Enhancing $\ell_1$-minimization estimates of polynomial chaos expansions using basis selection

John D. Jakeman; Khachik Sargsyan; Michael S. Eldred

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Enhancing $\ell_1$-minimization estimates of polynomial chaos expansions using basis selection

John D. Jakeman, Michael S. Eldred, Khachik Sargsyan

Numerical Analysis · Submitted 30 Jul 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Specifically, we show that for a given computational budget, basis selection produces a more accurate PCE than would be obtained if the basis is fixed a priori.

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arXiv ID1407.8093 (version 1)

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CitationJohn D. Jakeman, Khachik Sargsyan, Michael S. Eldred. "Enhancing $\ell_1$-minimization estimates of polynomial chaos expansions using basis selection." arXiv:1407.8093 (2014). https://pith.science/paper/F43IGYS7

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In this paper we present a basis selection method that can be used with $\ell_1$-minimization to adaptively determine the large coefficients of polynomial chaos expansions (PCE). The adaptive construction produces anisotropic basis sets that have more terms in important dimensions and limits the number of unimportant terms that increase mutual coherence and thus degrade the performance of $\ell_1$-minimization. The important features and the accuracy of basis selection are demonstrated with a number of numerical examples. Specifically, we show that for a given computational budget, basis selection produces a more accurate PCE than would be obtained if the basis is fixed a priori. We also demonstrate that basis selection can be applied with non-uniform random variables and can leverage gradient information.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper we present a basis selection method that can be used with $\ell_1$-minimization to adaptively determine the large coefficients of polynomial chaos expansions (PCE).	contribution 0.99	0.91	0.17	asserted 0.89
s1	The adaptive construction produces anisotropic basis sets that have more terms in important dimensions and limits the number of unimportant terms that increase mutual coherence and thus degrade the performance of $\ell_1$-minimization.	method 0.88	0.81	0.16	asserted 1.00
s2	The important features and the accuracy of basis selection are demonstrated with a number of numerical examples.	result 0.60	0.92	0.12	asserted 1.00
s3	Specifically, we show that for a given computational budget, basis selection produces a more accurate PCE than would be obtained if the basis is fixed a priori.	result 1.00	0.95	0.62	asserted 1.00
s4	We also demonstrate that basis selection can be applied with non-uniform random variables and can leverage gradient information.	contribution 0.64	0.92	0.07	asserted 0.99

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Original abstract

In this paper we present a basis selection method that can be used with $\ell_1$-minimization to adaptively determine the large coefficients of polynomial chaos expansions (PCE). The adaptive construction produces anisotropic basis sets that have more terms in important dimensions and limits the number of unimportant terms that increase mutual coherence and thus degrade the performance of $\ell_1$-minimization. The important features and the accuracy of basis selection are demonstrated with a number of numerical examples. Specifically, we show that for a given computational budget, basis selection produces a more accurate PCE than would be obtained if the basis is fixed a priori. We also demonstrate that basis selection can be applied with non-uniform random variables and can leverage gradient information.

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5 arXiv papers cite this in 9 sentences

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Cites it as support 1 paper

Recently, [ref] and [ref] proved that adaptive basis scheme can be more efficient than fixing a candidate basis a priori.

Hierarchical adaptive polynomial chaos expansions · HIERARCHICAL ADAPTIVE POLYNOMIAL CHAOS EXPANSIONS · 0.99

Cites it as background 3 papers

Approaches of this type have attracted interested in recent years in the area of computational uncertainty quantification for partial differential equations [ref] .

Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations · Background on Affine-Parametric Operator Equations · 0.99
Working towards a similar end, various approaches have also been considered to learn a better sparsity basis [ref] or to use additional gradient samples [ref] .

Compressed sensing approaches for polynomial approximation of high-dimensional functions · Existing literature · 0.94
Further developments which combine spectral expansions and compressive sensing ideas have lead to so-called sparse polynomial chaos expansions [ref] .

Polynomial-Chaos-based Kriging · Introduction · 0.80

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