On stable invertibility and global Newton convergence for convex monotonic functions
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We derive a simple criterion that ensures uniqueness, Lipschitz stability and global convergence of Newton's method for finite dimensional inverse problems with a continuously differentiable, componentwise convex and monotonic forward function. Our criterion merely requires to evaluate the directional derivative of the forward function at finitely many evaluation points and finitely many directions. Using a relation to monotonicity and localized potentials techniques for inverse coefficient problems in elliptic PDEs, we will then show that a discretized inverse Robin transmission problem always fulfills our criterion if enough measurements are being used. Thus our result enables us to determine those boundary measurements from which an unknown coefficient can be uniquely and stably reconstructed with a given desired resolution by a globally convergent Newton iteration.
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