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1. Variational multi-task MRI reconstruction: Joint reconstruction, registration and super-resolution

Author

Carola-Bibiane Schönlieb, Noémie Debroux, Angelica I. Aviles-Rivero, Veronica Corona, Carole Le Guyader, Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge [UK] (CAM), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), Institut Pascal (IP), Centre National de la Recherche Scientifique (CNRS)-Université Clermont Auvergne (UCA)-Institut national polytechnique Clermont Auvergne (INP Clermont Auvergne), Université Clermont Auvergne (UCA)-Université Clermont Auvergne (UCA), Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI), Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie), Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA)-Normandie Université (NU), Corona, Veronica [0000-0003-2160-5482], and Apollo - University of Cambridge Repository

Subjects
math.NA, Image Registration, Computer science, media_common.quotation_subject, Physics::Medical Physics, Motion Correction, ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION, MRI Reconstruction, Image registration, Fidelity, CPU time, Health Informatics, Motion (physics), 030218 nuclear medicine & medical imaging, Set (abstract data type), Motion, 03 medical and health sciences, Imaging, Three-Dimensional, 0302 clinical medicine, Image Super-resolution, Image Processing, Computer-Assisted, FOS: Electrical engineering, electronic engineering, information engineering, FOS: Mathematics, [INFO.INFO-IM]Computer Science [cs]/Medical Imaging, Humans, Radiology, Nuclear Medicine and imaging, Computer vision, Mathematics - Numerical Analysis, Representation (mathematics), cs.NA, media_common, ComputingMethodologies_COMPUTERGRAPHICS, Radiological and Ultrasound Technology, business.industry, Image and Video Processing (eess.IV), Numerical Analysis (math.NA), Electrical Engineering and Systems Science - Image and Video Processing, Magnetic Resonance Imaging, Computer Graphics and Computer-Aided Design, Term (time), Task (computing), eess.IV, Computer Vision and Pattern Recognition, Artificial intelligence, business, Algorithms, 030217 neurology & neurosurgery
Abstract

Motion degradation is a central problem in Magnetic Resonance Imaging (MRI). This work addresses the problem of how to obtain higher quality, super-resolved motion-free reconstructions from highly undersampled MRI data. In this work, we present for the first time a variational multi-task framework that allows joining three relevant tasks in MRI: reconstruction, registration and super-resolution. Our framework takes a set of multiple undersampled MR acquisitions corrupted by motion into a novel multi-task optimisation model, which is composed of an L2 fidelity term that allows sharing representation between tasks, super-resolution foundations and hyperelastic deformations to model biological tissue behaviors. We demonstrate that this combination yields significant improvements over sequential models and other bi-task methods. Our results exhibit fine details and compensate for motion producing sharp and highly textured images compared to state of the art methods while keeping low CPU time. Our improvements are appraised on both clinical assessment and statistical analysis.

Published
2021
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2. A Fokker-Planck approach to the study of robustness in gene expression

Author

Pierre Degond, Sepideh Mirrahimi, Maxime Herda, Department of Mathematics [Imperial College London], Imperial College London, Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), PD acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) under grants no. EP/M006883/1 and EP/N014529/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award no. WM130048 and by the National Science Foundation (NSF) under grant no. RNMS11-07444 (KI-Net). PD is on leave from CNRS, Institut de Mathématiques de Toulouse, France. MH acknowledges support by the Labex CEMPI (ANR-11-LABX-0007-01). SM acknowledges support by the CNRS–Royal Society exchange projects 'CODYN' and 'Segregation models in social sciences' and the Chaire Modélisation Mathématique et Biodiversité of Véolia Environment - Ecole Polytechnique - Museum National d’Histoire Naturelle - Fondation X., ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), Laboratoire Paul Painlevé - UMR 8524 (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), PD acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) under grants no. EP/M006883/1 and EP/N014529/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award no. WM130048 and by the National Science Foundation (NSF) under grant no. RNMS11-07444 (KI-Net). PD is on leave from CNRS, Institut de Math\'ematiques de Toulouse, France. MH acknowledges support by the Labex CEMPI (ANR-11-LABX-0007-01). SM acknowledges support by the CNRS–Royal Society exchange projects ``CODYN' and ``Segregation models in social sciences' and the Chaire Mod\'elisation Math\'ematique et Biodiversit\'e of V\'eolia Environment - \'Ecole Polytechnique - Museum National d’Histoire Naturelle - Fondation X., The Royal Society, Engineering & Physical Science Research Council (EPSRC), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
MICRORNAS, 0904 Chemical Engineering, 02 engineering and technology, Fokker-Planck Equation, 0903 Biomedical Engineering, Joint probability distribution, 0102 Applied Mathematics, Gene expression, 0202 electrical engineering, electronic engineering, information engineering, Gene Regulatory Networks, Statistical physics, inverse gamma distributions, microRNA, Applied Mathematics, 05 social sciences, General Medicine, Numerical Analysis (math.NA), Quantitative Biology::Genomics, Computational Mathematics, Modeling and Simulation, INEQUALITIES, 020201 artificial intelligence & image processing, Fokker–Planck equation, Marginal distribution, General Agricultural and Biological Sciences, Life Sciences & Biomedicine, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Biotechnology, Analysis of PDEs (math.AP), Micro RNA, math.NA, Bioinformatics, [SDV.BC]Life Sciences [q-bio]/Cellular Biology, Mathematics - Analysis of PDEs, 0502 economics and business, Gene expression level, QA1-939, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, cs.NA, math.AP, Science & Technology, Robustness (evolution), [SDV.BBM.BM]Life Sciences [q-bio]/Biochemistry, Molecular Biology/Molecular biology, Inverse Gamma distribution, Systems modeling, BRUNN-MINKOWSKI, 35Q84, 92C40, 92D20, 35Q92, 65M08, Mathematical & Computational Biology, Fokker-Planck equations, TP248.13-248.65, Mathematics, 050203 business & management
Abstract

