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

TL;DR

This paper demonstrates that in higher dimensions, a potential in the Schrödinger equation can be uniquely identified from finitely many boundary measurements if it lies in a known finite-dimensional space, advancing inverse boundary value problem theory.

Contribution

It establishes global uniqueness, stability, and reconstruction results for nonlinear inverse boundary value problems with finitely many measurements, a first in the field.

Findings

Unique determination of potentials in finite-dimensional subspaces

Lipschitz stability estimates provided

Explicit measurement count as a function of subspace dimension

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 $\dim \mathcal W$.