Non-intrusive Tensor Reconstruction for High-Dimensional Random PDEs

Martin Eigel · Johannes Neumann · Reinhold Schneider · Sebastian Wolf

14 citations0/0 files storedComputational Methods in Applied Mathematics

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Probabilistic and Robust Engineering Design Model Reduction and Neural Networks Tensor decomposition and applications

Abstract

Abstract This paper examines a completely non-intrusive, sample-based method for the computation of functional low-rank solutions of high-dimensional parametric random PDEs, which have become an area of intensive research in Uncertainty Quantification (UQ). In order to obtain a generalized polynomial chaos representation of the approximate stochastic solution, a novel black-box rank-adapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to (Quasi-)Monte Carlo sampling.

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References

1991Stochastic Finite Elements: A Spectral Approach 2007A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2008A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data 2014An Introduction to Computational Stochastic PDEs 2010A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2006Karhunen–Loève approximation of random fields by generalized fast multipole methods 2004Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation 2011Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs

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