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A domain decomposition method for stochastic elliptic differential equations

Banff International Research Station for Mathematical Innovation and Discovery

In this talk I will discuss the use of a Domain Decomposition method to reduced the computational complexity of classical problems arising in Uncertainty Quantification and stochastic Partial Differential equations. The first problem concerns the determination of the Karhunen-Loeve decomposition of a stochastic process given its covariance function. We propose to solve independently the decomposition problem over a set of subdomains, each with low complexity cost, and subsequently assemble a reduced problem to determined the global problem solution. We propose error estimates to control the resulting approximation error. Second, these ideas are extended to construct an efficient sampling approach for elliptic problems with stochastic coefficients expanded in a KL form. Here, we rely on the resolution of low complexity local stochastic elliptic problems to exhibit contributions to the condensed stochastic problem for the unknown boundary values at the internal subdomain boundaries. By relying intensively on local resolutions, that can be performed independently, the proposed approaches are naturally suited to parallel implementation and we will provide scalability results.

Mathematics; Probability theory and stochastic processes; Numerical analysis; Industrial mathematics / modelling and simulation

video/mp4

Author affiliation: Centre National de la Recherche Scientifique

2018-04-10

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/65246

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/