Accelerated projected steepest descent method for nonlinear inverse problems with sparsity constraints

Abstract

This paper is concerned with the construction of an iterative algorithm to solve nonlinear inverse problems with an 1 constraint. One extensively studied method to obtain a solution of such an 1 penalized problem is iterative soft-thresholding. Regrettably, such iteration schemes are computationally very intensive. A subtle alternative to iterative soft-thresholding is the projected gradient method that was quite recently proposed by Daubechies et.al. in [3]. The authors have shown that the proposed scheme is indeed numerically much thriftier. However, its current applicability is limited to linear inverse problems. In this paper we provide an extension of this approach to nonlinear problems. Adequately adapting the conditions on the (variable) thresholding parameter to the nonlinear nature, we can prove convergence in norm for this projected gradient method, with and without acceleration. A numerical verification is given in the context of nonlinear and non-ideal sensing. For this particular recovery problem we can achieve an impressive numerical performance (when comparing it to non-accelerated procedures).

Projected gradient iteration for nonlinear operator equation

“…The convergence results are given in Section 3. In Section 4, we present a numerical example confirming the theoretical results, and compare the projected gradient algorithm (5) used in this paper and the implicit iterative algorithm (6) used in [16].…”

“…In comparison with [16], this paper has interesting features. First, our proof of the result about strong convergence in Lemma 4 appears simpler than that of [16]. This is for the different approach we take.…”

“…When A is a nonlinear operator and p = 1, the projected gradient iteration is proposed, and some convergence results of the iteration sequence have been established. We refer to [14][15][16].…”

“…This is for the different approach we take. Second, the projected gradient iteration in [16] is different from (5). Given x (n) , [16] constructs x (n+1) by solving the minimization problem,…”

Sparsity and Compressed Sensing in Inverse Problems

“…Influenced by the huge impact of sparse signal representations and the practical feasibility of advanced sparse recovery algorithms, the combination of sparse signal recovery and inverse problems emerged in the last decade as a new growing area. Currently, there exist a great variety of sparse recovery algorithms for inverse problems (linear as well as for nonlinear operator equations) within this context, see, e.g., [5,6,7,14,15,16,25,26,41,44,45]. These recovery algorithms are successful for many applications and have lead to breakthroughs in many fields.…”