Regularization methods for inverse problems.

Abstract

Many applications in industry and science require the solution of an inverse problem. To obtain a stable estimate of the solution of such problems, it is often necessary to im- plement a regularization strategy. In the first part of the present work, a multiplicative regularization strategy is analyzed and compared with Tikhonov regularization. In the second part, an inverse problem that arises in financial mathematics is analyzed and its solution is regularized. Tikhonov regularization for the solution of discrete ill-posed problems is well doc- umented in the literature. The L-curve criterion is one of a few techniques that are preferred for the selection of the Tikhonov parameter. A more recent regularization ap- proach less well known is a multiplicative regularization strategy, which unlike Tikhonov regularization, does not require the selection of a parameter. We analyze a multiplica- tive regularization strategy for the solution of discrete ill-posed problems by comparing it with Tikhonov regularization aided with the L-curve criterion. We then proceed to analyze the stability of a method for estimating the risk-neutral density (RND) for the price of an asset from option prices. RND estimation is an inverse problem. The method analyzed first applies the principle of maximum entropy, where the maximum entropy solution (MES) corresponds to the estimated RND. Next, it pro- vides an effective characterization of the constraint qualification (CQ) under which the MES can be computed by solving the dual problem, where an explicit function in finitely many variables is minimized. In our analysis, we show that the MES is stable under pa- rameter perturbation, but the parameters are unstable under data perturbation. When noisy data are used, we show how to project the data so that the CQ is satisfied and the method can be used. To stabilize the method, we use Tikhonov regularization and choose the penalty parameter via the L-curve method. We demonstrate with numerical examples that the method becomes then much more stable to perturbation in data. Accordingly, we perform a convergence analysis of the regularized solution.