Silvia Gazzola

Regularization techniques for linear inverse problems

Inverse Problems arise in many Engineering and Scientific applications. Because of their inherent ill-posedness, solving inverse problems is always a delicate task and appropriate regularization methods should be employed for this purpose. This tutorial will introduce you to a variety of popular regularization methods that can be used when solving large-scale linear inverse problems. Specifically, topics to be addressed include:

·      the singular value decomposition of a matrix as a tool for analyzing linear inverse problems; 

·      popular spectral filtering methods, such as Tikhonov regularization;

·      popular iterative regularization methods for linear systems, such as Landweber method, CGLS and LSQR;

·      strategies for choosing the regularization parameter;

·      applications to imaging problems.

The tutorial will also comprise numerical illustrations, based on IR Tools, which is a MATLAB Package of Iterative Regularization Methods and Test Problems for Linear Inverse Problems. IR Tools is available through github: https://github.com/jnagy1/IRtools. 

Christian Rieger

Some remarks on solving partial differential equations on embedded manifolds
via meshless methods

This talk is about recent analytical progress in the field of meshless methods for partial differential equations on embedded manifolds.

As first method, we will discuss meshless Galerkin methods for second order elliptic equations on manifolds. Here, we will present a variant of the classical multigrid variants which allows for a rigorous analysis. Moreover, we will present techniques to reduce the computational complexity using localized Lagrange functions.

As next meshless method, we will discuss finite differnece methods where the weights are apadted to radial basis functions (RBF-FD) and progress on stability results for those methods.

Finally, we will briefly present recent analytical results for meshless Semi-Lagrangian methods.

This talk is based on joint works with W. Erb, T. Hangelbroek, F. J. Narcowich, J.D. Ward and G. B. Wright.