Parameter Estimation for Large-Scale Reconstruction

Steffen Brun Kjøller

AbstractThe focus of this thesis is on solving linear inverse problems, using Tikhonov Regularization. For the Tikhonov Regularization, a regularization parameter is needed, describing the balance between being true to the data and getting a smooth solution. When solving small problems, it is possible to compute a curve, called an L-Curve outlining this balance. A good regularization parameter can be determined as the corner point on such a curve.

We will come up with an estimated L-Curve from just a few points. This will make it possible to locate a corner point and thereby a good regularization parameter, for large-scale problems where it is not possible to compute the L-Curve.

It is important for this method to work, that the estimated L-Curve has the same properties as the exact L-Curve. We start by showing that the basic idea is useful by solving small inverse problems. Since the problems are small, we can compute the L-Curve, and thereby we compare the computed L-Curve and corner point, with the estimated L-Curve and the corner point from this.

Hereafter we will try some attempts to improve the method. Not all these attempts improve the method of estimating an L-Curve. Instead some of the attempts will be used for showing the limitations of the method. Among these are the choice of data points, which is important to get reasonable results. They have to represent the whole curve, so there is enough information to estimate the L-Curve.

We will describe the requirements of a good choice of data points, as well as, how to ensure that even for a badly chosen set of data points a reasonable solution can be obtained. That is, to add points, making sure that the data points are placed reasonably.

Finally two large-scale problems will be solved to demonstrate the method in use, before a conclusion on the results can be made. Moreover some thoughts on areas for future improvements are presented.