Stochastic collocation methods via minimization of Transformed L₁ penalty

2018/05/14 by Ling Guo, Jing Li, Guo, Ling +3
Decision Sciences · Mathematics · Engineering · #Probabilistic and Robust Engineering Design #Statistical Methods and Inference #Sparse and Compressive Sensing Techniques

paper · pdf · doi:10.48550/arxiv.1805.05416

Abstract

We study the properties of sparse reconstruction of transformed ℓ₁ (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.

Citations

• Nearly unbiased variable selection under minimax concave penalty
• Sparse Approximate Solutions to Linear Systems
• Stable signal recovery from incomplete and inaccurate measurements

Related

• Stochastic collocation methods via L₁ minimization using randomized quadratures
• Uniform Uncertainty Principle and signal recovery via Regularized Orthogonal Matching Pursuit
• Sparse Legendre expansions via ℓ₁ minimization
• An efficient adaptive sparse grid collocation method through derivative estimation
• Robust Sparse Analysis Regularization