Stochastic basis adaptation and spatial domain decomposition for partial differential equations with random coefficients

Tipireddy, Ramakrishna; Stinis, Panagiotis; Tartakovsky, Alexandre M.

doi:10.1137/16M1097134

Stochastic basis adaptation and spatial domain decomposition for partial differential equations with random coefficients

Journal Article · Tue Mar 13 00:00:00 EDT 2018 · SIAM/ASA Journal on Uncertainty Quantification

DOI:https://doi.org/10.1137/16M1097134· OSTI ID:1818972

^[1]; Stinis, Panagiotis ^[1]; Tartakovsky, Alexandre M. ^[1]

BATTELLE (PACIFIC NW LAB)

We present a novel approach for high-dimensional stochastic partial differential equations that decomposes the computational domain into smaller disjoint subdomains and reduces the random dimension in each subdomain to save computational costs. The proposed approach combines spatial domain decomposition and stochastic basis adaptation in each subdomain. We adapt the stochastic basis in each subdomain such that the local solution has a low-dimensional random space representation. To compute the stochastic solution in each subdomain, we use the Neumann-Neumann algorithm, where we first estimate the interface solution and then evaluate the interior solution in each subdomain using the interface solution as a boundary condition. The interior solutions in each subdomain are computed independently of each other, which reduces the operation count from $$\approx O(N^\alpha)$$ to $$\approx O(M^\alpha),$$ where $$N$$ is the total number of degrees of freedom, $$M$$ is the number of degrees of freedom in each subdomain, and the exponent $$\alpha>1$$ depends on the uncertainty quantification method used. In addition, the localized nature of solutions makes the proposed approach highly parallelizable. We illustrate the accuracy and efficiency of the approach for a steady-state diffusion equation with a random space-dependent diffusion coefficient.

Research Organization:: Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)

Sponsoring Organization:: USDOE

DOE Contract Number:: AC05-76RL01830

OSTI ID:: 1818972

Report Number(s):: PNNL-SA-121137

Journal Information:: SIAM/ASA Journal on Uncertainty Quantification, Journal Name: SIAM/ASA Journal on Uncertainty Quantification Journal Issue: 1 Vol. 6

Country of Publication:: United States

Language:: English

References (24)

