[Paper Review] Stability estimates for the Calder\'on problem with partial data

David Dos Santos Ferreira, Pedro Caro+1 more|arXiv (Cornell University)|May 6, 2014

Numerical methods in inverse problems16 references4 citations

TL;DR

This paper establishes global stability estimates of log-log type for the Calderón problem with partial data in dimensions n > 3, extending prior local results. Using complex geometric optics solutions in anisotropic settings, it proves that the potential in a Schrödinger equation can be stably reconstructed from partial Dirichlet-to-Neumann data, generalizing uniqueness results to a global stability framework.

ABSTRACT

This is a follow-up of a previous article where we proved local stability estimates for a potential in a Schrodinger equation on an open bounded set in dimension $n=3$ from the Dirichlet-to-Neumann map with partial data. The region under control was the penumbra delimited by a source of light outside of the convex hull of the open set. These local estimates provided stability of log-log type corresponding to the uniqueness results in Calderon's inverse problem with partial data proved by Kenig, Sjostrand and Uhlmann. In this article, we prove the corresponding global estimates in all dimensions higher than three. The estimates are based on the construction of solutions of the Schrodinger equation by complex geometrical optics developed in the anisotropic setting by Dos Santos Ferreira, Kenig, Salo and Uhlmann to solve the Calderon problem in certain admissible geometries.

Motivation & Objective

• To extend local stability estimates for the Calderón problem with partial data to global estimates in dimensions greater than three.
• To establish quantitative stability bounds of log-log type for the inverse problem of recovering a potential in a Schrödinger equation from partial Dirichlet-to-Neumann data.
• To generalize prior uniqueness results by Kenig, Sjöstrand, and Uhlmann to a global stability framework using advanced solution constructions.
• To apply the anisotropic complex geometric optics method to handle the partial data setting in higher dimensions.

Proposed method

• Construction of complex geometric optics solutions for the Schrödinger equation in anisotropic geometries, as developed by Dos Santos Ferreira, Kenig, Salo, and Uhlmann.
• Adaptation of the anisotropic framework to handle partial data by focusing on the Dirichlet-to-Neumann map restricted to a subset of the boundary.
• Use of Carleman estimates and asymptotic analysis to control the behavior of solutions and their traces on the boundary.
• Derivation of stability estimates by comparing solutions corresponding to different potentials and relating their differences to the difference in the Dirichlet-to-Neumann maps.
• Extension of the stability framework from local estimates (in the penumbra region) to global estimates across the entire domain in dimensions n > 3.
• Leveraging the structure of the anisotropic setting to ensure the required solvability and decay properties of the complex geometric optics solutions.

Experimental results

Research questions

• RQ1 Can global stability estimates of log-log type be established for the Calderón problem with partial data in dimensions greater than three?
• RQ2 How does the anisotropic complex geometric optics method enable the extension of local stability results to global ones?
• RQ3 What are the quantitative bounds relating the difference in potentials to the difference in partial Dirichlet-to-Neumann data in higher dimensions?
• RQ4 To what extent can the stability framework be generalized beyond the convex hull and penumbra region in the partial data setting?
• RQ5 What structural assumptions on the geometry are necessary to achieve global stability in the anisotropic setting?

Key findings

• Global stability estimates of log-log type are established for the Calderón problem with partial data in dimensions n > 3.
• The stability result extends prior local estimates that were confined to the penumbra region outside the convex hull of the domain.
• The construction of complex geometric optics solutions in the anisotropic setting enables the derivation of global stability bounds.
• The method relies on the anisotropic framework developed by Dos Santos Ferreira, Kenig, Salo, and Uhlmann to handle the partial data problem.
• The estimates provide a quantitative version of the uniqueness result for the Calderón problem with partial data in higher dimensions.
• The work generalizes the stability framework beyond the local regime, offering a global reconstruction stability guarantee.