Automatic regularization methods for inverse problems
10 September 2018
Campus Middelheim, G.010 - Middelheimlaan 1 - 2020 Antwerpen (route: UAntwerpen, Campus Middelheim
Organization / co-organization:
Department of Mathematics & Computer Science
PhD defence Nick Schenkels - Faculty of Science, Department of Mathematics and Computer Science
When a process is described by a mathematical model, then this model typically depends on a number of parameters. If these parameters are known, then the model can be used to predict the outcome of the process. In many applications, however, people are interested in solving the inverse problem: find the unknown parameters controlling the model based on the measured output of the process.
One of the main difficulties is that these mathematical models are often not invertible and – even if they are – that small measurement errors can lead to large mistakes is the reconstructed parameters. In order to deal with these issues so-called regularization methods have been developed, of which Tikhonov regularization is a widely used example. This method is based on the choice of a regularization parameter that balances solving the original mathematical problem – which is hard to invert – and enforcing certain restrictions which the solution should satisfy. Naturally, the value of this regularization parameter has a great influence on the resulting reconstruction and is therefore of crucial importance.
Since solving an inverse problem for a fixed value of the regularization parameter can have a significant computational cost, trying a few values using a trial-and-error approach is usually inefficient. The goal of the research presented in this thesis was therefore to develop numerical algorithms that iteratively find the solution of the inverse problem, as well as a suitable value for the regularization parameter. In order to do so, we started by studying the generalized Arnoldi-Tikhonov method for square linear inverse problems. In this algorithm the update for the solution is alternated with an update for the regularization parameter based on a linear approximation of the discrepancy curve in a low dimensional Krylov subspace.
As a first step, we generalize the idea behind the generalized Arnoldi-Tikhonov method to linear inverse problems with a non-square matrix and to non-linear problems. Subsequently, as a second step and based on similar ideas, we consider a system of equations that is non-linear in the solution of the inverse problem and the regularization parameter. We then show that it is possible to solve this system for both values simultaneously using a Newton method with a specific choice for the step size in each iteration.