Least squares formulations for some elliptic second order problems, feedforward neural network solutions and convergence results Jerome Pousin Journal of Computational Mathematics and Data Science, 2022 Recently some neural networks have been proposed for computing approximate solutions to partial differential equations. For second order elliptic or parabolic PDEs, this is possible by using penalized Least squares formulations of PDEs. In this article, for some second order elliptic PDEs we propose a theoretical setting, and we investigate the abstract convergence results between the solution and the computed one with neural networks. These results are obtained by minimizing appropriate loss functions made of a least squares formulation of the PDE augmented with a penalization term for accounting the Dirichlet boundary conditions. More precisely, it is shown that the error has two components, one due to the neural network and one due to the way the boundary conditions are imposed (via a penalization technic). The interplay between the two errors shows that the accuracy of the neural network has to be chosen accordingly with the accuracy of the boundary conditions.
Apparent diffusion coefficient measured by diffusion MRI of moving and deforming domains Imen Mekkaoui, Jérôme Pousin, Jan Hesthaven, Jing-Rebecca Li Journal of Magnetic Resonance, 2020 The modeling of the diffusion MRI signal from moving and deforming organs such as the heart is challenging due to significant motion and deformation of the imaged medium during the signal acquisition. Recently, a mathematical formulation of the Bloch-Torrey equation, describing the complex transverse magnetization due to diffusion-encoding magnetic field gradients, was developed to account for the motion and deformation. In that work, the motivation was to cancel the effect of the motion and deformation in the MRI image and the space scale of interest spans multiple voxels. In the present work, we adapt the mathematical equation to study the diffusion MRI signal at the much smaller scale of biological cells. We start with the Bloch-Torrey equation defined on a cell that is moving and deforming and linearize the equation around the magnitude of the diffusion-encoding gradient. The result is a second order signal model in which the linear term gives the imaginary part of the diffusion MRI signal and the quadratic term gives the apparent diffusion coefficient (ADC) attributable to the biological cell. We numerically validate this model for a variety of motions and deformations.
Probabilistic atlas construction of human cardiac fiber structure in DT-MRI Feng Yang, YueMin Zhu, Lihui Wang, Marc Robini, Jerome Pousin, Patrick Clarysse International Conference on Signal Processing Proceedings ICSP, 2019 We investigate the problem of constructing statistical atlas of human cardiac fiber structure from limited number of datasets. We introduced a Parzen-Gaussian model to construct the probabilistic atlas of human cardiac fiber structure due to the fact that it can generate a general atlas using very small number of samples. Experimental results showed that atlas using Parzen-Gaussian model can avoid FA/MD collapse, which can be introduced by atlas from registered DW images and registered tensor fields.
A weighted finite element mass redistribution method for dynamic contact problems F. Dabaghi, P. Krejčí, A. Petrov, J. Pousin, Y. Renard Journal of Computational and Applied Mathematics, 2019 This paper deals with a one-dimensional wave equation being subjected to a unilateral boundary condition. An approximation of this problem combining the finite element and mass redistribution methods is proposed. The mass redistribution method is based on a redistribution of the body mass such that there is no inertia at the contact node and the mass of the contact node is redistributed on the other nodes. The convergence as well as an error estimate in time are proved. The analytical solution associated with a benchmark problem is introduced and it is compared to approximate solutions for different choices of mass redistribution. However some oscillations for the energy associated with approximate solutions obtained for the second order schemes can be observed after the impact. To overcome this difficulty, an new unconditionally stable and a very lightly dissipative scheme is proposed.
Residual-based a posteriori error estimation for contact problems approximated by Nitsche's method Franz Chouly, Mathieu Fabre, Patrick Hild, Jérôme Pousin, Yves Renard IMA Journal of Numerical Analysis, 2018 We introduce a residual-based a posteriori error estimator for contact problems in two and three dimensional linear elasticity, discretized with linear and quadratic finite elements and Nitsche's method. Efficiency and reliability of the estimator are proved under a saturation assumption. Numerical experiments illustrate the theoretical properties and the good performance of the estimator.
