The Mathematics of Inverse Problems

This program is primarily for graduate students of IMA Participating Institutions. The NSF may provide support for a limited number of students at other US universities.

From Monday, June 15 through Friday, July 3, 2009, University of Delaware, Newark will be the host of the Institute for Mathematics and its Applications (IMA) Summer Graduate Program in Mathematics. The course will concentrate on The Mathematics of Inverse Problems.

Program Description: Inverse problems is a fast-growing area involving a broad range of disciplines from the most abstract and pure mathematics to practical engineering. The 2009 summer program on inverse problems covers three different types of inverse problems: inverse problems for hyperbolic PDE's, inverse scattering in the frequency domain, and variational inverse problems. The program will cover the techniques used to tackle problems at the cutting edge of mathematical research in each of these areas. This is a unique and timely synthesis of disciplines that will position future researchers for the next step in inverse scattering from waves that, we believe, will combine variational methods with direct qualitative techniques.

Week 1, June 15-19: William Symes/Rakesh, hyperbolic inverse problems. Bill Symes is Noah Harding Professor in the Department of Computational and Applied Mathematics at Rice University and Managing Editor of Inverse Problems. Symes and UD's Rakesh will focus on inverse problems for hyperbolic PDEs for one and higher space dimensions. They will consider theoretical and computational issues for inversion from the Dirichlet to Neumann map as well as from smaller subsets of this data, including formally determined data.

Week 2, June 22-26: John Sylvester/David Colton, inverse scattering problems in the frequency domain. John Sylvester of the University of Washington is one of the world leaders in the theory of inverse scattering in the frequency domain. He and UD's David Colton will give a series of lectures on the mathematical foundations of acoustic scattering theory together with qualitative methods in inverse scattering theorey. Supplementary lectures will be given by UD's Fioralba Cakoni and Peter Monk on regularization methods for ill-posed problems and inverse scattering for electromagnetic waves. Numerical methods in inverse scattering theory will be a common thread in all of the above lectures.

Week 3, June 29-July 3: Jonathan Borwein/Russell Luke, variational inverse problems. Jonathan Borwein is Canada Research Chair in IT at Dalhousie and Laureate Professor of Mathematics in Newcastle Australia. He is one of 250 of the most highly cited mathematicians from 1980-1999 (ISIHighlyCited) and the co-inventor of the Borwein-Preiss variational principle, among other achievements. Borwein and UD's Russell Luke will focus on fundamentals of variational analysis, notions of well-posedness and regularity, and finally ill-posed problems which are at the frontiers of analysis. Practical and timely applications to optics and crystallography will be explored.