Hybrid Krylov Subspace Iterative Methods for Inverse Problems

Inverse problems arise in many imaging applications, such as image
reconstruction (e.g., computed tomography), image deblurring, and
digital super-resolution. These inverse problems are very difficult
to solve; in addition to being large scale, the underlying
mathematical model is often ill-posed, which means that noise and
other errors in the measured data can be highly magnified in computed
solutions. Regularization methods are often used to overcome this
difficulty. In this talk we describe hybrid Krylov subspace based
regularization approaches that combine matrix factorization methods
with iterative solvers. The methods are very efficient for large scale
imaging problems, and can also incorporate methods to automatically
estimate regularization parameters. We also show how the approaches
can be adapted to enforce sparsity and nonnegative constraints.

We will use many imaging examples that arise in medicine and astronomy
to illustrate the performance of the methods, and at the same time
demonstrate a new MATLAB software package that provides an easy to use
interface to their implementations.

This is joint work with Silvia Gazzola (University of Bath) and
Per Christian Hansen (Technical University of Denmark).