Assessment of the effect of tissue motion in diffusion MRI: Derivation of new apparent diffusion coefficient formula Elie Bretin, , Imen Mekkaoui, Jérôme Pousin Inverse Problems and Imaging, 2018 We investigate in this paper the diffusion magnetic resonance imaging (MRI) in deformable organs such as the living heart. The difficulty comes from the hight sensitivity of diffusion measurement to tissue motion. Commonly in literature, the diffusion MRI signal is given by the complex magnetization of water molecules described by the Bloch-Torrey equation. When dealing with deformable organs, the Bloch-Torrey equation is no longer valid. Our main contribution is then to introduce a new mathematical description of the Bloch-Torrey equation in deforming media. In particular, some numerical simulations are presented to quantify the influence of cardiac motion on the estimation of diffusion. Moreover, based on a scaling argument and on an asymptotic model for the complex magnetization, we derive a new apparent diffusion coefficient formula. Finally, some numerical experiments illustrate the potential of this new version which gives a better reconstruction of the diffusion than using the classical one.
An overview of recent results on Nitsche’s method for contact problems Franz Chouly, Mathieu Fabre, Patrick Hild, Rabii Mlika, Jérôme Pousin, Yves Renard Lecture Notes in Computational Science and Engineering, 2017 We summarize recent achievements in applying Nitsche's method to some contact and friction problems. We recall the setting of Nitsche's method in the case of unilateral contact with Tresca friction in linear elasticity. Main results of the numerical analysis are detailed: consistency, well-posedness, fully optimal convergence in $H^1(\\Omega)$-norm, residual-based a posteriori error estimation. Some numerics and some recent extensions to multi-body contact, contact in large transformations and contact in elastodynamics are presented as well.
Quantifying the effect of tissue deformation on diffusion-weighted MRI: A mathematical model and an efficient simulation framework applied to cardiac diffusion imaging Imen Mekkaoui, Kevin Moulin, Pierre Croisille, Jerome Pousin, Magalie Viallon Physics in Medicine and Biology, 2016 Cardiac motion presents a major challenge in diffusion weighted MRI, often leading to large signal losses that necessitate repeated measurements. The diffusion process in the myocardium is difficult to investigate because of the unqualified sensitivity of diffusion measurements to cardiac motion. A rigorous mathematical formalism is introduced to quantify the effect of tissue motion in diffusion imaging. The presented mathematical model, based on the Bloch-Torrey equations, takes into account deformations according to the laws of continuum mechanics. Approximating this mathematical model by using finite elements method, numerical simulations can predict the sensitivity of the diffusion signal to cardiac motion. Different diffusion encoding schemes are considered and the diffusion weighted MR signals, computed numerically, are compared to available results in literature. Our numerical model can identify the existence of two time points in the cardiac cycle, at which the diffusion is unaffected by myocardial strain and cardiac motion. Of course, these time points depend on the type of diffusion encoding scheme. Our numerical results also show that the motion sensitivity of the diffusion sequence can be reduced by using either spin echo technique with acceleration motion compensation diffusion gradients or stimulated echo acquisition mode with unipolar and bipolar diffusion gradients.
Numerical study of convergence of the mass redistribution method for elastodynamic contact problems 11th World Congress on Computational Mechanics Wccm 2014 5th European Conference on Computational Mechanics Eccm 2014 and 6th European Conference on Computational Fluid Dynamics Ecfd 2014, 2014
On a derived non linear model in image restoration Proceedings of 2013 International Conference on Industrial Engineering and Systems Management IEEE Iesm 2013, 2013
Optimal extended optical flow and statistical constraint Ispa 2009 Proceedings of the 6th International Symposium on Image and Signal Processing and Analysis, 2009
