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AN O(N) DIRECT SOLVER FOR INTEGRAL EQUATIONS ON THE PLANE
"... Abstract. An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that offdiagonal blocks of certain dense matrices have numerically low ..."
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Abstract. An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that offdiagonal blocks of certain dense matrices have numerically low rank. Technically, the solver is inspired by previously developed direct solvers for integral equations based on “recursive skeletonization ” and “Hierarchically SemiSeparable” (HSS) matrices, but it improves on the asymptotic complexity of existing solvers by incorporating an additional level of compression. The resulting solver has optimal O(N) complexity for all stages of the computation, as demonstrated by both theoretical analysis and numerical examples. The computational examples further display good practical performance in terms of both speed and memory usage. In particular, it is demonstrated that even problems involving 107 unknowns can be solved to precision 10−10 using a simple Matlab implementation of the algorithm executed on a single core.
Front. Math. China
"... A direct solver with O(N) complexity for integral equations on onedimensional domains ..."
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A direct solver with O(N) complexity for integral equations on onedimensional domains
An O(N) algorithm for constructing the solution operator to elliptic boundary value problems in the absence
"... of body loads ..."
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"... A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method ..."
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A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method
Normpreserving discretization of integral equations for elliptic PDEs with internal layers I: the onedimensional case
"... We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations (PDEs) in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various Lp norms. We sh ..."
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We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations (PDEs) in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various Lp norms. We show that highorder accurate Nyström discretization leads to wellconditioned finitedimensional linear systems if and only if the discretization is both normpreserving in a correctly chosen Lp space and adaptively refined in the internal layer.
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"... Abstract. We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a lowrank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sp ..."
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Abstract. We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a lowrank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sparse one that allows fast factorization and application of the inverse. The algorithm extends the Martinsson–Rokhlin method developed for 2D boundary integral equations and proceeds in two phases: a precomputation phase, consisting of matrix compression and factorization, followed by a solution phase to apply the matrix inverse. For boundary integral equations which are not too oscillatory, e.g., based on the Green functions for the Laplace or lowfrequency Helmholtz equations, both phases typically have complexity O(N) in two dimensions, where N is the number of discretization points. In our current implementation, the corresponding costs in three dimensions are O(N3/2) and O(N logN) for precomputation and solution, respectively. Extensive numerical experiments show a speedup of ∼100 for the solution phase over modern fast multipole methods; however, the cost of precomputation remains high. Thus, the solver is particularly suited to problems where large numbers of iterations would be required. Such is the case with illconditioned linear systems or when the same system is to be solved with multiple righthand sides. Our algorithm is implemented in Fortran and freely available.