Inverse Problems: Computational Methods and Emerging Applications

September 8 - December 12, 2003

Pictures

Organizing Committee

Heinz Engl, Chair (Johannes Kepler University, Austria)
Mario Bertero (Univ of Genova, Italy)
Tony Chan (UCLA)
David Donoho (Stanford University)
Alfred Louis (Saarland University)
Joyce McLaughlin (Rensselaer Polytechnic Institute)
Eric Michielssen (University of Illinois at Urbana-Champaign)
Edward Pike (King's College, London)
Lothar Reichel (Kent State University)
Gunther Uhlmann (University of Washington)

Special Events

• Inverse problems: Orientation at Mays Landing. September 8, 2003.
• Inverse Problems: Computational Methods and Emerging Applications Tutorials. September 9 - 12, 2003.
• Inverse Problems: Industrial Problems Study Group. September 15 - 18, 2003.
• Biological and artificial swarms. October 3 - 4, 2003.
• Inverse Problems Workshop Series I. October 16 - 23, 2003.
• Statistical Methods for Inverse Problems. November 5 - 6, 2003.
• Inverse Problems Workshop Series II. November 12 - 20, 2003.
• Inverse Problems Culminating Workshop at Lake Arrowhead. December 7 - 12, 2003.

Participants

The long-term program will involve a community of researchers. The intent is for long-term participants to have an opportunity to learn about inverse problems from the perspectives of many different fields and to meet a diverse group of people and have an opportunity to form new collaborations. We anticipate an active program throughout the entire program period of research, seminars, and speakers. In addition to these activities, there will be opening tutorials, several workshops, an industrial projects study group and a culminating workshop at Lake Arrowhead.

Full and partial support for long-term participants is available, and those interested are encouraged to fill out an online application at the bottom of this page. Support for individual workshops is also available, and may be applied for through the online application for each workshop. We are especially interested in applicants who are interested in becoming core participants and participating in the entire program (September 8 - December 12, 2003), but give due consideration to applications for shorter periods. Funding for participants is available at all academic levels, though recent PhD's, graduate students, and researchers in the early stages of their career are especially encouraged to apply.

Encouraging the careers of women and minority mathematicians and scientists is an important component of IPAM's mission and we welcome their applications.

Scientific Overview

Inverse problems are problems where causes for a desired or an observed effect are to be determined. They have, nearly always driven by applications, been studied for nearly a century now. An important key feature, both theoretically and numerically, of inverse problems is their “ill-posedness”, i.e., they do not fulfill Hadamard's classical requirements of existence, uniqueness and stability, under data perturbations, of a solution: Solutions of an inverse problem might not exist for all data (e.g., a consistent temperature history exists only for a very smooth final temperature in the model of the classical heat equation), it might not be unique (which raises the practically relevant question of “identifiability”, i.e., the question if the data contain enough information to determine the desired quantity), and it might be unstable with respect to data perturbations. The last aspect is of course especially important, since in real-world problems, measurements always contain noise (another source of noise being errors in numerical procedures), and approximation methods for solving inverse problems which are as insensitive to noise as possible have to be constructed, so-called “regularization methods”.

In the last twenty years, the field of inverse problems has undergone rapid development: The enormous increase in computing power and the development of powerful numerical methods made it possible to simulate real-world “direct” problems of growing complexity. Since in many applications in science and engineering, the “inverse question” of determining causes for desired or observed effects is really the final question, this lead to a growing appetite in applications for posing and solving inverse problems, which in turn stimulated mathematical research e.g., on uniqueness questions and on developing stable and efficient numerical methods (regularization methods) for solving inverse problems. This began mainly for linear problems, but more recently it has also been done for nonlinear problems.

