Basis adaptation and domain decomposition for steady partial differential equations with random coefficients
Abstract
In this paper, we present a novel approach for solving steadystate stochastic partial differential equations (PDEs) with highdimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each subdomain. The basis adaptation is used to address the curse of dimensionality by constructing an accurate lowdimensional representation of the stochastic PDE solution (probability density function and/or its leading statistical moments) in each subdomain. Restricting the basis adaptation to a specific subdomain affords finding a locally accurate solution. Then, the solutions from all of the subdomains are stitched together to provide a global solution. We support our construction with numerical experiments for a steadystate diffusion equation with a random spatially dependent coefficient. Lastly, our results show that highly accurate global solutions can be obtained with significantly reduced computational costs.
 Authors:

 Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
 Publication Date:
 Research Org.:
 Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 OSTI Identifier:
 1379948
 Alternate Identifier(s):
 OSTI ID: 1395272; OSTI ID: 1549945
 Report Number(s):
 PNNLSA115134
Journal ID: ISSN 00219991; PII: S0021999117306484
 Grant/Contract Number:
 AC0576RL01830
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 351; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; basis adaptation; dimension reduction; domain decomposition; polynomial chaos; uncertainty quantification
Citation Formats
Tipireddy, R., Stinis, P., and Tartakovsky, A. M. Basis adaptation and domain decomposition for steady partial differential equations with random coefficients. United States: N. p., 2017.
Web. doi:10.1016/j.jcp.2017.08.067.
Tipireddy, R., Stinis, P., & Tartakovsky, A. M. Basis adaptation and domain decomposition for steady partial differential equations with random coefficients. United States. doi:10.1016/j.jcp.2017.08.067.
Tipireddy, R., Stinis, P., and Tartakovsky, A. M. Mon .
"Basis adaptation and domain decomposition for steady partial differential equations with random coefficients". United States. doi:10.1016/j.jcp.2017.08.067. https://www.osti.gov/servlets/purl/1379948.
@article{osti_1379948,
title = {Basis adaptation and domain decomposition for steady partial differential equations with random coefficients},
author = {Tipireddy, R. and Stinis, P. and Tartakovsky, A. M.},
abstractNote = {In this paper, we present a novel approach for solving steadystate stochastic partial differential equations (PDEs) with highdimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each subdomain. The basis adaptation is used to address the curse of dimensionality by constructing an accurate lowdimensional representation of the stochastic PDE solution (probability density function and/or its leading statistical moments) in each subdomain. Restricting the basis adaptation to a specific subdomain affords finding a locally accurate solution. Then, the solutions from all of the subdomains are stitched together to provide a global solution. We support our construction with numerical experiments for a steadystate diffusion equation with a random spatially dependent coefficient. Lastly, our results show that highly accurate global solutions can be obtained with significantly reduced computational costs.},
doi = {10.1016/j.jcp.2017.08.067},
journal = {Journal of Computational Physics},
number = ,
volume = 351,
place = {United States},
year = {2017},
month = {9}
}