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Title: Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance

Abstract

We introduce a general framework for approximating parabolic Stochastic Partial Differential Equations (SPDEs) based on fluctuation-dissipation balance. Using this approach we formulate Stochastic Discontinuous Galerkin Methods (SDGM). We show how methods with linear-time computational complexity can be developed for handling domains with general geometry and generating stochastic terms, handling both Dirichlet and Neumann boundary conditions. We demonstrate our approach on example systems and contrast with alternative approaches using direct stochastic discretizations based on random fluxes. We show how our Fluctuation-Dissipation Discretizations (FDD) framework allows us to compensate for discrepancies in dissipative properties between the continuous operators and their discretized versions. This allows us to handle general heterogeneous discretizations, accurately capturing statistical relations. Here, our FDD framework provides a general approach for formulating SDGM discretizations and other numerical methods for robust approximation of stochastic differential equations.

Authors:
; ;
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of California, Santa Barbara, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
OSTI Identifier:
1562583
Alternate Identifier(s):
OSTI ID: 1574443; OSTI ID: 1633911
Report Number(s):
SAND-2019-12136J
Journal ID: ISSN 2590-0374; S2590037419300688; 100068; PII: S2590037419300688
Grant/Contract Number:  
SC0009254; AC04-94AL85000
Resource Type:
Published Article
Journal Name:
Results in Applied Mathematics
Additional Journal Information:
Journal Name: Results in Applied Mathematics Journal Volume: 4 Journal Issue: C; Journal ID: ISSN 2590-0374
Publisher:
Elsevier
Country of Publication:
Netherlands
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Stochastic partial differential equations; Fluctuation-dissipation balance; Discontinuous Galerkin methods

Citation Formats

Pazner, W., Trask, N., and Atzberger, P. J. Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance. Netherlands: N. p., 2019. Web. doi:10.1016/j.rinam.2019.100068.
Pazner, W., Trask, N., & Atzberger, P. J. Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance. Netherlands. doi:10.1016/j.rinam.2019.100068.
Pazner, W., Trask, N., and Atzberger, P. J. Sun . "Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance". Netherlands. doi:10.1016/j.rinam.2019.100068.
@article{osti_1562583,
title = {Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance},
author = {Pazner, W. and Trask, N. and Atzberger, P. J.},
abstractNote = {We introduce a general framework for approximating parabolic Stochastic Partial Differential Equations (SPDEs) based on fluctuation-dissipation balance. Using this approach we formulate Stochastic Discontinuous Galerkin Methods (SDGM). We show how methods with linear-time computational complexity can be developed for handling domains with general geometry and generating stochastic terms, handling both Dirichlet and Neumann boundary conditions. We demonstrate our approach on example systems and contrast with alternative approaches using direct stochastic discretizations based on random fluxes. We show how our Fluctuation-Dissipation Discretizations (FDD) framework allows us to compensate for discrepancies in dissipative properties between the continuous operators and their discretized versions. This allows us to handle general heterogeneous discretizations, accurately capturing statistical relations. Here, our FDD framework provides a general approach for formulating SDGM discretizations and other numerical methods for robust approximation of stochastic differential equations.},
doi = {10.1016/j.rinam.2019.100068},
journal = {Results in Applied Mathematics},
number = C,
volume = 4,
place = {Netherlands},
year = {2019},
month = {12}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1016/j.rinam.2019.100068

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