Project description
Uncovering hidden structures with advanced maths
Medical and seismic imaging rely on external measurements to reveal hidden structures, such as the inside of the human body or the Earth’s subsurface. However, these measurements are often incomplete or distorted by noise. Solving such challenges requires mathematical techniques that can reconstruct missing details. Recent advances in geometry, computational methods, and wave analysis are opening new possibilities. In this context, the ERC-funded PDE-INVERSE project focuses on inverse problems for non-linear partial differential equations, using a method developed for the geometric wave equation. This approach harnesses the non-linear interaction of waves to achieve results that remain out of reach for linear equations. PDE-INVERSE is driving innovations in brain imaging, virus detection, and Earth sciences.
Objective
Inverse problems are a research field at the intersection of pure and applied mathematics. The goal in inverse problems is to recover information from indirect, incomplete or noisy observations. The problems arise in medical and seismic imaging where measurements made on the exterior of a body are used to deduce the properties of the inaccessible interior. We use mathematical methods ranging from microlocal analysis of partial differential equations and metric geometry to stochastics and computational methods to solve these problems.
The focus of the project are the inverse problems for non-linear partial differential equations. We attack these problems using a recent method that we developed originally for the geometric wave equation. This method uses the non-linear interaction of waves as a beneficial tool. Using it, we have been able to solve inverse problems for non-linear equations for which the corresponding problem for linear equations is still unsolved. We study the determination of a Lorentzian space-time from scattering measurements and the lens rigidity conjecture. We use geometric methods, originally developed for General Relativity, to analyze waves in a moving medium and to develop methods for medical imaging. By applying Riemannian geometry and our results in invisibility cloaking, we study counterexamples for non-linear inverse problems and use transformation optics to construct scatterers with exotic properties.
We also consider solution algorithms that combine the techniques used to prove uniqueness results for inverse problems, manifold learning and operator recurrent networks. Applications include new virus imaging methods using electron microscopy and the imaging of brains.
Practical algorithms based on the results of the research will be developed in collaboration with scientists working in medical imaging, optics, and Earth sciences.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
- natural sciencesbiological sciencesmicrobiologyvirology
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equations
- natural sciencesphysical sciencesopticsmicroscopy
- natural sciencesmathematicspure mathematicsgeometry
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Keywords
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Topic(s)
Funding Scheme
HORIZON-ERC - HORIZON ERC GrantsHost institution
00014 HELSINGIN YLIOPISTO
Finland