Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems
This work addresses the challenge of solving nonlinear inverse problems with non-convex regularization, offering a theoretically grounded and computationally efficient approach.
The paper presents a method for regularizing nonlinear inverse problems using Ivanov regularization, which involves minimizing a non-convex cost function under a norm constraint. The method iteratively solves quadratic trust region subproblems with possibly indefinite Hessians, and the authors provide a convergence analysis and numerical experiments.
In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a non-convex cost function under a norm constraint, where non-convexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (non-convex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz [10]. We here present a convergence analysis of the overall method as well as numerical experiments.