Objective This project will develop a mathematical theory of sample complexity, i.e. of finite measurements, for inverse problems in partial differential equations (PDE). Inverse problems are ubiquitous in science and engineering, and appear when a quantity has to be reconstructed from indirect measurements. Whenever physics plays a crucial role in the description of an inverse problem, the mathematical model is based on a PDE. Many imaging modalities belong to this category, including ultrasonography, electrical impedance tomography and photoacoustic tomography. Many different PDE appear, depending on the physical domain. Currently, there is a substantial gap between theory and practice: all theoretical results require infinitely many measurements, while in all applied studies and practical implementations, only a finite number of measurements are taken. We argue that this gap is crucial, since the number of measurements is usually not very large, and has important consequences, regarding the choice of measurements, the priors on the unknown and the reconstruction algorithms. Many safe and effective modalities have had very limited use due to low reconstruction quality. Within a multidisciplinary approach, by combining methods from PDE theory, numerical analysis, signal processing, compressed sensing and machine learning, we will bridge this gap by developing a theory of sample complexity for inverse problems in PDE. This will allow for the deriving of a new mathematical theory of inverse problems for PDE under realistic assumptions, which will impact the implementation of many modalities, guiding the choice of priors and measurements. Consequently, emerging imaging modalities will become closer to actual usage. As a by-product, we will also derive new compressed sensing results which are valid for a general class of problems, including nonlinear and ill-posed, and sparsity constraints. Collaborations with experts in the relevant fields will ensure the project’s success. Fields of science engineering and technologyelectrical engineering, electronic engineering, information engineeringelectronic engineeringsignal processingcompressed sensingnatural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equationsnatural sciencescomputer and information sciencesartificial intelligencemachine learningnatural sciencesmathematicsapplied mathematicsnumerical analysisnatural sciencesmathematicsapplied mathematicsmathematical model Programme(s) HORIZON.1.1 - European Research Council (ERC) Main Programme Topic(s) ERC-2021-STG - ERC STARTING GRANTS Call for proposal ERC-2021-STG See other projects for this call Funding Scheme HORIZON-AG - HORIZON Action Grant Budget-Based Host institution UNIVERSITA DEGLI STUDI DI GENOVA Net EU contribution € 1 153 125,00 Address VIA BALBI 5 16126 Genova Italy See on map Region Nord-Ovest Liguria Genova Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 153 125,00 Beneficiaries (1) Sort alphabetically Sort by Net EU contribution Expand all Collapse all UNIVERSITA DEGLI STUDI DI GENOVA Italy Net EU contribution € 1 153 125,00 Address VIA BALBI 5 16126 Genova See on map Region Nord-Ovest Liguria Genova Activity type Higher or Secondary Education Establishments Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 1 153 125,00
