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.

中文翻译：

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分层张量表示中对数正态系数的自适应随机 Galerkin FEM

众所周知，用于非仿射系数表示的随机伽辽金方法会导致理论和数值方面的重大困难。在这项工作中，导出了一种用于线性参数偏微分方程的自适应伽辽金有限元方法，其对数正态系数离散在 Hermite 混沌多项式中。它采用适应问题的函数空间来确保变分公式的可解性。通过使用分层张量表示，参数运算符固有的高计算复杂性变得易于处理。为此，推导出了对数正态系数的新张量序列格式并进行了数值验证。核心新颖性是可靠的基于残差的后验误差估计器的推导。这可以看作是随机伽辽金方法的独特之处。它允许使用自适应算法来引导物理网格和各向异性维纳混沌多项式的细化。为了使误差估计器的评估变得可行，开发了一种数值有效的张量格式离散化。具有无界对数正态系数场的基准示例说明了所提出的 Galerkin 离散化和完全自适应算法的性能。