Choice of the Parameters in A Primal-Dual Algorithm for Bregman Iterated Variational Regularization

TL;DR

This paper investigates how parameter choices affect the convergence and stability of a primal-dual algorithm for Bregman iterated variational regularization, with theoretical analysis and application to image processing.

Contribution

It provides a detailed analysis of parameter selection in primal-dual algorithms for Bregman TV regularization, including convergence rates and error estimates.

Findings

Convergence to minimum norm solution under variational source condition

Parameter choice significantly influences algorithm stabilization

Error bounds depend on operator size and noise level

Abstract

Focus of this work is solving a non-smooth constraint minimization problem by a primal-dual splitting algorithm involving proximity operators. The problem is penalized by the Bregman divergence associated with the non-smooth total variation (TV) functional. We analyse two aspects: Firstly, the convergence of the regularized solution of the minimization problem to the minimum norm solution. Second, the convergence of the iteratively regularized minimizer to the minimum norm solution by a primal-dual algorithm. For both aspects, we use the assumption of a variational source condition (VSC). This work emphasizes the impact of the choice of the parameters in stabilization of a primal-dual algorithm. Rates of convergence are obtained in terms of some concave, positive definite index function. The algorithm is applied to a simple two dimensional image processing problem. Sufficient error analysis profiles are provided based on the size of the forward operator and the noise level in the measurement.