A hyperelastic deformable template for cardiac segmentation in MRI Youssef Rouchdy, Jérôme Pousin, Joël Schaerer, Patrick Clarysse Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics, 2007
Solving the transport equation subject to an affine constraint Eccomas 2004 European Congress on Computational Methods in Applied Sciences and Engineering, 2004
Identification of the diffusion coefficient for a semi-linear hyperbolic-parabolic problem with discontinuous nonlinear terms Computational Geosciences, 1998
Some elliptic second order problems and neural network solutions: Existence and error estimates J Pousin Journal of Computational and Applied Mathematics 436, 115398 , 2024 2024 Citations: 1
A mathematical model of COVID-19: Analysis and identification of parameters for better decision making O Diallo, Y Kone, C Sanogo, J Pousin Applied Mathematics 13 (2), 205-214 , 2022 2022 Citations: 8
Least squares formulations for some elliptic second order problems, feedforward neural network solutions and convergence results J Pousin Journal of Computational Mathematics and Data Science 2, 100023 , 2022 2022 Citations: 5
Nearest Neighbors Strategy, Lagrange Interpolation, and Error Estimates for a Learning Neural Network K Benmansour, J Pousin SN Computer Science 2 (1), 38 , 2021 2021
Apparent diffusion coefficient measured by diffusion MRI of moving and deforming domains I Mekkaoui, J Pousin, J Hesthaven, JR Li Journal of Magnetic Resonance 318, 106809 , 2020 2020 Citations: 4
Effective diffusion tensor measured by diffusion MRI of moving and deforming domains I Mekkaoui, J Pousin, JS Hesthaven, JR Li Academic Press Inc-Elsevier Science , 2020 2020
A weighted finite element mass redistribution method for dynamic contact problems F Dabaghi, P Krejčí, A Petrov, J Pousin, Y Renard Journal of Computational and Applied Mathematics 345, 338-356 , 2019 2019 Citations: 15
Probabilistic Atlas Construction of Human Cardiac Fiber Structure in DT-MRI F Yang, YM Zhu, L Wang, M Robini, J Pousin, P Clarysse 2018 14th IEEE International Conference on Signal Processing (ICSP), 1154-1157 , 2018 2018 Citations: 1
Residual-based a posteriori error estimation for contact problems approximated by Nitsche’s method F Chouly, M Fabre, P Hild, J Pousin, Y Renard IMA Journal of Numerical Analysis 38 (2), 921-954 , 2018 2018 Citations: 41
An overview of recent results on Nitsche’s method for contact problems F Chouly, M Fabre, P Hild, R Mlika, J Pousin, Y Renard Geometrically Unfitted Finite Element Methods and Applications: Proceedings … , 2018 2018 Citations: 124
Assessment of the effect of tissue motion in diffusion MRI: Derivation of new apparent diffusion coefficient formula E Bretin, I Mekkaoui, J Pousin 2017 Citations: 5
A new method for determining the weights in multi-criteria decision making based on ordinal ranking of criteria and lagrange multiplier A Bouhedja, J Pousin Metallurgical and Mining Industry, 22-31 , 2017 2017 Citations: 5
An improved nonlinear model for image restoration S Boujena, E El Guarmah, O Gouasnouane, J Pousin Pure and Applied Functional 2 (4), 599-623 , 2017 2017 Citations: 8
Controlling the spurious oscillations in a least squares formulation of the transport equation approximated with space-time finite element K Benmansour, E Bretin, L Piffet, J Pousin 2016
Quantifying the effect of tissue deformation on diffusion-weighted MRI: a mathematical model and an efficient simulation framework applied to cardiac diffusion imaging I Mekkaoui, K Moulin, P Croisille, J Pousin, M Viallon Physics in Medicine & Biology 61 (15), 5662-5686 , 2016 2016 Citations: 22
A robust finite element redistribution approach for elastodynamic contact problems F Dabaghi, A Petrov, J Pousin, Y Renard Applied Numerical Mathematics 103, 48-71 , 2016 2016 Citations: 20
A model of elasticity taking into account the displacement orientation in the deformation A Azzayani, S Boujena, J Pousin Br. J. Math. Comput. Sci. , 2016 2016 Citations: 1
SMAI-JCM M Fabre, J Pousin, Y Renard 2016
A fictitious domain method for frictionless contact problems in elasticity using Nitsche’s method M Fabre, J Pousin, Y Renard The SMAI journal of computational mathematics 2, 19-50 , 2016 2016 Citations: 34
Motion Estimation and Analysis P Clarysse, J Pousin Multi‐Modality Cardiac Imaging: Processing and Analysis, 65-101 , 2015 2015
MOST CITED SCHOLAR PUBLICATIONS
