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.