Uniqueness and reconstruction for the fractional Calderón problem with a single measurement

By Tuhin Ghosh, Angkana Rüland, Mikko Salo, Gunther Uhlmann. Journal of Functional Analysis, 2018

147 citations recorded by the source corpus.

Abstract We show global uniqueness in the fractional Calderon problem with a single measurement and with data on arbitrary, possibly disjoint subsets of the exterior. The previous work [10] considered the case of infinitely many measurements. The method is again based on the strong uniqueness properties for the fractional equation, this time combined with a unique continuation principle from sets of measure zero. We also give a constructive procedure for determining an unknown potential from a single exterior measurement, based on constructive versions of the unique continuation result that involve different regularization schemes.

Subfields: Biomedical Engineering, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4b7d52ee2713b38c200474111682fe29177b5457

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Biomedical Engineering

1. Jesse Railo
2. Keijo Monkkonen
3. Giovanni Covi
4. Katya Krupchyk
5. Tuhin Ghosh
6. Ali Feizmohammadi
7. Gunther Uhlmann
8. Yi-Hsuan Lin

In Control and Systems Engineering

1. Jesse Railo
2. Gunther Uhlmann
3. Keijo Monkkonen
4. Giovanni Covi

In Mathematical Physics

1. Katya Krupchyk
2. Gunther Uhlmann
3. Jesse Railo
4. Keijo Monkkonen
5. Giovanni Covi
6. Tuhin Ghosh
7. Ali Feizmohammadi
8. Mikko Salo

Citations in the corpus

• The higher order fractional Calderón problem for linear local operators: Uniqueness (2020, background): “The fractional Calderón problem has also been solved under single measurement [14].”
• Fractional anisotropic Calderón problem on closed Riemannian manifolds (2021, background): “Following this work, inverse problems of recovering lower order terms for fractional elliptic equations have been studied extensively, see for example [15], [14], [35], [34], [1], [4], [5], [6], [7], [8], [9], [30], [31], [33] for some of the important contributions.”
• Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2020, methodology): “ and EˆV a measurable set with positive measure. Suppose that f 2Hr(Rn), fj E = 0 and ( ) sfj V = 0. Show that f0 or give a counterexample. The positive answer to question 2.2 is known when s 2(0;1) [26]. Question 2.2 with a potential qfrom a suitable class of functions is also an interesting and mo”