Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients

We propose a level set approach for elliptic inverse problems with piecewise constant coefficients. The geometry of the discontinuity of the coefficient is represented implicitly by level set functions. The inverse problem is solved using a variational augmented Lagrangian formulation with total variation regularization of the coefficient. The corresponding Euler–Lagrange equation gives the evolution equation for the level set functions and the constant values of the coefficients. We use a multiple level set representation which allows the coefficient to have multiple constant regions. Knowledge of the exact number of regions is not required, only an upper bound is needed. Numerical experiments show that the method can recover coefficients with rather complicated geometries of discontinuities under moderate amount of noise in the observation data. The method is also robust with respect to the initial guess for the geometry of the coefficient discontinuities.