We study several Fokker-Planck equations arising from a stochastic chemical kinetic system modeling a gene regulatory network in biology. The densities solving the Fokker-Planck equations describe the joint distribution of the messenger RNA and micro RNA content in a cell. We provide theoretical and numerical evidences that the robustness of the gene expression is increased in the presence of micro RNA. At the mathematical level, increased robustness shows in a smaller coefficient of variation of the marginal density of the messenger RNA in the presence of micro RNA. These results follow from explicit formulas for solutions. Moreover, thanks to dimensional analyses and numerical simulations we provide qualitative insight into the role of each parameter in the model. As the increase of gene expression level comes from the underlying stochasticity in the models, we eventually discuss the choice of noise in our models and its influence on our results., Comment: Minor revisions

Published
2020
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3. Comparison of nonlinear mappings for reduced-order modelling of vibrating structures: normal form theory and quadratic manifold method with modal derivatives

Author

Loic Salles, Cyril Touzé, Alessandra Vizzaccaro, Institut des Sciences de la mécanique et Applications industrielles (IMSIA - UMR 9219), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Saclay-EDF R&D (EDF R&D), Rolls-Royce Plc, and Engineering & Physical Science Research Council (E

Subjects
FOS: Computer and information sciences, Technology, Normal form, 02 engineering and technology, 01 natural sciences, Modal derivatives, 09 Engineering, Computational Engineering, Finance, and Science (cs.CE), [SPI]Engineering Sciences [physics], Quadratic equation, Engineering, 0203 mechanical engineering, Normal mode, [NLIN]Nonlinear Sciences [physics], Arch, Computer Science - Computational Engineering, Finance, and Science, 010301 acoustics, Mathematics, cs.CE, Reduced-order modelling, Applied Mathematics, Mathematical analysis, Numerical Analysis (math.NA), Invariant (physics), Engineering, Mechanical, 020303 mechanical engineering & transports, Quadratic manifold, math.NA, Aerospace Engineering, Ocean Engineering, SPECTRAL SUBMANIFOLDS, Mechanics, CIRCULAR CYLINDRICAL-SHELLS, Continuation, SYSTEMS, NUMERICAL COMPUTATION, 0103 physical sciences, FOS: Mathematics, OSCILLATIONS, Mathematics - Numerical Analysis, Electrical and Electronic Engineering, CONTINUATION, cs.NA, 01 Mathematical Sciences, NORMAL-MODES, Science & Technology, Mechanical Engineering, [SPI.MECA.VIBR]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Vibrations [physics.class-ph], Acoustics, Nonlinear system, REDUCTION, Modal, Control and Systems Engineering, PROPER ORTHOGONAL DECOMPOSITION, [SPI.MECA.STRU]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Structural mechanics [physics.class-ph], Phase space, LARGE-AMPLITUDE VIBRATIONS
Abstract

