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.
Highlights
In the thriving field of uncertainty quantification (UQ), efficient numerical methods for the approximate solution of random PDEs have been a topic of vivid research
Two important properties are the length of the expansion of random fields, which often directly translates to the number of independent random variables describing the variability in the model, and the type of dependence on these random variables
Remark 5.4 Note that an L2-integration of the residual, which is an element of the dual space Vθ∗, is possible since the solution consists of finite element functions
Summary
In the thriving field of uncertainty quantification (UQ), efficient numerical methods for the approximate solution of random PDEs have been a topic of vivid research. While popular sample-based Monte Carlo methods obtain dimension-independent convergence rates, these are rather low despite often encountered higher regularity of the parameter dependence Such methods can only be used to evaluate functionals of the solution (QoIs = quantities of interest) and an a posteriori error control usually is not feasible reliably. Collocation methods with pointwise evaluations in the parameter space are usually constructed either based on some a priori knowledge or by means of an iterative refinement algorithm which takes into account the hierarchical surplus on possible new discretization levels While these approaches work reasonably well, methods for a reliable error control do not seem immediate since the approximation relies only on interpolation properties. For the affine case and under certain assumptions, first ideas were recently presented in [28]
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