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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
47/2018

Adaptive Stochastic Galerkin FEM for Lognormal Coefficients in Hierarchical Tensor Representations

Martin Eigel, Manuel Marschall, Max Pfeffer and Reinhold Schneider

Abstract

Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residual-based a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm.

Received:
Jul 11, 2018
Published:
Jul 17, 2018
MSC Codes:
35R60, 47B80, 60H35, 65C20, 65N12, 65N22, 65J10

Related publications

inJournal
2020 Journal Open Access
Martin Eigel, Manuel Marschall, Max Pfeffer and Reinhold Schneider

Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations

In: Numerische Mathematik, 145 (2020) 3, pp. 655-692