Random finite-difference discretizations of the ambrosio–tortorelli functional with optimal mesh-size

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 δ, the discretized functionals Γ-converge to the Mumford–Shah functional only if δ ≼ ε, ε being the elliptic approximation parameter of the Ambrosio–Tortorelli functional. Discretizing with respect to stationary, ergodic, and isotropic random lattices we prove this Γ-convergence result also for δ ∼ ε, 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 Γ-limit is of Mumford–Shah type. Finally, we present some numerical results highlighting the isotropic behavior of our random discrete functionals.