Inverse problems arise in a variety of applications: image
processing, finance, mathematical biology, and more. Mathematical
models for these applications may involve integral equations, partial
differential equations, and dynamical systems, and solution schemes
are formulated by applying algorithms that incorporate regularization
techniques and/or statistical approaches. In most cases these
solutions schemes involve the need to solve a large-scale
ill-conditioned linear system that is corrupted by noise and other
errors. In this talk we describe and demonstrate capabilities of a
new MATLAB software package that consists of state-of-the-art
iterative methods for solving such systems, which includes approaches
that can automatically estimate regularization parameters, stopping
iterations, etc., making them very simple to use. Thus, the package
allows users to easily incorporate into their own applications (or
simply experiment with) different iterative methods and regularization
strategies with very little programming effort. On the other hand,
sophisticated users can also easily access various options to tune the
algorithms for certain applications. Moreover, the package includes
several test problems and examples to illustrate how the iterative
methods can be used on a variety of large-scale inverse problems.
The talk will begin with a brief introduction to inverse problems,
discuss considerations that are needed to compute an approximate
solution, and describe some details about new efficient hybrid Krylov
subspace methods that are implemented in our package. These methods
can guide users in automatically choosing regularization parameters,
and can be used to enforce various regularization schemes, such as
sparsity. We will use imaging examples that arise in medicine and
astronomy to illustrate the performance of the methods.
This is joint work with Silvia Gazzola (University of Bath) and
Per Christian Hansen (Technical University of Denmark).
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