# InvProbGeomPDE Inverse Problems in Partial Differential Equations and Geometry (InvProbGeomPDE)

Main funder

Funder's project number: 307023

Funds granted by main funder (€)

- 1 041 240,00

Funding program

Project timetable

Project start date: 01/11/2012

Project end date: 30/11/2017

Summary

Inverse problems research concentrates on the mathematical theory and practical interpretation of indirect measurements. Applications are found in virtually every research field involving scientific, medical, or industrial imaging and mathematical modelling. Familiar examples include X-ray Computed Tomography (CT) and Magnetic Resonance Imaging (MRI). Inverse problems methods make it possible to employ important advances in modern mathematics in a vast number of application areas. Also, applications inspire new questions that are both mathematically deep and have a close connection to other sciences. This has made inverse problems research one of the most important and topical fields of modern applied mathematics.

The research team proposes to study fundamental mathematical questions in the theory of inverse problems. Particular emphasis will be placed on questions involving the interplay of mathematical analysis, partial differential equations, and Riemannian geometry. A major topic in the research programme is the famous inverse conductivity problem due to Calderón forming the basis of Electrical Impedance Tomography (EIT), an imaging modality proposed for early breast cancer detection and nondestructive testing of industrial parts. The geometric version of the Calderón problem is among the outstanding unsolved questions in the field. The research team will attack this and other aspects of the problem field, partly based on substantial recent progress due to the PI and collaborators. The team will also work on integral geometry questions arising in Travel Time Tomography in seismic imaging and in differential geometry, building on the solution of the tensor tomography conjecture in two dimensions obtained by the PI and collaborators. The research will focus on fundamental theoretical issues, but the motivation comes from practical applications and thus there is potential for breakthroughs that may lead to important advances in medical and seismic imaging.

The research team proposes to study fundamental mathematical questions in the theory of inverse problems. Particular emphasis will be placed on questions involving the interplay of mathematical analysis, partial differential equations, and Riemannian geometry. A major topic in the research programme is the famous inverse conductivity problem due to Calderón forming the basis of Electrical Impedance Tomography (EIT), an imaging modality proposed for early breast cancer detection and nondestructive testing of industrial parts. The geometric version of the Calderón problem is among the outstanding unsolved questions in the field. The research team will attack this and other aspects of the problem field, partly based on substantial recent progress due to the PI and collaborators. The team will also work on integral geometry questions arising in Travel Time Tomography in seismic imaging and in differential geometry, building on the solution of the tensor tomography conjecture in two dimensions obtained by the PI and collaborators. The research will focus on fundamental theoretical issues, but the motivation comes from practical applications and thus there is potential for breakthroughs that may lead to important advances in medical and seismic imaging.

Principal Investigator

Primary responsible unit

Fields of science

Keywords (YSO)

Related publications

- Radial symmetry of minimizers to the weighted Dirichlet energy (2021) Koski, Aleksis; et al.; A1; OA
- Improved Hölder regularity for strongly elliptic PDEs (2020) Astala, Kari; et al.; A1; OA
- Quantitative approximation properties for the fractional heat equation (2020) Rüland, Angkana; et al.; A1; OA
- Sobolev homeomorphic extensions onto John domains (2020) Koskela, Pekka; et al.; A1; OA
- The Calderón problem for the fractional Schrödinger equation (2020) Ghosh, Tuhin; et al.; A1; OA
- The fractional Calderón problem : Low regularity and stability (2020) Rüland, Angkana; et al.; A1; OA
- The Poisson embedding approach to the Calderón problem (2020) Lassas, Matti; et al.; A1; OA
- The Radó-Kneser-Choquet theorem for p-harmonic mappings between Riemannian surfaces (2020) Adamowicz, Tomasz; et al.; A1; OA
- Uniqueness and reconstruction for the fractional Calderón problem with a single measurement (2020) Ghosh, Tuhin; et al.; A1; OA
- Monotonicity and local uniqueness for the Helmholtz equation (2019) Harrach, Bastian; et al.; A1; OA