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.
Highlights
Various problems in science and engineering lead to nonlocal operators and corresponding operator equations
We consider (i) a boundary integral equation arising from the polarizable continuum model in quantum chemistry as a classical example for a nonlocal operator equation, (ii) a parabolic problem for the fractional Laplacian, and (iii) the fast numerical simulation of Gaussian random fields as an important example from computational uncertainty quantification
The representation of the system matrix corresponding to a nonlocal operator with respect to an appropriate wavelet basis leads to a quasi-sparse matrix, i.e. a matrix with many small entries which can be neglected without compromising accuracy
Summary
Various problems in science and engineering lead to nonlocal operators and corresponding operator equations. Direct solver, Wavelet matrix compression, Polarizable continuum model, Fractional Laplacian, Gaussian random fields. Besides the rigorously controllable error for the matrix compression in the wavelet format and the roundoff errors in the computation of the matrix factorization, no additional approximation errors are introduced This is a major difference to other approaches for the discretization and the arithmetics of nonlocal operators, e.g. by means of hierarchical matrices. We consider (i) a boundary integral equation arising from the polarizable continuum model in quantum chemistry as a classical example for a nonlocal operator equation, (ii) a parabolic problem for the fractional Laplacian, and (iii) the fast numerical simulation of Gaussian random fields as an important example from computational uncertainty quantification. C D is defined as D C, while we write C ∼ D, iff C D and C D
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