Identification of Discontinuous Coefficients From Elliptic Problems Using
Total Variation Regularization

Author: Tony F. Chan and Xue-Cheng Tai

Abstract: We propose several formulations for recovering
discontinous coefficient of elliptic problems by using
total variation (TV) regularization. The motivation for
using TV is its well-established ability to recover
sharp discontinuities. We employ an augmented Lagrangian
variational formulation for solving the output-least-squares
inverse problem. 
In addition to the basic output-least-squares formulation, 
we introduce two new techniques to handle large observation errors.
First, we use a filtering step to remove as much of the
observation error as possible.
Second, we introduce two extensions of the output-least-squares
model; one model employs
observations of the gradient of the state variable while the other
utilizes the flux. 
Numerical experiments indicate that the combination of these
two techniques enables us to successfully recover highly discontinous
coefficient even under observation errors as high as $100\%$
in the $L^2$ norm.