离散椭圆Robin传输反问题的唯一性、稳定性和整体收敛
Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem
摘要：
对于连续可微、逐点凸、单调函数的有限维零点寻找问题，我们导出了一个简单的判别准则，保证了牛顿法的唯一性、Lipschitz稳定性和整体收敛.我们的准则仅仅要求在多个评价点和多个方向上评价前向函数的方向导数。然后，我们证明了这个结果可以用来证明唯一性，稳定性和整体收敛的反系数问题的许多测量。本文研究了椭圆型偏微分方程中确定未知逆Robin透射系数的问题。利用单调性和局部化势的关系，证明了在先验已知边界的先验已知分区上的一个分段常数系数是唯一确定的，并且它可以由全局收敛的牛顿迭代唯一稳定地重构.我们推导出一种构造性的方法来识别这些边界测量，计算稳定常数，并给出一个数值例子。
英文摘要：
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.
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