationally expensive to solve for large problems. This expense is further amplified when a significant number of iterations is required in an iterative Krylov solver such as generalized minimial residual method (GMRES). In this paper, we apply a fast inexact direct solver, the inverse fast multipole method, as an efficient preconditioner for GMRES. This solver is based on the framework of H2‐matrices and uses low‐rank compressions to approximate certain matrix blocks. It has a tunable accuracy ε and a computational cost that scales as scriptOfalse(N2ptnormallnormalonormalg21false/εfalse). We discuss various numerical benchmarks that validate the accuracy and confirm the efficiency of the proposed method. We demonstrate with several types of boundary conditions that the preconditioner is capable of significantly accelerating the convergence of GMRES when compared to a simple block‐diagonal preconditioner, especially for pipe flow problems involving many pores.

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“…The error analysis of the proxy surface method, however, is only briefly discussed in [7] without much detail and the selection of Y p to discretize the proxy surface is heuristic in previous applications [7,5,10,11]. In this paper, we provide a detailed error analysis of the proxy surface method for the 3D Laplace kernel.…”

Section: Introductionmentioning

confidence: 99%

“…Ref. [7,10] suggest using |Y p | ∼ O(|X 0 |) and [5] claims correctly but without an explanation that for the Laplace kernel, proxy surfaces of different sizes can be discretized using a constant number of points and this constant only depends on the compression precision.…”

Section: Introductionmentioning

confidence: 99%

Error analysis of an accelerated interpolative decomposition for 3D Laplace problems

Xing

Chow

2020

Applied and Computational Harmonic Analysis

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In constructing the H 2 representation of dense matrices defined by the Laplace kernel, the interpolative decomposition of certain off-diagonal submatrices that dominates the computation can be dramatically accelerated using the concept of a proxy surface. We refer to the computation of such interpolative decompositions as the proxy surface method. We present an error bound for the proxy surface method in the 3D case and thus provide theoretical guidance for the discretization of the proxy surface in the method.

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An Adaptive Fast Direct Solver for Boundary Integral Equations in Two Dimensions

Cited by 13 publications

References 0 publications

``Compress and Eliminate” Solver for Symmetric Positive Definite Sparse Matrices

``Compress and Eliminate” Solver for Symmetric Positive Definite Sparse Matrices

An efficient preconditioner for the fast simulation of a 2D stokes flow in porous media

Error analysis of an accelerated interpolative decomposition for 3D Laplace problems

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