Periodic Reporting for period 4 - IPTheoryUnified (Inverse boundary problems: toward a unified theory)
Période du rapport: 2022-11-01 au 2023-10-31
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. In particular, Objectives 1 and 2 progressed largely according to the plan, whereas Objective 3 was unexpectedly successful and progressed much faster than expected.
In Objective 1 (normal forms, controllability and pseudo-linearization), a large part related to studying inverse problems for real principal type operators by propagation of singularities (key questions 1a) and 1b) in the DoA) was 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 to appear in American Journal of Mathematics). 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 to appear in Annales de l'ENS) and uniqueness for arbitrary connections in two dimensions was finally established in Paternain-Salo (arXiv:2006.02257 to appear in Journal of Differential Geometry), 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, which sums up the results in Objective 2, appeared in 2023 (joint with Paternain and Uhlmann). Just before the end of the ERC project, an important followup work (Mazzucchelli-Salo-Tzou, arXiv:2306.05906) to the real principal type article mentioned in Objective 1 was completed, giving a partial answer to key question 2d) and yielding solutions to the relevant integral geometry problems, thus connecting Objectives 1 and 2.
Objective 3 (nonlocal and nonlinear models) was more successful than expected. The methods outlined in the ERC proposal quickly led to fairly complete results for Calderón type inverse problems for nonlocal models, including uniqueness (Ghosh-Salo-Uhlmann, Analysis & PDE (2020)), 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 then shifted to nonlinear models, where our initial results (Lassas-Liimatainen-Lin-Salo, J. Math. Pures Appl. 2021 and Rev. Mat. Iberoamericana 2021) launched a new research direction in inverse problems for elliptic PDE. The methods introduced in the nonlocal and nonlinear cases within the ERC project have generated great activity and numerous followup works by different research groups. In Google Scholar (in January 2024), our main nonlocal article from 2020 has 144 citations and first nonlinear article from 2021 has 98 citations, which are large citation numbers in mathematics for such a short time.
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 with many of the collaborators of the ERC project attending was organized in Helsinki.
In Objective 1 (normal forms, controllability and pseudo-linearization), a large part related to studying inverse problems for real principal type operators by propagation of singularities (key questions 1a) and 1b) in the DoA) was 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 to appear in American Journal of Mathematics). 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 to appear in Annales de l'ENS) and uniqueness for arbitrary connections in two dimensions was finally established in Paternain-Salo (arXiv:2006.02257 to appear in Journal of Differential Geometry), 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, which sums up the results in Objective 2, appeared in 2023 (joint with Paternain and Uhlmann). Just before the end of the ERC project, an important followup work (Mazzucchelli-Salo-Tzou, arXiv:2306.05906) to the real principal type article mentioned in Objective 1 was completed, giving a partial answer to key question 2d) and yielding solutions to the relevant integral geometry problems, thus connecting Objectives 1 and 2.
Objective 3 (nonlocal and nonlinear models) was more successful than expected. The methods outlined in the ERC proposal quickly led to fairly complete results for Calderón type inverse problems for nonlocal models, including uniqueness (Ghosh-Salo-Uhlmann, Analysis & PDE (2020)), 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 then shifted to nonlinear models, where our initial results (Lassas-Liimatainen-Lin-Salo, J. Math. Pures Appl. 2021 and Rev. Mat. Iberoamericana 2021) launched a new research direction in inverse problems for elliptic PDE. The methods introduced in the nonlocal and nonlinear cases within the ERC project have generated great activity and numerous followup works by different research groups. In Google Scholar (in January 2024), our main nonlocal article from 2020 has 144 citations and first nonlinear article from 2021 has 98 citations, which are large citation numbers in mathematics for such a short time.
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 with many of the collaborators of the ERC project attending was organized in Helsinki.
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 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 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 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 models