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

作者：Liu Chein Shan^*; Kuo Chung Lun

来源：Computer Modeling in Engineering & Sciences, 2011, 76(2): 109-132.

• Ill-posed systems
• Tikhonov regularization
• Dynamical Tikhonov regularization
• Optimal vector driven algorithm (OVDA)
• Landweber-Scherzer iterative algorithm

The Tikhonov method is a famous technique for regularizing ill-posed systems. In this theory a regularization parameter alpha 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 alpha(k) 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.

• 出版日期2011-6