The Special Semester at IPAM will focus on new challenges that have appeared recently in the field of inverse problems:

1.) New application fields:

• Imaging Science including Image Processing, Computer Graphics and Computer Vision. Many imaging problems are by their nature inverse problems, which suggests the use of regularization methods for their solution. On the other hand, specific methods developed in imaging like bounded-variation-regularization and diffusion filtering are being applied to other inverse problems. From the applications side e.g. in medical imaging, new techniques are emerging like elastographics and optical tomography, which in turn pose new mathematical and computational questions.
• Inverse problems in life sciences: Life sciences are a real growth field for mathematical modeling. An important step in modeling is to determine parameters from measurements. In life sciences, this usually leads to large-scale inverse problems, e.g., the simultaneous determination of hundreds of rate constants in very large reaction diffusion systems. Other, already a bit more classical, inverse problems in life sciences include inverse folding problems. Since many mathematical models in the life sciences are just now being developed, this will be the right time to bring experts in life sciences who develop such models and experts on inverse problems together in a kind of exploratory workshop.
• Inverse problems in industry: Knowledge about mathematical and numerical methods for inverse problems has diffused faster into the scientific community than into industry. On the other hand, many mathematical problems of interest to industry are in essence inverse problems. At the IPAM Special Semester, such problems will be studied and worked on in a “study group” format.
• Inverse problems in physical sciences: Many measurements in the physical sciences are indirect, thus their interpretation is an inverse problem whose ill-posedness is not always appropriately addressed. An example are deconvolution problems for ground based telescopes; deconvolution also appears in many other applications like in spectroscopy or in the interpretation of time-resolved fluorescence data with a variety of applications in medicine and biology. Also, modeling of epitaxial growth or other growth processes involves various inverse problems like the determination of growth rates from measured data or shape optimization problems; a methodological link to inverse problems is the use of level set methods.

2.) Methodological challenges

• In recent years, extremely powerful numerical methods have been developed for solving complex direct problems, e.g., multi-field problems in three dimension, both static and dynamic. Such methods include multigrid or, more general, multi-level methods and domain decomposition. When solving inverse problems for such complex problems, new questions arise also for the numerical treatment of the inverse problem, which include the optimal coupling of regularization methods with direct solvers in order to achieve overall optimal performance.
• A powerful numerical method whose main advantage is that it can easily handle changes in the topology is the “level set method”. It has recently also been applied to inverse problems.
• Over the years, two major approaches have been followed in the inverse problems community: statistical and functional-analysis based approaches. A full understanding of the relations between these approaches is still lacking; this is also important for the issue of "uncertainty".

During this Special Semester, special emphasis will be placed on some of these and other emerging challenges, although more classical topics will not be neglected. The Special Semester is intended to bring together scientists and engineers with applied and pure mathematicians interested in inverse problems.

Although these events form the core of the proposed Special Semester, there will also be ongoing activities throughout the semester by visitors interacting on specific research problems with colleagues at UCLA and neighboring universities and with each other. In due course, a call for applications for long-term participants will be made, but tentative expressions of interest are already welcome now.

The Chair of the Program Committee is Prof. Heinz W. Engl (Industrial Mathematics Institute, Johannes Kepler Universität Linz, Austria). 

Below is a list of related seminars being held during this program. Note that the times and locations of the seminar series are general and you should check the IPAM Events Calendar for the status of specific seminars. Seminars are not held during workshops.

• Inverse Problems in Medical Imaging
  Organizer: Guillaume Bal
  Time: Mondays 11 - 12
  Location: IPAM Building, Room 1200

• General Colloquium
  Organizer: Ronny Ramlau
  Time: Mondays 4 - 5, Thursdays 3 - 4
  Location: IPAM Building, Room 1200

• Working Group on Elastography
  Organizer: Joyce McLaughlin
  Time: Tuesday 10 - 12
  Location: IPAM Building, Room 1200

• Inverse Problems - Jr. Seminar Series
  Time: Wednesdays 2 - 4
  Location: IPAM Building, Room 1200
  Presentations:

  • Brian Borchers, Linear Regression and its Discontents (PDF File).

  • Brian Borchers, A Bayesian Approach to the Discrete Linear Inverse Problem (PDF File).

• Seminar on Wave Guides
  Organizers: Joyce McLaughlin, Liliana Borcea
  Time: Thursdays 10 - 12
  Location: IPAM Building, Room 1200

• Learning Seminar on Inverse Problems
  Time: Fridays 1 - 2
  Location: IPAM Building, Room 1200

Lectures and Discussions on "Channels: A Specific Inverse Problem in Molecular Biology"
Friday October 24, 2003
Schedule
Organizer: Robert Eisenberg (Rush University)

Ion Channels: Devices that Control Biological Function
Inverse Problems in Ion Channels, Proteins, and Molecular Biology

Contact Us:

Institute for Pure and Applied Mathematics (IPAM)
Attn: INV2003
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/inv2003/