Innovating Works

PDE-INVERSE

Financiado
Geometric Methods in Inverse Problems for Partial Differential Equations
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 s... 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. ver más
31/10/2028
2M€
Duración del proyecto: 65 meses Fecha Inicio: 2023-05-23
Fecha Fin: 2028-10-31

Línea de financiación: concedida

El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2023-05-23
Línea de financiación objetivo El proyecto se financió a través de la siguiente ayuda:
ERC-2022-ADG: ERC ADVANCED GRANTS
Cerrada hace 2 años
Presupuesto El presupuesto total del proyecto asciende a 2M€
Líder del proyecto
HELSINGIN YLIOPISTO No se ha especificado una descripción o un objeto social para esta compañía.
Perfil tecnológico TRL 4-5