Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Electrical Impedance Tomography
SIAM Review
Fast simulation of 3D electromagnetic problems using potentials
Journal of Computational Physics
Computational Methods for Inverse Problems
Computational Methods for Inverse Problems
Journal of Computational Physics
A multilevel, level-set method for optimizing eigenvalues in shape design problems
Journal of Computational Physics
SIAM Journal on Numerical Analysis
On Effective Methods for Implicit Piecewise Smooth Surface Recovery
SIAM Journal on Scientific Computing
Electrical impedance tomography using level set representation and total variational regularization
Journal of Computational Physics
An extended level set method for shape and topology optimization
Journal of Computational Physics
Dual evolution of planar parametric spline curves and T-spline level sets
Computer-Aided Design
A semi-implicit level set method for structural shape and topology optimization
Journal of Computational Physics
Journal of Computational Physics
Parameter estimation in flow through partially saturated porous materials
Journal of Computational Physics
Level set method for the inverse elliptic problem in nonlinear electromagnetism
Journal of Computational Physics
A Regularized Gauss-Newton Trust Region Approach to Imaging in Diffuse Optical Tomography
SIAM Journal on Scientific Computing
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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.