We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton's method for the finite dimensional zero-finding problem of a continuously differentiable, pointwise convex and monotonic function. Our criterion merely requires to evaluate the directional derivative of the forward function at finitely many evaluation points and for finitely many directions. We then demonstrate that this result can be used to prove uniqueness, stability and global convergence for an inverse coefficient problem with finitely many measurements. We consider the problem of determining an unknown inverse Robin transmission coefficient in an elliptic PDE. Using a relation to monotonicity and localized potentials techniques, we show that a piecewise-constant coefficient on an a-priori known partition with a-priori known bounds is uniquely determined by finitely many boundary measurements and that it can be uniquely and stably reconstructed by a globally convergent Newton iteration. We derive a constructive method to identify these boundary measurements, calculate the stability constant and give a numerical example.

中文翻译：

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离散逆椭圆Robin传输问题的唯一性、稳定性和全局收敛性

我们推导出一个简单的标准，以确保牛顿方法的唯一性、Lipschitz 稳定性和全局收敛性，用于解决连续可微、逐点凸函数和单调函数的有限维寻零问题。我们的标准只需要在有限多个评估点和有限多个方向上评估前向函数的方向导数。然后，我们证明该结果可用于证明具有有限多个测量值的逆系数问题的唯一性、稳定性和全局收敛性。我们考虑确定椭圆偏微分方程中未知的逆罗宾传输系数的问题。使用与单调性和局部电位技术的关系，我们表明，具有先验已知边界的先验已知分区上的分段常数系数由有限多个边界测量唯一确定，并且可以通过全局收敛牛顿迭代唯一且稳定地重建。我们推导出一种构造方法来识别这些边界测量值，计算稳定常数并给出一个数值例子。