A global minimization algorithm for Tikhonov functionals with sparsity constraints

Abstract

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 α j , where the approximation obtained with α j serves as the starting value for the dual iteration with parameter α 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.

Variational regularization of complex deautoconvolution and phase retrieval in ultrashort laser pulse characterization

“…with penalty term given by (32). The corresponding functions in X = L 2 C (0, 1) are obtained as f σ,δ α,β = γ[x σ,δ α,β ].…”

“…where the NURBS penalty term is as defined by (32) and α = 10 −6 . Similarly, the relative data misfit term we have weighted the brightness of the plot of arg f σ,δ α,β * and arg(f † ) depending on their absolute values.…”

“…with penalty term R(x) defined by (32) and α = 10 −6 . For the TIGRA-type approach with respect to the weight β, we used β 0 = 10 2 and q = 0.25.…”

“…To obtain a better approximation of the global optimizer, we employ a strategy known as TIGRA (the name being derived from TIkhonov-GRAdient method), which was introduced by Ramlau [25]. This method was proven to converge globally for suitable problems and with suitably chosen parameters in [25,26,32].…”

“…The numerical solution of the deautoconvolution problem 1.1 as well as of phase retrieval problems 1.2 requires discretization of the complex-valued autoconvolution equation with kernel function (6). The most natural choice are discretizations using piecewise constant functions, either in terms of step functions {χ [i/n,(i+1)/n) } i=0,...,n−1 (see, e.g., [11,21,6,5]) or by means of Haar wavelets (e.g., [1,32]). While step functions yield simple (and computationally efficient) formulae, essentially reducing the continuous autoconvolution to its discrete counterpart, Haar wavelets are particularly suitable for the reconstruction of functions in L 2 (0, 1) as they yield an orthonormal basis both of the infinite dimensional Lebesgue space as well as of its truncated, finite-dimensional approximations.…”