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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - IPTheoryUnified (Inverse boundary problems: toward a unified theory)

Teaser

Summary

This project is concerned with the mathematical theory of inverse problems. This is a vibrant research field at the intersection of pure and applied mathematics, drawing techniques from partial differential equations, geometry, and harmonic analysis as well as generating new research questions inspired by applications. Prominent questions include the CalderÃ³n 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 generalising X-ray computed tomography in medical imaging.

Recently, exciting new connections between these different topics have begun to emerge, such as

- the explicit appearance of the geodesic X-ray transform in the CalderÃ³n problem
- an unexpected connection between the CalderÃ³n and Gelâ€™fand problems involving control theory
- the introduction of pseudo-linearization and microlocal normal forms in inverse problems for partial differential equations

These examples 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 the use of normal forms and pseudo-linearization as a unified point of view, including reductions to questions in integral geometry and control theory, the solution of integral geometry problems including the analysis of X-ray transforms, and a theory of inverse problems for nonlocal and nonlinear models.

Work performed

We describe progress in the three objectives stated in the DoA during the reporting period. In particular, Objectives 1 and 2 progress quite according to the plan, whereas Objective 3 has been unexpectedly successful and progressed faster than expected.

In Objective 1 (normal forms, controllability and pseudo-linearization), the first large part related to studying inverse problems for real principal type operators by propagation of singularities (key questions 1a) and 1b) in the DoA) is now basically complete. The results have been announced in talks at Berkeley, Santa Barbara and Santa Cruz, and will be submitted before the end of 2019. In Objective 2 (integral geometry problems), the first results on a Carleman estimate approach to integral geometry appeared in Paternain-Salo (arXiv 2018) addressing key questions 2c) and 2e). Objective 3 (nonlocal and nonlinear models) has been very successful: Calderon type inverse problems for nonlocal operators have now been analysed in terms of stability and sharp regularity (RÃ¼land-Salo, Inverse Problems (2018) and Nonlinear Analysis (to appear)), reconstruction (Ghosh-RÃ¼land-Salo-Uhlmann, arXiv 2018), and other equations (PhD student Covi, Nonlinear Analysis (to appear) and arXiv 2019). There has also been progress in nonlinear models (Lassas-Liimatainen-Lin-Salo, arXiv:1903.12562 and arXiv:1905.02764).

The first workshop related to the ERC project was organized in August 2018 in JyvÃ¤skylÃ¤, with the main collaborators attending, and it provided an excellent start for the project.

Final results

Expected results include

- Unified statements for inverse problems in transport theory and hyperbolic equations
- Systematic application of propagation of singularities to inverse problems for principal type operators
- A thorough analysis of several convex foliations and consequences for integral geometry problems
- A Carleman estimate approach to geodesic X-ray transforms on negatively curved manifolds
- Quantitative controllability, stability and reconstruction in inverse problems for fractional equations
- Inverse problems for nonlinear fractional models