A regularization method based on an augmented Lagrangian approach for parameter identification problems

AGNELLI J.P.; LEITAO A.; DE CEZARO A.

Helsinki

Congreso; 24th Inverse Days; 2018

Finnish Inverse Problems Society - Aalto University

We propose and analyse a solution method for parameter identification problems modeled by ill-posed nonlinear operator equations, where the parameter function to be identified is known to be a piecewise constant function.A level-set approach is used to represent the unknown parameter, and a corresponding Tikhonov functional is defined. Additionally, a suitable constraint is enforced, resulting that our Tikhonov functional is to be minimized over a set of piecewise constant level-set functions. Thus, the original parameter identification problem is rewritten in the form of a constrained optimization problem, which is solved using an augmented Lagrangian type method.We prove existence of zero duality gaps and existence of generalized Lagrangian multipliers. Moreover, we prove convergence and stability of the parameter identification method, i.e. the solutionmethod is a regularization method.Additionally, a primal-dual algorithm is proposed to compute approximate solutions of the original inverse problem, and its convergence is proved. Numerical examples applied to a 2D diffuse optical tomography benchmark problem demonstrate the viability of the proposed approach.