In this paper, we consider the inverse problem of recovering an isotropic elastic tensor from the Neumann-to-Dirichlet map. To this end, we prove a Lipschitz stability estimate for Lame parameters with certain regularity assumptions. In addition, we assume that the Lame parameters belong to a known finite subspace with a priori known bounds and that they fulfill a monotonicity property. The proof relies on a monotonicity result combined with the techniques of localized potentials. To numerically solve the inverse problem, we propose a Kohn-Vogelius-type cost functional over a class of admissible parameters subject to two boundary value problems. The reformulation of the minimization problem via the Neumann-to-Dirichlet operator allows us to obtain the optimality conditions by using the Frechet differentiability of this operator and its inverse. The reconstruction is then performed by means of an iterative algorithm based on a quasi-Newton method. Finally, we give and discuss several numerical examples.

中文翻译：

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线性弹性中拉梅参数的 Lipschitz 稳定性估计和重建

在本文中，我们考虑从 Neumann-to-Dirichlet 映射恢复各向同性弹性张量的逆问题。为此，我们证明了具有某些规律性假设的 Lame 参数的 Lipschitz 稳定性估计。此外，我们假设 Lame 参数属于具有先验已知边界的已知有限子空间，并且它们满足单调性。证明依赖于单调性结果与局部电位技术相结合。为了数值求解逆问题，我们提出了一个 Kohn-Vogelius 类型的成本函数，该函数针对一类受两个边界值问题影响的可接受参数。通过 Neumann-to-Dirichlet 算子对最小化问题的重新表述使我们能够通过使用该算子的 Frechet 可微性及其逆来获得最优条件。然后通过基于拟牛顿法的迭代算法进行重建。最后，我们给出并讨论了几个数值例子。