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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Approximate approximations: recent developments in the computation of high dimensional potentials

Authors: F. Lanzara and G. Schmidt
Original publication: Algebra i Analiz, tom 31 (2019), nomer 2.
Journal: St. Petersburg Math. J. 31 (2020), 355-370
MSC (2010): Primary 65D32; Secondary 65-05
Published electronically: February 4, 2020
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Abstract: The paper is devoted to a fast method of an arbitrary high order for approximating volume potentials that is successful also in the high dimensional case. The cubature formulas have been obtained by using the basis functions introduced in the theory of approximate approximations. As basis functions, we choose products of Gaussians and special polynomials, for which the action of integral operators can be written as one-dimensional integrals with a separable integrand, i.e., a product of functions depending only on one of the variables. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a separable representation of the integral operator. Since only one-dimensional operations are used, the resulting method is efficient also in the high dimensional case. We show how this new approach can be applied to the cubature of polyharmonic potentials, to potentials of elliptic differential operators acting on densities on hyper-rectangular domains, and to parabolic problems.

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Additional Information

F. Lanzara
Affiliation: Department of Mathematics, Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy

G. Schmidt
Affiliation: Lichtenberger Str. 12, 10178 Berlin, Germany

Keywords: Higher dimensions, separated representation, high order approximation, convection-diffusion potential, heat potential, polyharmonic potential
Received by editor(s): October 31, 2018
Published electronically: February 4, 2020
Dedicated: Dedicated to Vladimir Maz’ya on the occasion of his $80$th birthday
Article copyright: © Copyright 2020 American Mathematical Society