Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem
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Numerische Mathematik
Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem Bastian Harrach1 Received: 13 December 2019 / Revised: 5 October 2020 / Accepted: 26 October 2020 © The Author(s) 2020
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. Mathematics Subject Classification 35R30 · 65M32 · 58C15
1 Introduction New technologies for medical imaging, non-destructive testing, or geophysical exploration are often based on mathematical inverse coefficient problems, where the coefficient of a partial differential equation is to be reconstructed from (partial) knowledge of its solutions. A prominent example is the emerging technique of electrical impedance tomography (EIT), cf. [1,12,14,23,24,31,62,63,83,86,88,90,95,101,103], and the references therein for a broad overview. Inverse coefficient problems are usually highly non-linear and ill-posed, and uniqueness and stability questions, as well
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Bastian Harrach [email protected] Institute for Mathematics, Goethe-University Frankfurt, Frankfurt, Germany
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as the design and the theoretical study of reconstruction algorithms are extremely challenging topics in theoretical and applied research. In practical applications, only finitely many measurements can be made, the unknown coefficient has to be parametrized with finitely many variables (e.g., by assuming piecewise-constantness on a given partition), and physical considerations typically limit the unknown coefficient to fall between certain a-priori known bounds. Thus, after shifting and rescaling, a practical inverse coefficient problem can be put in the form of finding the zero x ∈ [0, 1]n of a non-linear function F : Rn → Rm , F(x) = 0. It is of utmost importance to determine what measurements make this finite-dimensional inverse (or zero-finding) problem uniquely solvable, to evaluate the stability of the solution with respect to model and meas
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