Nonlinear regularization methods for ill-posed problems with piecewise constant or strongly varying solutions

Abstract

In this paper we consider nonlinear ill-posed problems with piecewise constant or strongly varying solutions. A class of nonlinear regularization methods is proposed, in which smooth approximations to the Heavyside function are used to reparameterize functions in the solution space by an auxiliary function of levelset type. The analysis of the resulting regularization methods is carried out in two steps: first, we interpret the algorithms as nonlinear regularization methods for recovering the auxiliary function. This allows us to apply standard results from regularization theory, and we prove convergence of regularized approximations for the auxiliary function; additionally, we obtain the convergence of the regularized solutions, which are obtained from the auxiliary function by the nonlinear transformation. Second, we analyze the proposed methods as approximations to the levelset regularization method analyzed in [Frühauf F, Scherzer O and Leitão A 2005 Analysis of regularization methods for the solution of ill-posed problems involving discontinuous operators SIAM J. Numer. Anal. 43 767–86], which follows as a limit case when the smooth functions used for the nonlinear transformations converge to the Heavyside function. For illustration, we consider the application of the proposed algorithms to elliptic Cauchy problems, which are known to be severely ill-posed, and typically allow only for limited reconstructions. Our numerical examples demonstrate that the proposed methods provide accurate reconstructions of piecewise constant solutions also for these severely ill-posed benchmark problems.