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

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

Rio de Janeiro

Congreso; International Congress of Mathematicians; 2018

We propose and analyze a solution method for parameter identification problems modeled by ill-posed nonlinear operator equations in Banach spaces, 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 (on the space of level-set functions).Additionally, a suitable constraint is enforced, resulting that our Tikhonov functional is to be minimized over a set of piecewise constant level-set functions. Summarizing, the original parameter identification problem is rewritten as in the form of a constrained optimization problem, which is solved here by means of an augmented Lagrangian type method.We prove existence of zero duality gaps and existence of generalized Lagrangian multipliers for our Lagrangian approach. Moreover, we prove convergence and stability of our parameter identification method, i.e. our solution method is a regularization method.Additionally, a primal-dual algorithm is proposed to compute approximate solutions of the original inverse problem, and its convergence is provided. Numerical examples applied to a 2D diffuse optical tomography benchmark problem demonstrate the viability of the proposed approach.