[HTML][HTML] A fast direct solver for nonlocal operators in wavelet coordinates
H Harbrecht, M Multerer - Journal of computational physics, 2021 - Elsevier
Journal of computational physics, 2021•Elsevier
In this article, we consider fast direct solvers for nonlocal operators. The pivotal idea is to
combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with
the nested dissection ordering scheme. The latter drastically reduces the fill-in during the
factorization of the system matrix by means of a Cholesky decomposition or an LU
decomposition, respectively. This way, we end up with the exact inverse of the compressed
system matrix with only a moderate increase of the number of nonzero entries in the matrix …
combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with
the nested dissection ordering scheme. The latter drastically reduces the fill-in during the
factorization of the system matrix by means of a Cholesky decomposition or an LU
decomposition, respectively. This way, we end up with the exact inverse of the compressed
system matrix with only a moderate increase of the number of nonzero entries in the matrix …
Abstract
In this article, we consider fast direct solvers for nonlocal operators. The pivotal idea is to combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The latter drastically reduces the fill-in during the factorization of the system matrix by means of a Cholesky decomposition or an LU decomposition, respectively. This way, we end up with the exact inverse of the compressed system matrix with only a moderate increase of the number of nonzero entries in the matrix.
To illustrate the efficacy of the approach, we conduct numerical experiments for different highly relevant applications of nonlocal operators: We consider (i) the direct solution of boundary integral equations in three spatial dimensions, issuing from the polarizable continuum model, (ii) a parabolic problem for the fractional Laplacian in integral form and (iii) the fast simulation of Gaussian random fields.
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