Method of Green’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. Here, 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.
- Authors:
-
[1];
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Middle Tennessee State Univ., Murfreesboro, TN (United States)
- Publication Date:
- Research Org.:
- Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
- OSTI Identifier:
- 1665995
- Grant/Contract Number:
- AC05-00OR22725; DMS1620280
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Scientific Computing
- Additional Journal Information:
- Journal Volume: 84; Journal Issue: 3; Journal ID: ISSN 0885-7474
- Publisher:
- Springer
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Green's function; Green's potential; boundary integral equations; random boundaries; multilevel Monte Carlo
Citation Formats
Reshniak, Viktor, and Melnikov, Yuri. Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures. United States: N. p., 2020.
Web. doi:10.1007/s10915-020-01296-9.
Reshniak, Viktor, & Melnikov, Yuri. Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures. United States. https://doi.org/10.1007/s10915-020-01296-9
Reshniak, Viktor, and Melnikov, Yuri. Wed .
"Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures". United States. https://doi.org/10.1007/s10915-020-01296-9. https://www.osti.gov/servlets/purl/1665995.
@article{osti_1665995,
title = {Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures},
author = {Reshniak, Viktor and Melnikov, Yuri},
abstractNote = {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. Here, 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.},
doi = {10.1007/s10915-020-01296-9},
journal = {Journal of Scientific Computing},
number = 3,
volume = 84,
place = {United States},
year = {2020},
month = {8}
}
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