Random Finite-Difference Discretizations Of The Ambrosio-Tortorelli Functional With Optimal Mesh-Size

Abstract

We propose and analyze a finite-difference discretization of the Ambrosio-Tortorelli functional. It is known that if the discretization is made with respect to an underlying periodic lattice of spacing ffi, the discretized functionals Gamma-converge to the Mumford-Shah functional only if delta <= epsilon. epsilon being the elliptic approximation parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to stationary, ergodic, and isotropic random lattices we prove this Gamma-convergence result also for delta similar to epsilon, a regime at which the discretization with respect to a periodic lattice converges instead to an anisotropic version of the Mumford{Shah functional. Moreover, we show that this scaling is optimal in the sense that it is the largest possible discretization scale for which the Gamma-limit is of Mumford{Shah type. Finally, we present some numerical results highlighting the isotropic behavior of our random discrete functionals