Parallel Domain Decomposition Strategies for Stochastic Elliptic Equations. Part A: Local Karhunen--Loève Representations

Abstract

Abstract. This work presents a method to efficiently determine the dominant Karhunen-Loève (KL) modes of a random process with known covariance function. The truncated KL expansion is one of the most common techniques for the approximation of random processes, primarily because it is an optimal representation, in the mean squared error sense, with respect to the number of random variables in the representation. However, finding the KL expansion involves solving integral problems, which tends to be computationally demanding. This work addresses this issue by means of a work-subdivision strategy based on a domain decomposition approach, enabling the efficient computation of a possibly large number of dominant KL modes. Specifically, the computational domain is partitioned into smaller nonoverlapping subdomains, over which independent local KL decompositions are performed to generate local bases which are subsequently used to discretize the global modes over the entire domain. The latter are determined by means of a Galerkin projection. The procedure leads to the resolution of a reduced Galerkin problem, whose size is not related to the dimension of the underlying discretization space but is actually determined by the desired accuracy and the number of subdomains. It can also be easily implemented in parallel. Extensive numerical tests are used to validate the methodology and assess its serial and parallel performance. The resulting expansion is exploited in Part B to accelerate the solution of the stochastic partial differential equations using a Monte Carlo approach.

Parallel Domain Decomposition Strategies for Stochastic Elliptic Equations Part B: Accelerated Monte Carlo Sampling with Local PC Expansions

“…This results in a condensed problem for the nodal values at the subdomains' interfaces. Given this discretization, the preprocessing stage starts by breaking the condensed problem into individual contributions from each subdomain and computing local KL expansions over each subdomain (as described in [10]). Then, using the local KL expansion and taking advantage of the reduced stochastic dimension of the local problems, PC expansions of the local contributions to the condensed problem are constructed.…”

“…This paper (Part B) and its prequel (Part A) focus on the first two steps. In Part A [10], we discussed a domain decomposition strategy to approximate random fields (input uncertainty) using local reduced bases and local coordinates. Now (in Part B), the structure of local representations is exploited to accelerate the Monte Carlo (MC) sampling of the solution.…”

Fast sampling of parameterised Gaussian random fields

“…We refer to [27,35,36,43,61,67] for applications and reviews of reduced basis surrogates for parameterised eigenproblems with PDEs. For non-parametric KL eigenproblems we mention that reduced basis methods have been combined with domain decomposition ideas [14]. In this situation we need to solve several low-dimensional eigenproblems on the subdomains in the physical space.…”

A Domain Decomposition Model Reduction Method for Linear Convection-Diffusion Equations with Random Coefficients

“…In [18], Hou, et al combine the DD strategy with multi-scale finite element methods to solve elliptic PDEs with high-contrast random medium. In [11], Contreras, et al developed a new approach to capture the correlation structure of local random variables in different sub-domains, without performing expensive global KL expansion; and such strategy was incorporated into the DD framework to solve stochastic elliptic PDEs in [12]. In [38], Tipireddy, et al employed the local KL expansion to reduce local dimensions, and then combined basis adaptation and Hilbert-space KL expansion to approximate local solutions in the sub-domains.…”