Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem
TLDR
It is shown 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.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.read more
Citations
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Calder\'on's Inverse Problem with a Finite Number of Measurements.
TL;DR: In this paper, it was shown that a potential in the Schrodinger equation in three and higher dimensions can be determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace.
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Lipschitz stability estimate and reconstruction of Lam\'e parameters in linear elasticity
TL;DR: In this article, the authors consider the inverse problem of recovering an isotropic elastic tensor from the Neumann-to-Dirichlet map and prove a Lipschitz stability estimate for Lame parameters with certain regularity assumptions.
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Lipschitz stability estimate and reconstruction of Lamé parameters in linear elasticity
TL;DR: A Lipschitz stability estimate for Lamé parameters with certain regularity assumptions is prove to prove to solve the inverse problem of recovering an isotropic elastic tensor from the Neumann-to-Dirichlet map.
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Inverse problems on low-dimensional manifolds
TL;DR: In this paper, the authors consider inverse problems between infinite-dimensional Banach spaces and derive a globally-convergent reconstruction algorithm from a finite number of measurements, and prove uniqueness and Holder and Lipschitz stability.
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An Introduction to Finite Element Methods for Inverse Coefficient Problems in Elliptic PDEs
TL;DR: This text shows how to efficiently implement both using a standard FEM package and prove convergence of the FEM approximations against their true-solution counterparts, and numerically demonstrates the challenges that arise from non-uniqueness, non-linearity and instability issues.
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