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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 ofpolynomial chaos expansions (PCE). The adaptive construction producesanisotropic basis sets that have more terms in important dimensions and limitsthe number of unimportant terms that increase mutual coherence and thus degradethe performance of $\ell_1$-minimization. The important features and theaccuracy of basis selection are demonstrated with a number of numericalexamples. Specifically, we show that for a given computational budget, basisselection produces a more accurate PCE than would be obtained if the basis isfixed a priori. We also demonstrate that basis selection can be applied withnon-uniform random variables and can leverage gradient information.

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