A fast direct solver for nonlocal operators in wavelet coordinates

Harbrecht, Helmut
Departement Mathematik und Informatik, Universität Basel, Switzerland

Multerer, Michael
Istituto di scienza computazionale (ICS), Facoltà di scienze informatiche, Università della Svizzera italiana, Svizzera
Published in:
 Journal of computational physics.  2021, vol. 428, p. 15 p
English
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 fillin 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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https://susi.usi.ch/usi/documents/319364
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