Random finite-difference discretizations of the Ambrosio-Tortorelli functional with optimal mesh size
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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.
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