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