A local hybrid surrogate‐based finite element tearing interconnecting dual‐primal method for nonsmooth random partial differential equations

Abstract

A domain decomposition approach for high-dimensional random partial differential equations exploiting the localization of random parameters is presented. To obtain high efficiency, surrogate models in multielement representations in the parameter space are constructed locally when possible. The method makes use of a stochastic Galerkin finite element tearing interconnecting dual-primal formulation of the underlying problem with localized representations of involved input random fields. Each local parameter space associated to a subdomain is explored by a subdivision into regions where either the parametric surrogate accuracy can be trusted or where instead one has to resort to Monte Carlo. A heuristic adaptive algorithm carries out a problem-dependent hp-refinement in a stochastic multielement sense, anisotropically enlarging the trusted surrogate region as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration for the surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, for example, in a multiphysics setting, or when the Karhunen-Loève expansion of a random field can be localized. The efficiency of the proposed hybrid technique is assessed with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and nontrusted sampling regions. K E Y W O R D S domain decomposition, FETI, nonsmooth elliptic partial differential equations, partial differential equations with random coefficients, stochastic finite element method, uncertainty quantification 1 INTRODUCTION In uncertainty quantification (UQ), numerical methods typically are either based on pointwise sampling, which is applicable to quite general problems but rather inefficient, or they rely on (an often analytic) smoothness of the parameter This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Adaptive experimental design for multi‐fidelity surrogate modeling of multi‐disciplinary systems

“…The single‐fidelity version of the method we propose, and the preceding works we cited, possess similarities to domain decomposition 35‐38 and localized model reduction (LMR) 39‐42 . These methods efficiently solve partial differential equations (PDEs) by solving independent local problems on subdomains and computing a global solution via an appropriate coupling of the subdomains; LMR is domain decomposition technique that uses a localized reduced basis in each subdomain.…”