Journal article

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 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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