A global minimization algorithm for Tikhonov functionals with sparsity constraints

Abstract

In this paper we present a globally convergent algorithm for the computation of a minimizer of the Tikhonov functional with sparsity promoting penalty term for nonlinear forward operators in Banach space. The dual TIGRA method uses a gradient descent iteration in the dual space at decreasing values of the regularization parameter ${\alpha }_j$, where the approximation obtained with ${\alpha }_j$ serves as the starting value for the dual iteration with parameter ${\alpha }_{j+1}$. With the discrepancy principle as a global stopping rule the method further yields an automatic parameter choice. We prove convergence of the algorithm under suitable step-size selection and stopping rules and illustrate our theoretic results with numerical experiments for the nonlinear autoconvolution problem.