Blood compartmental metabolism of docosahexaenoic acid (DHA) in humans after ingestion of a single dose of [13C] DHA in phosphatidylcholine D Lemaitre-Delaunay, C Pachiaudi, M Laville, J Pousin, M Armstrong, ... Journal of lipid research 40 (10), 1867-1874 , 1999 1999 Citations: 194
Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems J Pousin, J Rappaz Numerische Mathematik 69 (2), 213-231 , 1994 1994 Citations: 153
An overview of recent results on Nitsche’s method for contact problems F Chouly, M Fabre, P Hild, R Mlika, J Pousin, Y Renard Geometrically Unfitted Finite Element Methods and Applications: Proceedings … , 2018 2018 Citations: 124
Human plasma albumin transports [13C] docosahexaenoic acid in two lipid forms to blood cells N Brossard, M Croset, S Normand, J Pousin, J Lecerf, M Laville, JL Tayot, ... Journal of lipid research 38 (8), 1571-1582 , 1997 1997 Citations: 116
A dynamic elastic model for segmentation and tracking of the heart in MR image sequences J Schaerer, C Casta, J Pousin, P Clarysse Medical Image Analysis 14 (6), 738-749 , 2010 2010 Citations: 109
Diffusion and dissolution in a reactive porous medium: Mathematical modelling and numerical simulations P Moszkowicz, J Pousin, F Sanchez Journal of Computational and Applied Mathematics 66 (1-2), 377-389 , 1996 1996 Citations: 54
Inégalité de Poincaré courbe pour le traitement variationnel de l'équation de transport P Azérad, J Pousin Comptes rendus de l'Académie des sciences. Série 1, Mathématique 322 (8 … , 1996 1996 Citations: 43
Residual-based a posteriori error estimation for contact problems approximated by Nitsche’s method F Chouly, M Fabre, P Hild, J Pousin, Y Renard IMA Journal of Numerical Analysis 38 (2), 921-954 , 2018 2018 Citations: 41
Solutions for linear conservation laws with velocity fields in O Besson, J Pousin Archive for Rational Mechanics and Analysis 186 (1), 159-175 , 2007 2007 Citations: 36
A fictitious domain method for frictionless contact problems in elasticity using Nitsche’s method M Fabre, J Pousin, Y Renard The SMAI journal of computational mathematics 2, 19-50 , 2016 2016 Citations: 34
A nonlinear elastic deformable template for soft structure segmentation: application to the heart segmentation in MRI Y Rouchdy, J Pousin, J Schaerer, P Clarysse Inverse Problems 23 (3), 1017-1035 , 2007 2007 Citations: 34
Convergence of mass redistribution method for theone-dimensional wave equation with a unilateral constraint at the boundary F Dabaghi, A Petrov, J Pousin, Y Renard ESAIM: Mathematical Modelling and Numerical Analysis 48 (4), 1147-1169 , 2014 2014 Citations: 24
Quantifying the effect of tissue deformation on diffusion-weighted MRI: a mathematical model and an efficient simulation framework applied to cardiac diffusion imaging I Mekkaoui, K Moulin, P Croisille, J Pousin, M Viallon Physics in Medicine & Biology 61 (15), 5662-5686 , 2016 2016 Citations: 22
Evaluation of the dynamic deformable elastic template model for the segmentation of the heart in MRI sequences C Casta, P Clarysse, J Schaerer, J Pousin MIDAS J-Card MR Left Ventricle Segmentation Challenge , 2009 2009 Citations: 21
A robust finite element redistribution approach for elastodynamic contact problems F Dabaghi, A Petrov, J Pousin, Y Renard Applied Numerical Mathematics 103, 48-71 , 2016 2016 Citations: 20
Bond graph formulation of an optimal control problem for linear time invariant systems W Marquis-Favre, O Mouhib, B Chereji, D Thomasset, J Pousin, M Picq Journal of the Franklin Institute 345 (4), 349-373 , 2008 2008 Citations: 19
Diffusion and dissolution/precipitation in an open porous reactive medium E Maisse, J Pousin Journal of computational and applied mathematics 82 (1-2), 279-290 , 1997 1997 Citations: 19
FEM implementation for the asymptotic partial decomposition F Fontvieille, GP Panasenko, J Pousin Applicable Analysis 86 (5), 519-536 , 2007 2007 Citations: 17
Simultaneous segmentation of the left and right heart ventricles in 3D cine MR images of small animals J Schaerer, Y Rouchdy, P Clarysse, B Hiba, P Croisille, J Pousin, ... Computers in Cardiology, 2005, 231-234 , 2005 2005 Citations: 17
Infinitely fast kinetics for dissolution and diffusion in open reactive systems J Pousin Nonlinear Analysis-Series A Theory and Methods and Series B Real World … , 2000 2000 Citations: 16
