Open Access

ARTICLE

A Dynamical Tikhonov Regularization Method for Solving Nonlinear Ill-Posed Problems

by Chein-Shan Liu¹, Chung-Lun Kuo²

1 Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liucs@ntu.edu.tw
2 Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung, Taiwan

Computer Modeling in Engineering & Sciences 2011, 76(2), 109-132. https://doi.org/10.3970/cmes.2011.076.109

Abstract

The Tikhonov method is a famous technique for regularizing ill-posed systems. In this theory a regularization parameter α needs to be determined. Based-on an invariant-manifold defined in the space of (x,t) and from the Tikhonov minimization functional, we can derive an optimal vector driven system of nonlinear ordinary differential equations (ODEs). In the Optimal Vector Driven Algorithm (OVDA), the optimal regularization parameter αk is presented in the iterative solution of x, which means that a dynamical Tikhonov regularization method is involved in the solution of nonlinear ill-posed problem. The OVDA is an extension of the Landweber-Scherzer iterative algorithm. Numerical examples of nonlinear ill-posed systems under noise are examined, revealing that the present OVDA has a good computational efficiency and accuracy.

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Keywords

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