Advanced numerical techniques for inverse problems,

with applications in imaging science and applied geophysics

Cagliari, July 17-21, 2017

**An Overview of Ill-Posed Inverse Problems:**

Many questions in Applied Mathematics, Science, and Engineering lead to
Inverse Problems. These problems often arise when one is interested in
determining the cause of an observed effect in

- inverse helioseismology, where one seeks to determine the structure of
the sun by measurements from earth or space,

- medical imaging, e.g., electrocardiographic imaging, computerized
tomography,

- image restoration, where one is interested in determining an unavailable
exact image from an available contaminated version,

- adaptive optics, where one is interested in determining the shape of
mirrors that provide high resolution of a contaminated image.

Many inverse problems are said to be ill-posed, because they either might
not have a solution, the solution might not be unique, or small
perturbations in the available data can give rise to very large
perturbations in the computed solution. The data generally is obtained by
measurement and, therefore, is typically contaminated by an error due to
measurement inaccuracies. One is faced with the problem of determining a
meaningful solution to the inverse ill-posed problem in the presence of
error in the data.

The difficulties in solving ill-posed problems can be reduced by replacing
the given problem by a nearby problem, whose solution is less sensitive to
errors in the data. This replacement is referred to as regularization. The
most commonly used regularization technique is due to Tikhonov. It replaces
the original problem by a penalized least squares problem. Many other
regularization techniques have been developed. All regularization techniques
require the determination of a regularization parameter that determines how
much the original problem is modified. It is important to choose both a
suitable regularization method and an appropriate value of the
regularization parameter to obtain accurate solutions of inverse ill-posed
problems.

Techniques of Numerical Linear Algebra can be used to investigate the
problems to be solved and to devise regularization methods. For instance,
the singular value decomposition (SVD) is a useful tool from linear algebra
for the investigation and regularization of small to medium-sized inverse
ill-posed problems. However, the SVD cannot be applied to large-scale
problems due to its high computational cost. Instead Krylov subspace
iterative methods can be used to reduce the given large problem to smaller
size. The latter can then be solved with the aid of the SVD.

** Purpose of the Summer School:**

The last 20 years has seen significant development in methods for analyzing
and solving inverse ill-posed problems. Many of these methods can be
expressed with the tools of Numerical Linear Algebra. The course will
provide an overview of many of established and new techniques for the
analysis and solution of inverse ill-posed problems. The theory presented is
illustrated with computed examples. Participants in the summer school will
be assigned homework that expands the theory that is presented in lectures,
and programming exercises that will show how the methods discussed perform.