AN ANALYSIS OF MULTIPLICATIVE REGULARIZATION

Inverse problems arise in many branches of science and engineering. In order to get a good approximation of the solution of this kind of problems, the use of regularization methods is required. Tikhonov regularization is one of the most popular methods for estimating the solutions of inverse problems. This method needs a regularization parameter and the quality of the approximate solution depends on how good the regularization parameter is. The L-curve method is a convenient parameter choice strategy for selecting the Tikhonov regularization parameter and it works well most of the time. There are some problems in which the L-curve criterion does not perform properly. Multiplicative regularization is a method for solving inverse problems and does not require any parameter selection strategies. However, it turns out that there is a close connection between multiplicative regularization and Tikhonov regularization; in fact, multiplicative regularization can be regarded as defining a parameter choice rule for Tikhonov regularization. In this work, we have analyzed multiplicative regularization for finite-dimensional problems. We also have presented some preliminary theoretical results for infinite-dimensional problems. Furthermore, we have demonstrated with numerical experiments that the multiplicative regularization method produces a solution that is usually very similar to the solution obtained by the L-curve method. This method is guaranteed to define a positive regularization parameter under some conditions. Computationally, this method is not expensive and is easier to analyze compared to the L-curve method.