Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures

Viktor Reshniak; Yuri Melnikov

doi:10.1007/s10915-020-01296-9

Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures

Viktor Reshniak, Yuri Melnikov

Data Analysis and Machine Learning

Research output: Contribution to journal › Article › peer-review

Abstract

Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient conventional schemes that rely on discretization of the whole domain. In this paper, we study elliptic PDEs in domains with boundaries comprised of a deterministic part and random apertures, and apply the method of modified potentials with Green’s kernels defined on the deterministic part of the domain. This approach allows to reduce the dimension of the original differential problem by reformulating it as a boundary integral equation posed on the random apertures only. The multilevel Monte Carlo method is then applied to this modified integral equation and its optimal ϵ^{- 2} asymptotical complexity is shown. Finally, we provide the qualitative analysis of the proposed technique and support it with numerical results.

Original language	English
Article number	42
Journal	Journal of Scientific Computing
Volume	84
Issue number	3
DOIs	https://doi.org/10.1007/s10915-020-01296-9
State	Published - Sep 1 2020

Funding

This material is based upon work supported in part by: the U.S. Department of Energy, Office of Science, Early Career Research Program under Award Number ERKJ314; U.S. Department of Energy, Office of Advanced Scientific Computing Research under Award Numbers ERKJ331 and ERKJ345; the National Science Foundation, Division of Mathematical Sciences, Computational Mathematics program under Contract Number DMS1620280; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC., for the U.S. Department of Energy under Contract DE-AC05-00OR22725. This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan ( http://energy.gov/downloads/doe-public-access-plan ).

Funders	Funder number
National Science Foundation
U.S. Department of Energy
Division of Mathematical Sciences	DMS1620280
Office of Science	ERKJ314
Advanced Scientific Computing Research	ERKJ345, ERKJ331
Laboratory Directed Research and Development	DE-AC05-00OR22725

Keywords

Boundary integral equations
Green’s function
Green’s potential
Multilevel Monte Carlo
Random boundaries

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10.1007/s10915-020-01296-9

Cite this

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title = "Method of Green{\textquoteright}s Potentials for Elliptic PDEs in Domains with Random Apertures",

abstract = "Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient conventional schemes that rely on discretization of the whole domain. In this paper, we study elliptic PDEs in domains with boundaries comprised of a deterministic part and random apertures, and apply the method of modified potentials with Green{\textquoteright}s kernels defined on the deterministic part of the domain. This approach allows to reduce the dimension of the original differential problem by reformulating it as a boundary integral equation posed on the random apertures only. The multilevel Monte Carlo method is then applied to this modified integral equation and its optimal ϵ- 2 asymptotical complexity is shown. Finally, we provide the qualitative analysis of the proposed technique and support it with numerical results.",

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Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures

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