WIAS Preprint No. 2515, (2018)

Adaptive stochastic Galerkin FEM for log-normal coefficients in hierarchical tensor representations



Authors

  • Eigel, Martin
  • Marschall, Manuel
    ORCID: 0000-0003-0648-1936
  • Pfeffer, Max
  • Schneider, Reinhold

2010 Mathematics Subject Classification

  • 35R60 47B80 60H35 65C20 65N12 65N22 65J10

Keywords

  • Partial differential equations with random coefficients, tensor representation, tensor train, uncertainty quantification, stochastic finite element methods, log-normal, adaptive methods, ALS, low-rank, reduced basis methods

DOI

10.20347/WIAS.PREPRINT.2515

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 log-normal 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.

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