Stochastic model reduction for chaos representations Doostan, Alireza; Ghanem, Roger G.; Red-Horse, John Computer Methods in Applied Mechanics and Engineering, Vol. 196, Issue 37-40 https://doi.org/10.1016/j.cma.2006.10.047	journal	August 2007
A Transformed Pressure Head-Based Approach to Solve Richards' Equation for Variably Saturated Soils Pan, Lehua; Wierenga, Peter J. Water Resources Research, Vol. 31, Issue 4 https://doi.org/10.1029/94WR03291	journal	April 1995
Addressing the curse of dimensionality in SSFEM using the dependence of eigenvalues in KL expansion on domain size Pranesh, Srikara; Ghosh, Debraj Computer Methods in Applied Mechanics and Engineering, Vol. 311 https://doi.org/10.1016/j.cma.2016.08.023	journal	November 2016
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data Nobile, F.; Tempone, R.; Webster, C. G. SIAM Journal on Numerical Analysis, Vol. 46, Issue 5 https://doi.org/10.1137/060663660	journal	January 2008
Minimal dynamical systems from PDEs using Sobolev eigenfunctions Kirby, Michael Physica D: Nonlinear Phenomena, Vol. 57, Issue 3-4 https://doi.org/10.1016/0167-2789(92)90014-E	journal	August 1992
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data Babuška, Ivo; Nobile, Fabio; Tempone, Raúl SIAM Review, Vol. 52, Issue 2 https://doi.org/10.1137/100786356	journal	January 2010
A Two-Scale Nonperturbative Approach to Uncertainty Analysis of Diffusion in Random Composites Xiu, Dongbin; Tartakovsky, Daniel M. Multiscale Modeling & Simulation, Vol. 2, Issue 4 https://doi.org/10.1137/03060268X	journal	January 2004
Numerical Studies of Three-dimensional Stochastic Darcy’s Equation and Stochastic Advection-Diffusion-Dispersion Equation Lin, G.; Tartakovsky, A. M. Journal of Scientific Computing, Vol. 43, Issue 1 https://doi.org/10.1007/s10915-010-9346-5	journal	January 2010
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations Xiu, Dongbin; Karniadakis, George Em SIAM Journal on Scientific Computing, Vol. 24, Issue 2 https://doi.org/10.1137/S1064827501387826	journal	January 2002
Local Polynomial Chaos Expansion for Linear Differential Equations with High Dimensional Random Inputs Chen, Yi; Jakeman, John; Gittelson, Claude SIAM Journal on Scientific Computing, Vol. 37, Issue 1 https://doi.org/10.1137/140970100	journal	January 2015
The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals Cameron, R. H.; Martin, W. T. The Annals of Mathematics, Vol. 48, Issue 2 https://doi.org/10.2307/1969178	journal	April 1947
Basis adaptation in homogeneous chaos spaces Tipireddy, Ramakrishna; Ghanem, Roger Journal of Computational Physics, Vol. 259 https://doi.org/10.1016/j.jcp.2013.12.009	journal	February 2014
Hydrogeophysical Approach for Identification of Layered Structures of the Vadose Zone from Electrical Resistivity Data Tartakovsky, Alexandre M.; Bolster, Diogo; Tartakovsky, Daniel M. Vadose Zone Journal, Vol. 7, Issue 4 https://doi.org/10.2136/vzj2008.0009	journal	November 2008
Exact PDF equations and closure approximations for advective-reactive transport Venturi, D.; Tartakovsky, D. M.; Tartakovsky, A. M. Journal of Computational Physics, Vol. 243 https://doi.org/10.1016/j.jcp.2013.03.001	journal	June 2013
On solving elliptic stochastic partial differential equations Babuška, Ivo; Chatzipantelidis, Panagiotis Computer Methods in Applied Mechanics and Engineering, Vol. 191, Issue 37-38 https://doi.org/10.1016/S0045-7825(02)00354-7	journal	August 2002
Evaluation of Proper Orthogonal Decomposition--Based Decomposition Techniques Applied to Parameter-Dependent Nonturbulent Flows Christensen., Erik Adler; Brøns, Morten; Sørensen, Jens Nørkær SIAM Journal on Scientific Computing, Vol. 21, Issue 4 https://doi.org/10.1137/S1064827598333181	journal	January 1999
An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media Lin, G.; Tartakovsky, A. M. Advances in Water Resources, Vol. 32, Issue 5 https://doi.org/10.1016/j.advwatres.2008.09.003	journal	May 2009
Uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids Lin, G.; Tartakovsky, A. M.; Tartakovsky, D. M. Journal of Computational Physics, Vol. 229, Issue 19 https://doi.org/10.1016/j.jcp.2010.05.036	journal	September 2010
The Nonlinear Gaussian Spectrum of Log-Normal Stochastic Processes and Variables Ghanem, R. Journal of Applied Mechanics, Vol. 66, Issue 4 https://doi.org/10.1115/1.2791806	journal	December 1999
Hilbert-space Karhunen–Loève transform with application to image analysis Levy, A.; Rubinstein, J. Journal of the Optical Society of America A, Vol. 16, Issue 1 https://doi.org/10.1364/JOSAA.16.000028	journal	January 1999
A non-adapted sparse approximation of PDEs with stochastic inputs Doostan, Alireza; Owhadi, Houman Journal of Computational Physics, Vol. 230, Issue 8 https://doi.org/10.1016/j.jcp.2011.01.002	journal	April 2011
Smoothed functional principal components analysis by choice of norm Silverman, Bernard W. The Annals of Statistics, Vol. 24, Issue 1 https://doi.org/10.1214/aos/1033066196	journal	February 1996
The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows Berkooz, G.; Holmes, P.; Lumley, J. L. Annual Review of Fluid Mechanics, Vol. 25, Issue 1 https://doi.org/10.1146/annurev.fl.25.010193.002543	journal	January 1993
A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations Nouy, Anthony Computer Methods in Applied Mechanics and Engineering, Vol. 196, Issue 45-48 https://doi.org/10.1016/j.cma.2007.05.016	journal	September 2007

Similar Records

Basis adaptation and domain decomposition for steady-state partial differential equations with random coefficients

Journal Article · Thu Dec 14 23:00:00 EST 2017 · Journal of Computational Physics · OSTI ID:22701644

Basis adaptation and domain decomposition for steady partial differential equations with random coefficients

Journal Article · Mon Sep 04 00:00:00 EDT 2017 · Journal of Computational Physics · OSTI ID:1379948

Stochastic multiscale flux basis for Stokes-Darcy flows

Journal Article · Thu Oct 17 00:00:00 EDT 2019 · Journal of Computational Physics · OSTI ID:1800166

Related Subjects

uncertainty quantification
basis adaptation

Stochastic basis adaptation and spatial domain decomposition for partial differential equations with random coefficients

Citation Formats

References (24)

Similar Records

Related Subjects