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Title: Algorithms for propagating uncertainty across heterogeneous domains

Abstract

We address an important research area in stochastic multi-scale modeling, namely the propagation of uncertainty across heterogeneous domains characterized by partially correlated processes with vastly different correlation lengths. This class of problems arise very often when computing stochastic PDEs and particle models with stochastic/stochastic domain interaction but also with stochastic/deterministic coupling. The domains may be fully embedded, adjacent or partially overlapping. The fundamental open question we address is the construction of proper transmission boundary conditions that preserve global statistical properties of the solution across different subdomains. Often, the codes that model different parts of the domains are black-box and hence a domain decomposition technique is required. No rigorous theory or even effective empirical algorithms have yet been developed for this purpose, although interfaces defined in terms of functionals of random fields (e.g., multi-point cumulants) can overcome the computationally prohibitive problem of preserving sample-path continuity across domains. The key idea of the different methods we propose relies on combining local reduced-order representations of random fields with multi-level domain decomposition. Specifically, we propose two new algorithms: The first one enforces the continuity of the conditional mean and variance of the solution across adjacent subdomains by using Schwarz iterations. The second algorithm ismore » based on PDE-constrained multi-objective optimization, and it allows us to set more general interface conditions. The effectiveness of these new algorithms is demonstrated in numerical examples involving elliptic problems with random diffusion coefficients, stochastically advected scalar fields, and nonlinear advection-reaction problems with random reaction rates.« less

Authors:
; ; ;
Publication Date:
Research Org.:
Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1255408
Report Number(s):
PNNL-SA-111664
Journal ID: ISSN 1064-8275; KJ0401000
DOE Contract Number:  
AC05-76RL01830
Resource Type:
Journal Article
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 37; Journal Issue: 6; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English

Citation Formats

Cho, Heyrim, Yang, Xiu, Venturi, D., and Karniadakis, George E. Algorithms for propagating uncertainty across heterogeneous domains. United States: N. p., 2015. Web. doi:10.1137/140992060.
Cho, Heyrim, Yang, Xiu, Venturi, D., & Karniadakis, George E. Algorithms for propagating uncertainty across heterogeneous domains. United States. doi:10.1137/140992060.
Cho, Heyrim, Yang, Xiu, Venturi, D., and Karniadakis, George E. Wed . "Algorithms for propagating uncertainty across heterogeneous domains". United States. doi:10.1137/140992060.
@article{osti_1255408,
title = {Algorithms for propagating uncertainty across heterogeneous domains},
author = {Cho, Heyrim and Yang, Xiu and Venturi, D. and Karniadakis, George E.},
abstractNote = {We address an important research area in stochastic multi-scale modeling, namely the propagation of uncertainty across heterogeneous domains characterized by partially correlated processes with vastly different correlation lengths. This class of problems arise very often when computing stochastic PDEs and particle models with stochastic/stochastic domain interaction but also with stochastic/deterministic coupling. The domains may be fully embedded, adjacent or partially overlapping. The fundamental open question we address is the construction of proper transmission boundary conditions that preserve global statistical properties of the solution across different subdomains. Often, the codes that model different parts of the domains are black-box and hence a domain decomposition technique is required. No rigorous theory or even effective empirical algorithms have yet been developed for this purpose, although interfaces defined in terms of functionals of random fields (e.g., multi-point cumulants) can overcome the computationally prohibitive problem of preserving sample-path continuity across domains. The key idea of the different methods we propose relies on combining local reduced-order representations of random fields with multi-level domain decomposition. Specifically, we propose two new algorithms: The first one enforces the continuity of the conditional mean and variance of the solution across adjacent subdomains by using Schwarz iterations. The second algorithm is based on PDE-constrained multi-objective optimization, and it allows us to set more general interface conditions. The effectiveness of these new algorithms is demonstrated in numerical examples involving elliptic problems with random diffusion coefficients, stochastically advected scalar fields, and nonlinear advection-reaction problems with random reaction rates.},
doi = {10.1137/140992060},
journal = {SIAM Journal on Scientific Computing},
issn = {1064-8275},
number = 6,
volume = 37,
place = {United States},
year = {2015},
month = {12}
}