Level set regularization for highly ill-posed distributed parameter

February 28, 2006
Location: SFU, ASB 9705
Presenter: Dr. Kees van den Doel

The recovery of a distributed parameter surface with discontinuities from inverse problems with elliptic forward PDEs is fraught with theoretical and practical difficulties. Better results are obtained for problems where the solution may take on at each point only one of two values, thus yielding a shape recovery problem.

In this talk, after introducing the subject, I will describe my recent work with Uri Ascher on level set regularization for such problems. Rather than explicitly integrating a time embedded PDE to steady state, which typically requires thousands of iterations, methods based on Gauss-Newton are applied directly. One of these can be viewed as damped Gauss-Newton utilized to approximate the steady state equations which in turn are viewed as the necessary conditions of a Tikhonov-type regularization with a sharpening sub-step at each iteration. In practice this method is eclipsed, however, by a special "finite time" trust region (or Levenberg-Marquardt) method which we call dynamic regularization applied to the output least squares formulation.

The regularization functional is applied to the (smooth) level set function rather than the discontinuous surface to be recovered, and the second focus of this work is on selecting this functional. Typical choices may lead to (or at least not avoid) flat level sets which in turn cause ill-conditioning in that a small point-wise change in the level set function causes a large change in its 0-level and hence in the recovered surface. But the regularization should also be selected so that its evolution is smooth, a particularly important concern when large iteration updates are contemplated. We propose a new, quartic, non-local regularization term and compare its performance to more usual choices.

Two numerical test cases are considered: a potential problem and the classical EIT/DC resistivity problem.

Bio: Kees van den Doel received a Ph.D. in theoretical physics from the University of California at Santa Cruz in 1984. After two years of postdoctoral research at the University of Tel-Aviv, he pursued his interest in music and spent four years studying music composition at the University of British Columbia. He obtained a second Ph.D. in Computer Science from the University of British Columbia in 1998. After a number of years in industry he returned to UBC in 2001 as a research associate. His main research interest are computer sound and scientific computation. He is currently working on physically-based audio synthesis for speech, and on level set methods for inverse modeling. In his spare time he plays traditional Persian music on ney and occasionally some Bach on the recorder.

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