On level set regularization for highly ill-posed distributed parameter estimation problems

Abstract

The recovery of a distributed parameter function 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. This article considers level set regularization for such problems. However, rather than explicitly integrating a time embedded PDE to steady state, which typically requires thousands of iterations, methods based on Gauss–Newton are applied more 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” or Levenberg–Marquardt-type method which we call dynamic regularization applied to the output least squares formulation. Our stopping criterion for the iteration does not involve knowledge of the true solution. The regularization functional is applied to the (smooth) level set function rather than to the discontinuous function to be recovered, and the second focus of this article is on selecting this functional. Typical choices may lead to flat level sets that in turn cause ill-conditioning. We propose a new, quartic, non-local regularization term that penalizes flatness and produces a smooth level set evolution, and compare its performance to more usual choices. Two numerical test cases are considered: a potential problem and the classical EIT/DC resistivity problem.