  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.
