On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification

TL;DR

This paper introduces a continuation approach to Tikhonov regularization that improves stability and avoids local minima in inverse problems, demonstrated through magnetic induction tomography examples.

Contribution

The paper proposes a novel continuation method for Tikhonov regularization, enabling separate treatment of stabilization and a priori knowledge incorporation.

Findings

Enables stable solution of ill-posed problems

Prevents convergence to local minima in inverse problems

Effective in magnetic induction tomography applications

Abstract

We present a new approach to convexification of the Tikhonov regularization using a continuation method strategy. We embed the original minimization problem into a one-parameter family of minimization problems. Both the penalty term and the minimizer of the Tikhonov functional become dependent on a continuation parameter. In this way we can independently treat two main roles of the regularization term, which are stabilization of the ill-posed problem and introduction of the a priori knowledge. For zero continuation parameter we solve a relaxed regularization problem, which stabilizes the ill-posed problem in a weaker sense. The problem is recast to the original minimization by the continuation method and so the a priori knowledge is enforced. We apply this approach in the context of topology-to-shape geometry identification, where it allows to avoid the convergence of gradient-based methods to a local minima. We present illustrative results for magnetic induction tomography which is an example of PDE constrained inverse problem.