Lorentzian Calderon problem: visibility and invisibility

Project Information

LoCal

Grant agreement ID: 101086697

DOI

10.3030/101086697

EC signature date 19 February 2023

Start date 1 August 2023

End date 31 July 2028

Funded under

• European Research Council (ERC)

Total cost

€ 1 372 986,00

EU contribution

€ 1 372 986,00

Coordinated by

HELSINGIN YLIOPISTO
Finland

Periodic Reporting for period 1 - LoCal (Lorentzian Calderon problem: visibility and invisibility)

Reporting period: 2023-08-01 to 2026-01-31

Summary of the context and overall objectives of the project

The purpose of this proposal is to attack one of the most fundamental inverse problems, called the Calderón problem, from multiple angles. The Calderón problem models recovery of information from indirect observations. Indirect observations are characteristic, for example, in medical imaging, exploration geophysics, and non-destructive testing. The Lorentzian Calderón problem gives a mathematical model that captures many essential features of such physical problems when traveling waves are used. Its sister, the elliptic, or Riemannian, version of the Calderón problem models probing with fixed frequency waves.

The overall objectives are

1. Solve the Lorentzian Calderón problem without curvature bounds
2. Solve inverse problems for nonlinear elliptic equations in non-product geometries
3. Construct counterexamples to the Lorentzian Calderón problem

More specific goals, as formulated in B2 part of the proposal, are

A. Solve the Lorentzian Calderón problem in geometries without null cut points
B. Solve inverse problems for nonlinear elliptic equations in near-Euclidean geometries
C. Find a counterexample to the Lorentzian Calderón problem with full data.

From the geometrical point of view, it could be said that the theory of inverse problems is mature in the case of nonlinear hyperbolic equations, while it is still in its youth in the case of non-linear elliptic and linear hyperbolic equations, the Lorentzian Calderón problem being a prototype of the latter case. The disparity between these cases can be explained heuristically by saying that the richer is the set of solutions to a partial differential equation, the easier is the associated inverse problem. A guiding principle in the project is to proceed from richer solution sets to poorer ones. We hope that advances in relation to the objectives 1 and 2 will eventually help us to make progress in the context of the Riemannian Calderón problem, whose the solution set is poorer than those for the linear hyperbolic and non-linear elliptic problems. The Riemannian problem has remained open for more than 30 years, while the Lorentzian version generalizes the even more classical inverse problem studied by Gelfand and Levitan.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Objective 1/Goal A. We solved a rigidity version of the Lorentzian Calderon problem, see the preprint arxiv.org/abs/2409.18604. Here the Minkowski geometry is compared to a general globally hyperbolic geometry. Remarkably, we don't need to assume that the general geometry has no null cut points. In this sense the result is stronger than goal A above. However, work remains to generalize the result so that we can allow for comparison to geometries beyond the Minkowski geometry, thus passing from a rigidity type result to a more traditional uniqueness result.

Objective 2/Goal B. We solved inverse problems for nonlinear elliptic equations in general geometries, see doi.org/10.1090/proc/16455. As the result is not confined to near-Euclidean geometries, it is stronger than goal B above.

A variation of Lorentzian Calderon problem considers lower order perturbations of the canonical Lorentzian wave equation. From the geometric and gauge theoretic point of view, it is natural to model first order perturbations using connections, and in the methodology section of part B2 of the proposal, we formulated a problem to recover such a connection. We have solved this problem and our result was announced very recently in a presentation by Sean Gomes, a postdoc in the project, at Applied Inverse Problems 2025 conference (August, Rio de Janeiro).

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Our solution to the rigidity version of the Lorentzian Calderon problem is based on a completely new method using distorted plane wave solutions and a combination of geometric, topological and unique continuation arguments. Further work is needed in order to understand how it can be adapted to other problems.

Our new techniques concerning objectives 1 and 2 have given new ideas on how to attack the Riemannian Calderón problem, but so far the problem has resisted our attacks.