科研之友
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摘要
In this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via l(q) minimization. The main results include: 1) By using the norm inequality between l(q) and l(2) and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via l(q) minimization; 2) We then combine this method with the stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability results for both sparse polynomial functions and general non-sparse functions. We also present various numerical experiments to show the performance of the l(q) algorithm. We first present some benchmark tests to demonstrate the ability of l(q) minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard l(1) and reweighted l(1) minimization. All the numerical results indicate that the l(q) method performs better than standard l(1) and reweighted l(1) minimization.
