# Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem

## TL;DR

This paper establishes conditions for uniqueness, stability, and convergence in an inverse elliptic Robin transmission problem, enabling reliable reconstruction of unknown coefficients from limited boundary measurements.

## Contribution

It introduces a simple criterion for stability and convergence that applies to inverse elliptic problems, and demonstrates its use in reconstructing Robin coefficients from finitely many measurements.

## Key findings

- A criterion based on directional derivatives ensures uniqueness and stability.
- The inverse Robin coefficient can be uniquely reconstructed with finitely many measurements.
- The method achieves global convergence in the reconstruction process.

## Abstract

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.