Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer

TL;DR

This paper introduces a novel, efficient numerical algorithm for accurately locating crack tips in Mumford-Shah minimizers, utilizing conformal mapping and an iteration method that avoids Jacobian computations.

Contribution

It presents a new fast iteration algorithm for crack-tip detection that transforms the problem via conformal mapping and employs the augmented immersed interface method.

Findings

Numerical results align with analytical predictions.

The method offers improved speed over traditional iterative approaches.

Provides insights into free discontinuity problems.

Abstract

A new algorithm to determine the position of the crack (discontinuity set) of certain minimizers of Mumford-Shah functional in situations when a crack-tip occurs is introduced. The conformal mapping $w=\sqrt{z}$ in the complex plane is used to transform the free discontinuity problem to a new type of free boundary problem, where the symmetry of the free boundary is an additional constraint of a non-local nature. Instead of traditional Jacobi or Newton iterative methods, we propose a simple iteration method which does not need the Jacobian but is way fast than the Jacobi iteration. In each iteration, a Laplace equation needs to be solved on an irregular domain with a Dirichlet boundary condition on the fixed part of the boundary; and a Neumann type boundary condition along the free boundary. The augmented immersed interface method is employed to solve the potential problem. The numerical results agree with the analytic analysis and provide insight into some open questions in free discontinuity problems.