Inverse Problems: Computational Methods and Emerging Applications

Inverse Problems Workshop Series II

November 12 - 20, 2003

Schedule and Presentations

Pictures

Inverse Problems in Materials Science
November 12 - 14, 2003

Organizing Committee:

Russel Caflisch, Chair (UCLA)

Scientific Introduction:

Inverse problems abound in materials science and engineering. Examples include determination of material properties from scattering data or other measurements, design of new materials with optimized properties, determination of operational parameters to meet system performance goals. This workshop will focus on inverse problem methods and results that have a particular material science aspect.

Invited Speakers:

Level Set Methods for Inverse and Optimal Design Problems
November 12 - 14, 2003

Organizing Committee:

Stanley Osher, Chair (IPAM)
Martin Burger (UCLA)
Fadil Santosa (University of Minnesota)

Scientific Introduction:

There are many inverse and optimal design problems in which the desired unknown is the geometry of a structure. Such problems arise, for example, in inverse scattering involving obstacles, and in design of optimal structures. These types of problem can in general be cast as optimization problems where one would minimize data misfit in the case of inverse problems, or a cost functional, in the case of optimal design.

A powerful approach for solving such problems is to parametrize the unknown geometry using a level set function. An evolution for the level set function can be created by considering an iterative approach for solving the minimization. When the iterations are considered as time-evolution, one obtains a coupled system of partial differential equations with a Hamilton-Jacobi flavor. Optimization can be interpreted as choosing velocities driving the evolution to a minimum.

Many numerical, modeling and theoretical issues arise here. This procedure has been used quite successfully recently. The workshop will deal with these and related problems.

Invited Speakers:

Computational Methods for Inverse Problems and Applications
November 15, 17 - 19, 2003

Organizing Committee:

Heinz Engl, Chair (Johannes Kepler University, Austria)
David Colton (University of Delaware)
Peter Deuflhard (Freie Universitat, Berlin)
David Donoho (Stanford University)
Eric Michielssen (University of Illinois at Urbana-Champaign)
Lothar Reichel (Kent State University)

Scientific Introduction:

Inverse problems are usually ill-posed, which reflects itself in the fact that numerical methods for solving inverse problems are highly sensitive to noise. A "naive" approach like just discretizing the problem and using traditional methods leads to unpleasant surprises: e.g., refining the discretization leads to more noise in the results, taking more iterations in an iterative methods may have the same effect. Methods to be used are termed "regularization methods"; their theory is well-developed and by now classical for linear inverse problems, current research is focussed on nonlinear problems. Also, large-scale inverse problems like parameter identification problems for three-dimensional partial differential equations pose new methodological problems like the efficient coupling of forward and inverse solvers. In this context, multigrid methods for inverse problems are of interest.

This workshop focuses on the aspects mentioned above, where both functional analytic and numerical linear algebra apsects will be discussed. Some of the talks focus on methods and their convergence theory per se, while others treat specific applications with an emphasis on numerical methods. The relationship between functional analytic and statistical approaches will also be discussed. The workshop is embedded into a group of related workshops, where the relation to level sets and their use as regularization methods for inverse problems is especially close. Recently, inverse problems techniques have also been used in mathematically analyzing learning techniques, and neural nets have been used (without much mathematical theory) for solving inverse problems; this provides a close link to the workshop on Inverse problems and Learning Theory and Algorithms.

Invited Speakers:

Inverse problems and learning theory and algorithms
November 20, 2003

Organizing Committee:

Tomaso Poggio, Chair (Massachusetts Institute of Technology)

Scientific Introduction:

Learning from examples can be regarded as an inverse and ill-posed problem.  In fact some of the main techniques used in learning are based on regularization (eg Radial Basis Functions, Support Vector Machines...).  However, modern, "classical" learning theory so far has focused on generalization (eg predictivity) rather than on stability properties of the learning algorithms. The workshop will focus on the problem of learning as an inverse problem, on regularization techniques to solve it and more fundamentally on the close relation between generalization and stability.

Invited Speakers:

Application/Registration

We are no longer accepting applications for financial support too attend this workshop. You may still register online by going to:

Contact Us:

Institute for Pure and Applied Mathematics (IPAM)
Attn: INVWS2
460 Portola Plaza
Los Angeles CA 90095-7121
Phone: 310 825-4755
Fax: 310 825-4756
Email: ipam@ucla.edu
Website: http://www.ipam.ucla.edu/programs/invws2/