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We study the properties of sparse reconstruction of transformed $\ell_1$(TL1) minimization and present improved theoretical results about therecoverability and the accuracy of this reconstruction from undersampledmeasurements. We then combine this method with the stochastic collocationapproach to identify the coefficients of sparse orthogonal polynomialexpansions for uncertainty quantification. In order to implement the TL1minimization, we use the DCA-TL1 algorithm which was introduced by Zhang andXin. In particular, when recover non-sparse functions, we adopt an adaptiveDCA-TL1 method to guarantee the sparest solutions. Various numerical examples,including sparse polynomial functions recovery and non-sparse analyticalfunctions recovery are presented to demonstrate the recoverability andefficiency of this novel method and its potential for problems of practicalinterests.

Stochastic collocation methods via minimization of Transformed $L_1$ penalty