Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs*

By Christoph Schwab, Claude Jeffrey Gittelson. Acta Numerica, 2011

301 citations recorded by the source corpus.

Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of best N-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficients.

Subfields: Environmental Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/79d8bd5a0d5458ca4ef2ea29f324ab9c8b2922cb

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Citations in the corpus

Hyperbolic Cross Approximation (2016, methodology): “See to [62, 159, 296] for surveys and bibliography on different aspects in approximation and numerical methods for the problem (10.”
The geometry of algorithms using hierarchical tensors (2013, background): “A prototypical example is the Schrödinger equation in quantum dynamics [1], but there is an abundance of other high-dimensional problems like the ones in [2] [3] [4] [5].”
Efficient Analysis of High Dimensional Data in Tensor Formats (2012, unknown): “we require that both κ(x, ω) and 1/κ(x, ω) are essentially bounded (have finite L∞ norm) [2, 37, 34, 44]: κ(x, ω) > 0 a.”
Low-rank tensor completion by Riemannian optimization (2014, background): “, [24] for a detailed discussion of this procedure.”
Approximation of high-dimensional parametric PDEs * (2015, result): “uation of nwidths are from [25]. These results establish for the ﬁrst time convergence rates immune to the curse of dimensionality, in the sense that they hold with inﬁnitely many variables, see also [44] for a survey dealing in particular with these issues. In a similar inﬁnite dimensional framewor”