New model function methods for determining regularization parameters in linear inverse problems

Authors:
Zewen Wang;Jijun Liu
Affiliations:
Department of Mathematics, Southeast University, Nanjing, 210096, PR China and School of Mathematics and Informational Science, East China Institute of Technology, Fuzhou, Jiangxi Province, 344000 ...;Department of Mathematics, Southeast University, Nanjing, 210096, PR China
Venue:
Applied Numerical Mathematics
Year:
2009

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Abstract

When the damped Morozov discrepancy principle is used to determine the Tikhonov regularization parameter, one should theoretically solve a nonlinear equation by some iteration process, which is generally of local convergence with large amount of computations. This paper considers an approximation of the regularization parameter under the model function framework, which solves an approximate Morozov equation with an explicit expression iteratively. For this approximation, three kinds of new model functions are proposed. The corresponding new algorithms for determining the regularization parameters are also established, with the rigorous proof of global convergence under a unified framework. Our work is a generalization and improvement of the earlier model function method [J.L. Xie, J. Zou, Inverse Problems 18 (5) (2002) 631-643]. Numerical implementations for some ill-posed problems are presented to illustrate the validity of the proposed algorithms.