The objective of this contribution is to compare two methods proposed recently in order to build efficient reduced-order models for geometrically nonlinear structures. The first method relies on the normal form theory that allows one to obtain a nonlinear change of coordinates for expressing the reduced-order dynamics in an invariant-based span of the phase space. The second method is the modal derivative (MD) approach, and more specifically the quadratic manifold defined in order to derive a second-order nonlinear change of coordinates. Both methods share a common point of view, willing to introduce a nonlinear mapping to better define a reduced order model that could take more properly into account the nonlinear restoring forces. However the calculation methods are different and the quadratic manifold approach has not the invariance property embedded in its definition. Modal derivatives and static modal derivatives are investigated, and their distinctive features in the treatment of the quadratic nonlinearity is underlined. Assuming a slow/fast decomposition allows understanding how the three methods tend to share equivalent properties. While they give proper estimations for flat symmetric structures having a specific shape of nonlinearities and a clear slow/fast decomposition between flexural and in-plane modes, the treatment of the quadratic nonlinearity makes the predictions different in the case of curved structures such as arches and shells. In the more general case, normal form approach appears preferable since it allows correct predictions of a number of important nonlinear features, including for example the hardening/softening behaviour, whatever the relationships between slave and master coordinates are., 43 pages, 12 figures, published in Nonlinear Dynamics

Published
2020
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4. Finite volume approximations of the Euler system with variable congestion

Author

Degond, Pierre, Minakowski, Piotr, Navoret, Laurent, Zatorska, Ewelina, Department of Mathematics [Imperial College London], Imperial College London, Interdisciplinary Center for Scientific Computing (IWR), Universität Heidelberg [Heidelberg] = Heidelberg University, TOkamaks and NUmerical Simulations (TONUS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), The Royal Society, Engineering & Physical Science Research Council (EPSRC), and Universität Heidelberg [Heidelberg]

Subjects
math.NA, singular pressure, Applied Mathematics, Numerical Analysis (math.NA), 0915 Interdisciplinary Engineering, Euler equations, free boundary, Asymptotic-Preserving scheme, 0102 Applied Mathematics, FOS: Mathematics, fluid model of crowd, Mathematics - Numerical Analysis, finite volume, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 0913 Mechanical Engineering
Abstract

We are interested in the numerical simulations of the Euler system with variable congestion encoded by a singular pressure (Degond et al., 2016). This model describes for instance the macroscopic motion of a crowd with individual congestion preferences. We propose an asymptotic preserving (AP) scheme based on a conservative formulation of the system in terms of density, momentum and density fraction. A second order accuracy version of the scheme is also presented. We validate the scheme on one-dimensionnal test-cases and compare it with a scheme previously proposed in Degond et al. (2016) and extended here to higher order accuracy. We finally carry out two dimensional numerical simulations and show that the model exhibit typical crowd dynamics.

Published
2018
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5. Preconditioned ADMM with Nonlinear Operator Constraint

Author

Florian Knoll, Martin Benning, Carola-Bibiane Schönlieb, Tuomo Valkonen, New York University [New York] (NYU), NYU System (NYU), University of Cambridge [UK] (CAM), University of Liverpool, Lorena Bociu, Jean-Antoine Désidéri, Abderrahmane Habbal, TC 7, and Apollo - University of Cambridge Repository

Subjects
Nonlinear inverse problem, math.NA, Computer science, 02 engineering and technology, 030218 nuclear medicine & medical imaging, Parallel MRI, Operator splitting, 03 medical and health sciences, 0302 clinical medicine, FOS: Mathematics, 0202 electrical engineering, electronic engineering, information engineering, Applied mathematics, Parallel magnetic resonance imaging, [INFO]Computer Science [cs], Mathematics - Numerical Analysis, Mathematics - Optimization and Control, Nonlinear inverse problems, Iterative Bregman method, math.OC, Regular polygon, Numerical Analysis (math.NA), Primal-dual, Proximal point method, Constraint (information theory), Nonlinear system, Optimization and Control (math.OC), 020201 artificial intelligence & image processing, ADMM, Gradient method, Nonlinear operators
Abstract

This is the author accepted manuscript. The final version is available from Springer via https://doi.org/10.1007/978-3-319-55795-3_10., We are presenting a modification of the well-known Alternating Direction Method of Multipliers (ADMM) algorithm with additional preconditioning that aims at solving convex optimisation problems with nonlinear operator constraints. Connections to the recently developed Nonlinear Primal-Dual Hybrid Gradient Method (NL-PDHGM) are presented, and the algorithm is demonstrated to handle the nonlinear inverse problem of parallel Magnetic Resonance Imaging (MRI)., MB, CS and TV acknowledge EPSRC grant EP/M00483X/1. FK ackowledges National Institutes of Health grant NIH P41 EB017183.

Published
2015
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