Talk by Nick Schenkels, 10/12/2018

Monday December 10 2018 at 14:00 in Grace Hopper 2

Title: Automatic regularization methods for inverse problems


Processes described by a mathematical model typically depend 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, it is the inverse problem that is of greater interest: 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 solution. 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, but since solving the 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.

In this presentation we will present a number of algorithms that can iteratively find the Tikhonov solution of the inverse problem, as well as a suitable value for the regularization parameter based on the discrepancy principle. We will start by introducing the generalized Arnoldi-Tikhonov method and show how it can be adapted for non-square linear systems. We will then show how the ideas behind the method can be applied in a non-linear setting. Finally, we will discuss an alternative approach based on solving the system of non-linear equations found by combining the Tikhonov normal equations and the discrepancy principle.

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