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

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.

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Cited by in corpus (9)

• Infinite-dimensional inverse problems with finite measurements
• Lipschitz stability estimate and reconstruction of Lamé parameters in linear elasticity
• An introduction to finite element methods for inverse coefficient problems in elliptic PDEs
• Inverse problems on low-dimensional manifolds
• Calderón's Inverse Problem with a Finite Number of Measurements II: Independent Data
• Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming
• Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks
• Reconstruction of small and extended regions in EIT with a Robin transmission condition
• Feynman's inverse problem