SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 2275-2318, January 2021.
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 $\delta$, the discretized functionals $\Gamma$-converge to the Mumford--Shah functional only if $\delta\ll\varepsilon$, $\varepsilon$ 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 $\delta\sim\varepsilon$, 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.

中文翻译：

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具有最佳网格大小的Ambrosio-Tortorelli函数的随机有限差分离散

SIAM数学分析杂志，第53卷，第2期，第2275-2318页，2021年1月。
我们提出并分析了Ambrosio-Tortorelli函数的有限差分离散化。众所周知，如果离散化是针对间隔为$ \ delta $的底层周期性晶格进行的，则仅当$ \ delta \ ll \ varepsilon $， $ \ varepsilon $是Ambrosio-Tortorelli函数的椭圆近似参数。关于平稳，遍历和各向同性随机晶格的离散化，我们也证明了该$ \ Gamma $收敛结果也适用于$ \ delta \ sim \ varepsilon $，在该状态下，关于周期晶格的离散化收敛为各向异性版的Mumford-Shah功能。而且，我们表明，在$ \ Gamma $限制为Mumford-Shah类型的情况下，此缩放比例是最大可能的离散化比例，这是最佳的。最后，我们提出了一些数值结果，突出了我们的随机离散泛函的各向同性行为。