A non-adapted sparse approximation of PDEs with stochastic inputs

By Alireza Doostan, Houman Owhadi. Journal of Computational Physics, 2010

465 citations recorded by the source corpus.

We propose a method for the approximation of solutions of PDEs with stochastic coefficients based on the direct, i.e., non-adapted, sampling of solutions. This sampling can be done by using any legacy code for the deterministic problem as a black box. The method converges in probability (with probabilistic error bounds) as a consequence of sparsity and a concentration of measure phenomenon on the empirical correlation between samples. We show that the method is well suited for truly high-dimensional problems.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/dc24237ca830da4249ffd17e37b86799753921f4

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Citations in the corpus

Computational Aspects of Stochastic Collocation with Multifidelity Models (2014, unknown): “, [1, 2, 3, 4, 5, 7, 8, 12, 15, 17, 19, 21].”
Approximation of high-dimensional parametric PDEs * (2015, methodology): “Other non-intrusive methods are based on least-square regression as discussed in [16, 34, 35, 69], or on pseudo-spectral approximation as discussed in [91, 26].”
Polynomial chaos expansions for dependent random variables (2019, methodology): “Stochastic collocation methods include pseudospectral projection [12, 14], sparse grid interpolation [4, 8], least orthogonal interpolation [43], least squares [40, 59], compressive sensing [17, 65, 29] and low-rank tensor decompositions [10, 18, 26].”
Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression (2014, methodology): “Such a sampling, dubbed here standard sampling, is commonly used in PC regression, [7, 2, 27, 28].”
Stochastic Collocation Algorithms Using l1-Minimization for Bayesian Solution of Inverse Problems (2015, background): “In the recent work [12], the idea of compressive sensing (CS) has been extended to stochasD ow nl oa de d 06 /1 0/ 15 to 1 32 .”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, background): “The relevance of the sparse PC approximation of PDEs with random inputs has been discussed in several work including [11, 20, 27].”
Least squares polynomial chaos expansion: A review of sampling strategies (2017, methodology): “Sampling-based techniques such as least squares approximation (LSA) [51, 44], pseudo-spectral collocation [72, 21], Monte Carlo sampling [72], and compressive sampling [27, 46] have been employed in the literature to identify the PC coefﬁcients.”
Polynomial-Chaos-based Kriging (2015, unknown): “Further developments which combine spectral expansions and compressive sensing ideas have lead to so-called sparse polynomial chaos expansions Blatman and Sudret (2008, 2010a,b); Doostan and Owhadi (2011); Blatman and Sudret (2011); Doostan et al. (2013); Jakeman et al. (2014).”
Surrogate accelerated sampling of reservoir models with complex structures using sparse polynomial chaos expansion (2015, methodology): “However, as the number of dimension increases, a major drawback of the OMP method lies in determining an appropriate value for the threshold as different threshold values could lead to different results [55].”
Metamodel-based sensitivity analysis: polynomial chaos expansions and Gaussian processes (2016, unknown): “Doostan and Owhadi [21]; Sargsyan et al.”
Efficient Bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates (2014, methodology): “Doostan and Owhadi [28] combined the idea of sparse calibration and gPC to approximate the solution of PDEs with stochastic coefficients.”