Inverse Problems: Computation and Applications

Inverse problems are concerned with the determination of causes of observed effects. Their investigation and solution continue to receive considerable attention by mathematicians, researchers in applied sciences, as well as engineers. A reason for this is the widespread availability of inexpensive computing power, which makes it possible for many scientists and engineers to solve more of the many inverse problems that arise in the applied sciences and engineering. This in turn spurs analyses of new inverse problems and the development of new numerical methods for their solution. The conference ‘‘Inverse Problems: Computation and Applications’’ held at the Centre International de Rencontres Mathèmatiques, Luminy, May 31–June 4, 2010, focused on inverse problems that are ill-posed in the sense of Hadamard’s famous definition: they might not have a solution, the solution might not be unique, and the solution might depend discontinuously on the data. The solution of Fredholm integral equations of the first kind with a smooth kernel is a classical ill-posed problem that often arises in the context of inverse problems. Parameter estimation problems for differential equations also are inverse problems that often are ill-posed. Applications where inverse problems arise include astronomical imaging, computerized tomography, remote sensing, and the restoration of images that have been contaminated by blur and noise. In most practical applications where inverse problems arise, the available data is contaminated by noise. This feature is a major contributor to the numerical difficulty in solving ill-posed inverse problems. The noise may stem from inaccuracies in the measurements of the data, transmission errors, and discretization. The high sensitivity of ill-posed problems to perturbations in the data makes it necessary to replace the given inverse problem by a nearby problem that is less sensitive to errors in the available data. This replacement is known as regularization. The most well-known regularization method is due to A.N. Tikhonov. Regularization by truncated singular value decomposition and by truncated iteration are other popular approaches to determine nearby problems that are less sensitive to the errors in the data and to round-off errors introduced during the computations. It was the aim of the conference Inverse Problems: Computation and Applications to: • promote interaction between researchers, who work on theoretical and computational aspects of inverse problems, • facilitate exchange between mathematicians, researchers in the applied sciences, and engineers, • enable researchers, who use different solution and analysis techniques, to exchange their experiences, and • introduce post-docs and Ph.D. students to inverse problems. The conference was organized with support from the Laboratoire de Mathématiques Pures et Appliquées at the University of Littoral Côte d’Opale. The conference had 54 participants, many of whom can be seen in Fig. 1. A selection of the papers presented at the conference are published in this special issue of J. Comput. Appl. Math. The guest editors would like to thank the authors for submitting their papers, the referees for their thoughtful reports, as well as Professor Luc Wuytack, editor for this journal, and Palavi Das, Elena Grinari, and Stella Yan at Elsevier for their support and help with this special issue. We would like to express our gratitude to all conference participants for their contributions and for making the conference a very interesting and pleasant event. Finally, we also would like to thank the staff at CIRM for their kind help with this conference.