Stochastic collocation methods via minimization of Transformed $L_1$ penalty
For researchers in uncertainty quantification, this work offers a novel method to enhance sparse reconstruction in polynomial chaos expansions, though it is incremental as it adapts existing TL1 minimization to a new application domain.
The paper proposes combining transformed L1 (TL1) minimization with stochastic collocation for sparse polynomial expansion in uncertainty quantification, showing improved recovery accuracy and efficiency for both sparse and non-sparse functions through numerical examples.
We study the properties of sparse reconstruction of transformed $\ell_1$ (TL1) minimization and present improved theoretical results about the recoverability and the accuracy of this reconstruction from undersampled measurements. We then combine this method with the stochastic collocation approach to identify the coefficients of sparse orthogonal polynomial expansions for uncertainty quantification. In order to implement the TL1 minimization, we use the DCA-TL1 algorithm which was introduced by Zhang and Xin. In particular, when recover non-sparse functions, we adopt an adaptive DCA-TL1 method to guarantee the sparest solutions. Various numerical examples, including sparse polynomial functions recovery and non-sparse analytical functions recovery are presented to demonstrate the recoverability and efficiency of this novel method and its potential for problems of practical interests.