# Inverse boundary problems - toward a unified theory (IPTheoryUnified)

Main funder

Funder's project number: 770924

Funds granted by main funder (€)

- 920 880,00

Funding program

Project timetable

Project start date: 01/05/2018

Project end date: 31/10/2023

Summary

*********************** CAUTION: VERY ROUGH FIRST DRAFT ***********************

This proposal is concerned with the mathematical theory of inverse problems. This is a vibrant research field in the intersection of pure and applied mathematics, drawing techniques from several different areas and generating new research questions inspired by applications. Prominent questions include the Calderón problem related to electrical imaging, the Gel'fand problem related to seismic imaging, and geometric inverse problems such as inversion of the geodesic X-ray transform.

Recently, exciting new connections between these different topics have begun to emerge in the work of the PI and others, such as

- the explicit appearance of the geodesic X-ray transform in the Calderón problem

- an unexpected connection between the Calderón and Gel’fand problems involving control theory

- pseudo-linearization as a potential unifying principle for reducing nonlinear problems to linear ones

- the introduction of microlocal normal forms in inverse problems for PDE

These examples strongly suggest that there is a larger picture behind various different inverse problems, which remains to be fully revealed.

This project will explore the possibility of a unified theory for several inverse boundary problems. Particular objectives include:

1. The use of normal forms and pseudo-linearization as a unified point of view to inverse boundary problems, including reductions to questions in integral geometry and control theory

2. The solution of integral geometry problems, including the analysis of convex foliations, invertibility of the geodesic X-ray transform, and a systematic Carleman estimate approach to uniqueness results

3. A theory of inverse problems for nonlocal and nonlinear models based on control theory arguments

Such a unified theory could have remarkable consequences for practical reconstructions and open questions, such as control theory methods in inverse transport, the solution of the boundary rigidity problem, or a general pseudo-linearization approach for global inverse problems.

This proposal is concerned with the mathematical theory of inverse problems. This is a vibrant research field in the intersection of pure and applied mathematics, drawing techniques from several different areas and generating new research questions inspired by applications. Prominent questions include the Calderón problem related to electrical imaging, the Gel'fand problem related to seismic imaging, and geometric inverse problems such as inversion of the geodesic X-ray transform.

Recently, exciting new connections between these different topics have begun to emerge in the work of the PI and others, such as

- the explicit appearance of the geodesic X-ray transform in the Calderón problem

- an unexpected connection between the Calderón and Gel’fand problems involving control theory

- pseudo-linearization as a potential unifying principle for reducing nonlinear problems to linear ones

- the introduction of microlocal normal forms in inverse problems for PDE

These examples strongly suggest that there is a larger picture behind various different inverse problems, which remains to be fully revealed.

This project will explore the possibility of a unified theory for several inverse boundary problems. Particular objectives include:

1. The use of normal forms and pseudo-linearization as a unified point of view to inverse boundary problems, including reductions to questions in integral geometry and control theory

2. The solution of integral geometry problems, including the analysis of convex foliations, invertibility of the geodesic X-ray transform, and a systematic Carleman estimate approach to uniqueness results

3. A theory of inverse problems for nonlocal and nonlinear models based on control theory arguments

Such a unified theory could have remarkable consequences for practical reconstructions and open questions, such as control theory methods in inverse transport, the solution of the boundary rigidity problem, or a general pseudo-linearization approach for global inverse problems.

Principal Investigator

Primary responsible unit

Related publications

- Fixed angle inverse scattering in the presence of a Riemannian metric (2022) Ma, Shiqi; et al.; A1; OA
- Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities (2022) Lu, Shuai; et al.; A1; OA
- Refined instability estimates for some inverse problems (2022) Kow, Pu-Zhao; et al.; A1; OA
- Unique Continuation Results for Certain Generalized Ray Transforms of Symmetric Tensor Fields (2022) Agrawal, Divyansh; et al.; A1; OA
- A sharp stability estimate for tensor tomography in non-positive curvature (2021) Paternain, Gabriel P.; et al.; A1; OA
- Free boundary methods and non-scattering phenomena (2021) Salo, Mikko; et al.; A1; OA
- On some partial data Calderón type problems with mixed boundary conditions (2021) Covi, Giovanni; et al.; A1; OA
- Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems (2021) Covi, Giovanni; et al.; A1; OA
- An inverse problem for the fractional Schrödinger equation in a magnetic ﬁeld (2020) Covi, Giovanni; A1; OA
- Applications of Microlocal Analysis in Inverse Problems (2020) Salo, Mikko; A2; OA