A Dynamical Tikhonov Regularization for Solving Ill-posed Linear Algebraic Systems

The Tikhonov method is a famous technique for regularizing ill-posed linear problems, wherein a regularization parameter needs to be determined. This article, based on an invariant-manifold method, presents an adaptive Tikhonov method to solve ill-posed linear algebraic problems. The new method consists in building a numerical minimizing vector sequence that remains on an invariant manifold, and then the Tikhonov parameter can be optimally computed at each iteration by minimizing a proper merit function. In the optimal vector method (OVM) three concepts of optimal vector, slow manifold and Hopf bifurcation are introduced. Numerical illustrations on well known ill-posed linear problems point out the computational efficiency and accuracy of the present OVM as compared with classical ones.