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Calder\'on's Inverse Problem with a Finite Number of Measurements

Authors :
Alberti, Giovanni S.
Santacesaria, Matteo
Source :
Forum of Mathematics, Sigma, 7 (2019), E35
Publication Year :
2018

Abstract

We prove that an $L^\infty$ potential in the Schr\"odinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace $\mathcal W$. As a corollary, we obtain a similar result for Calder\'on's inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces $\mathcal W$, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of $\dim \mathcal W$.<br />Comment: 15 pages

Details

Database :
arXiv
Journal :
Forum of Mathematics, Sigma, 7 (2019), E35
Publication Type :
Report
Accession number :
edsarx.1803.04224
Document Type :
Working Paper
Full Text :
https://doi.org/10.1017/fms.2019.31