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Talk Abstract

A novel approach to numerical methods in diffuse and acoustical imaging

A novel approach to numerical methods in diffuse and acoustical imaging

In the past several years researchers have become increasingly
more interested in effective numerical methods for image reconstruction
which would deal with highly scattered radiation, as opposed
to the conventional techniques of computed tomography. A major
practical attraction of such methods lies in a number of important
applications which range from the early female breast cancer
diagnosis and underwater mine search (* i.e.* diffuse imaging)
to the ``classical'' inverse problems in geophysics (*i.e.*
acoustical imaging).

These methods are expected to work with *n*-dimensional
*(n=2,3)*, rather than with 1-dimensional Inverse Scattering
Problems (ISP). Obviously, the multi-dimensionality is a major
obstacle in the development of such algorithms.

In this talk, we will present a novel approach to this challenging problem derived recently by ourselves [1,2]. We call this procedure ``Carleman's Weight Method'' (CWM). CWM works with ISPs for both hyperbolic and parabolic equations which model perfectly acoustical and diffuse imaging respectively. One of the attractive features of CWM is that rigorous global convergence results are proven.

Also, we will present our numerical resutls for a locally
convergent version of CWM (*i.e.* Newton's Method) which
deals with diffuse image reconstruction. It should be pointed
out that these results were obtained for two ranges of parameters
which perfectly model the situation of both an early breast
cancer diagnosis and underwater mine search using ultrafast
laser pulse propagation. Our computational experience shows
a good potential of CWM. In particular, its computational complexity
is of order several magnitudes less than that of many competing
algorithms.

This is joint work with Thomas R. Lucas and Robert M. Frank.

- M.V. Klibanov and O.V. Ioussoupova, SIAM J. Math Anal.,
**26**(1995), 147--179. - S. Gutman, M.V. Klibanov, and A.V. Tikhonavov, IMA J. Appl. Math., (1996).