Adaptive choice of the Tikhonov regularization parameter to solve ill-posed linear algebraic equations via Liapunov Optimizing Control

Numerical problems, including linear algebraic equations, are frequently ill-posed, which means that small perturbations in the observation data can have large effects on the computed solutions. In these cases some kind of regularization method must be used to find a feasible solution. Regularization implies the modification of the original ill-posed problem, so that it becomes well-posed, and the Tikhonov regularization method is the most often used. This method requires the choice of a regularization parameter, and this choice is a crucial issue. Recently, several papers were presented proposing adaptive Tikhonov regularization parameters, instead of constant ones. This paper revisits all these adaptive choices and shows that they can be interpreted in a unified and systematic manner using the Liapunov Optimizing Control (LOC) method, which, in addition, opens up the possibility of new choices of the regularization parameter. Examples of matrices from real applications show that one of the methods designed by the LOC method, called the OVM method by its discoverer, can even outperform the CG method for some matrices, although the CG method performs better for a larger class of matrices.