Tikhonov functionals incorporating tolerances in discrepancy term for inverse problems

Abstract

This thesis contributes to the field of inverse problems and their regularization through Tikhonov-type regularization schemes. Tikhonov-type regularization schemes use a discrepancy term on the operator evaluation and a regularization term on the parameter. One can change the penalty term to incorporate different a-priori information about the true parameter. For example, most research focuses on the regularization term, such as applying sparsity constraints instead of $L_2$-penalty. This work takes a different approach and adds an $\varepsilon$-insensity to the discrepancy term. This insensitivity does add another regularization and accounts for confidence bands. We can obtain these bands through multiple measurements, for example.

Besides the mathematical framework, this work explores possible numerical solvers for the altered Tikhonov functional. Depending on the chosen norm of the discrepancy term and the type of penalty term, the altered Tikhonov functional may not be differentiable. In this case, a non-smooth solver is necessary. This thesis compares existing non-smooth solvers with a newly introduced subgradient method with adaptive step size.

Finally, we apply the theory to a parameter identification problem. The example is from micro-milling and the resulting surface. First, an existing cutting process model is taken and expanded to account for wear on the cutting tool during the process. Then we use the model for the parameter identification.