Tikhonov Functionals Incorporating Tolerances

Abstract

Tikhonov functionals are a well known method for solving inverse problems. They consist of a discrepancy and a penalty term. The first term evaluates the deviation of simulated data from measured data. We alternate this term by incorporating tolerances, which neglects small deviations from the data within a prescribed tolerance. This approach adapts ideas from support vector regression, which utilizes such a tolerance for identity operators and semi discrete problems. Furthermore, the application for inverse problems is motivated by applications where such tolerances naturally occur, e.g. application with multiple measurements. In this case instead of one measurement a confidence interval for the measurement can be used.In this work we provide an overview on the necessary analysis and alternation of Tikhonov functionals incorporating tolerances. In addition, an example of applications are shown and discussed.