IJPAM: Volume 43, No. 4 (2008)

CONVERGENCE RATES FOR MULTI-PARAMETER
REGULARIZATION IN BANACH SPACES

Torsten Hein
Faculty of Mathematics
Technical University of Chemnitz
Chemnitz, 09107, GERMANY
e-mail: torsten.hein@mathematik.tu-chemnitz.de


Abstract.In this paper we present a multi-parameter regularization approach for solving nonlinear ill-posed problems when a finite-dimensional 'vector' of data is given. Based on the the convergence analysis for nonlinear Tikhonov regularization we show stability and convergence of the method in reflexive Banach spaces. Additionally we prove convergence rates results by using Bregman distances and discuss numerical algorithms for solving the underlying minimization problem in an efficient way. The advantage of considering multi-parameter regularization approaches is illustrated by an example arising in mathematical finance.

Received: March 4, 2008

AMS Subject Classification: 47J06, 49N45, 65J20, 91B28

Key Words and Phrases: inverse problem, nonlinear ill-posed problem, multi-parameter regularization, Bregman distance, convergence rates, Lagrangian methods

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 43
Issue: 4