Periodic Reporting for period 3 - IPTheoryUnified (Inverse boundary problems: toward a unified theory)
Okres sprawozdawczy: 2021-05-01 do 2022-10-31
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 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 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 Calderón problem
- an unexpected connection between the Calderón and Gel’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.
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 Calderón problem
- an unexpected connection between the Calderón and Gel’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.
We describe progress in the three objectives stated in the DoA during the reporting period. In particular, Objectives 1 and 2 progress largely 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) has been completed. The results were announced in talks at Berkeley, Santa Barbara and Santa Cruz in late 2019 and became available in Oksanen-Salo-Stefanov-Uhlmann (arXiv:2001.07599). Another large work in this direction, related to stability and instability properties of rather general inverse problems in a unified setting, appeared in Koch-Rüland-Salo (Ars Inveniendi Analytica 2021).
In Objective 2 (integral geometry problems), the first results on a Carleman estimate approach to integral geometry appeared in Paternain-Salo (arXiv:1805.02163) and uniqueness for arbitrary connections in two dimensions was finally established in Paternain-Salo (arXiv:2006.02257) addressing key questions 2c) and 2e). The two-dimensional theory has now become fairly complete, and a Cambridge University Press monograph devoted to this theory written together with Paternain and Uhlmann will appear in January 2023.
Objective 3 (nonlocal and nonlinear models) has been more successful than expected. The methods outlined in the ERC proposal have already led to fairly complete results for Calderón type inverse problems for nonlocal models, including sharp stability and regularity (Rüland-Salo, Inverse Problems (2018) and Nonlinear Analysis (2020)), reconstruction (Ghosh-Rüland-Salo-Uhlmann, Journal of Functional Analysis 2020), and other equations (PhD student Covi, Nonlinear Analysis (2020) and Inverse Problems (2020)). The focus in Objective 3 has now shifted to nonlinear models, where our initial results (Lassas-Liimatainen-Lin-Salo, J. Math. Pures Appl. 2021 and Rev. Mat. Iberoamericana 2021) have already generated followup works by several research groups.
The first workshop related to the ERC project was organized in August 2018 in Jyväskylä, with the main collaborators attending, and it provided an excellent start for the project. The second workshop was scheduled for August 2020 but had to be postponed because of the COVID-19 pandemic. Online workshops were organized in August 2020 and August 2021 instead. In August 2022, an in-person workshop was organized in Helsinki.
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) has been completed. The results were announced in talks at Berkeley, Santa Barbara and Santa Cruz in late 2019 and became available in Oksanen-Salo-Stefanov-Uhlmann (arXiv:2001.07599). Another large work in this direction, related to stability and instability properties of rather general inverse problems in a unified setting, appeared in Koch-Rüland-Salo (Ars Inveniendi Analytica 2021).
In Objective 2 (integral geometry problems), the first results on a Carleman estimate approach to integral geometry appeared in Paternain-Salo (arXiv:1805.02163) and uniqueness for arbitrary connections in two dimensions was finally established in Paternain-Salo (arXiv:2006.02257) addressing key questions 2c) and 2e). The two-dimensional theory has now become fairly complete, and a Cambridge University Press monograph devoted to this theory written together with Paternain and Uhlmann will appear in January 2023.
Objective 3 (nonlocal and nonlinear models) has been more successful than expected. The methods outlined in the ERC proposal have already led to fairly complete results for Calderón type inverse problems for nonlocal models, including sharp stability and regularity (Rüland-Salo, Inverse Problems (2018) and Nonlinear Analysis (2020)), reconstruction (Ghosh-Rüland-Salo-Uhlmann, Journal of Functional Analysis 2020), and other equations (PhD student Covi, Nonlinear Analysis (2020) and Inverse Problems (2020)). The focus in Objective 3 has now shifted to nonlinear models, where our initial results (Lassas-Liimatainen-Lin-Salo, J. Math. Pures Appl. 2021 and Rev. Mat. Iberoamericana 2021) have already generated followup works by several research groups.
The first workshop related to the ERC project was organized in August 2018 in Jyväskylä, with the main collaborators attending, and it provided an excellent start for the project. The second workshop was scheduled for August 2020 but had to be postponed because of the COVID-19 pandemic. Online workshops were organized in August 2020 and August 2021 instead. In August 2022, an in-person workshop was organized in Helsinki.
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
- 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