Dynamic level set regularization for large distributed parameter estimation problems

This paper considers inverse problems of shape recovery from noisy boundary data, where the forward problem involves the inversion of elliptic PDEs. The piecewise constant solution, a scaling and translation of a characteristic function, is described in terms of a smoother level set function. A fast and simple dynamic regularization method has been recently proposed that has a robust stopping criterion and typically terminates after very few iterations. Direct linear algebra methods have been used for the linear systems arising in both forward and inverse problems, which is suitable for problems of moderate size in 2D. For larger problems, especially in 3D, iterative methods are required. In this paper we extend our previous results to large-scale problems by proposing and investigating iterative linear system solvers in the present context. Perhaps contrary to one's initial intuition, the iterative methods are particularly useful for the inverse rather than the forward linear systems. Moreover, only very few preconditioned conjugate gradient iterations are applied towards the solution of the linear system for the inverse problem, allowing the regularizing effects of such iterations to take centre stage. The efficacy of the obtained method is demonstrated.