Jean-Marie Mirebeau
Jean-Marie Mirebeau
Soutenance de mon habilitation à diriger les recherches,
Schémas Numériques pour les EDPs Anisotropes sur Grille Cartésienne
Le 18 mai 2018 à 15h, Amphithéatre Jean-Christophe Yoccoz, Laboratoire de Mathématiques d’Orsay.
Preliminary HDR manuscript. Accès à l’institut de Mathématiques d’Orsay.
Winner of the 8th Popov Prize, for Outstanding Research Achievements in Approximation Theory !
Prize received at the 15th International Conference on Approximation Theory, May 2016.
Member of the ANR project MAGA, Monge Ampere and Algorithmic Geometry, led by Quentin Merigot.
Leader of the ANR project NS-LBR, for the development of Numerical Schemes using Lattice Basis Reduction. Ended in February 2017.
Main developer of the Hamiltonian-Fast-Marching C++ library. See the introductory Python notebooks, as well as the included Matlab(R), Mathematica(R) and Python examples.
Research Interests
I develop numerical schemes for Partial Differential Equations (PDEs), focusing on issues related to Anisotropy. I also investigate their applications, in particular to medical image processing.
Keywords : Anisotropic Partial Differential Equations,
Anisotropic Diffusion, Anisotropic Eikonal equations, Monge-Ampere equations, Convexity Constraint.
Anisotropic Eikonal Equations
The eikonal equation is a Partial Differential Equation (PDE) which characterizes the geodesic distance on a (Finsler or Riemannian) manifold. It has numerous applications, including medical image processing.
I have developed solvers for this PDE which are particularly fast and accurate in the context of large anisotropies. Slides, Introductory notebooks.
These methods have been used to find the optimal solutions to control problems, including the Reed-Shepp car and the Euler/Mumford elasticae, which be approximated by shortest path problems with respect to strongly anisotropic metrics. The approach allows for data-driven terms in the PDE, and is mostly applied medical image segmentation. Slides
Journal papers:
-M., Fast Marching Methods for Curvature Penalized Shortest Paths, Preprint, 2017
-M., Anisotropic Fast Marching on Cartesian Grids using Voronoi’s First Reduction of Quadratic Forms, Preprint, 2017
- R. Duits, S. Meesters, M, J. Portegies, Optimal paths for variants of the 2D and 3D Reeds-Shepp Car with Applications in Image Analysis, Preprint, 2017
-M., Efficient Fast Marching with Finsler Metrics, Numerische Math, 2014
-M., Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction, SINUM, 2014
Conference proceedings:
- M, J. Dreo, Automatic differentiation of non-holonomic fast marching for computing most threatening trajectories under sensors surveillance, GSI 2017
- D. Chen, M, L. Cohen, Finsler Geodesic Evolution Model for Region based Active Contours, BMVC 2016
- D. Chen, L. Cohen, M., A New Finsler Minimal Path Model with Curvature Penalization for Image Segmentation and Closed Contour Detection, CVPR 2016
-D. Chen, L. Cohen, M., Global Minimum for Curvature Penalized Minimal Path Method, BMVC 2015
- G. Sanguinetti, E. Bekkers, R. Duits, M. Janssen, A. Mashtakov, M., Sub-Riemannian Fast Marching in SE2, Preprint, 2015
- D. Chen, L. Cohen, M.,Vessel Extraction using Anisotropic Minimal Paths and Path Score, ICIP 2014
Open source code:
Hamiltonian Fast Marching code, for Riemannian and non-holonomic models.
Anisotropic Fast Marching in ITK, The Insight Journal, 2015. Version with a Matlab(r) interface.
Monge-Ampere Equations and the Constraint of Convexity
The Monge-Ampere equation is a fully non-linear PDE, which requires specific discretizations. I study two classes of numerical schemes, based on the general method of monotone wide stencil schemes, and the more specific approach of semi-discrete optimal transport based on weighted Voronoi diagrams, implemented in CGAL.
Slides on semi discrete optimal transport, and volume preserving maps (by Q. Merigot)
Slides on the constraint of convexity.
PDE discretizations based on wide stencil schemes:
-M., Adaptive, Anisotropic and Hierarchical cones of Discrete Convex Functions, Numerische Math, 2015
-Benamou, Collino, M., Monotone and Consistent Discretization of the Monge Ampere Operator, Math of Comp, 2015
-M., Discretization of the 3D Monge-Ampere Operator, between Wide Stencils and Power Diagrams, M2AN, 2015
Applications of semi-discrete optimal transport:
-M., Numerical resolution of Euler equations, through semi-discrete optimal transport, Proceedings des journées EDP de Roscoff 2015
-Merigot, M., Minimal geodesics along volume-preserving maps, through semi-discrete optimal transport, Preprint
Anisotropic Diffusion
Nonlinear Anisotropic Diffusion is involved in state of the art image processing methods, as well as in physical phenomena. We developed a discretization for this PDE, on grids, which combines low numerical cost and strong mathematical guarantees.
Scheme description and analysis:
-Fehrenbach, M., Sparse Non-Negative Stencils for Anisotropic Diffusion, JMIV, 2014
-M., Minimal Stencils for Monotony or Causality Preserving Discretizations of Anisotropic PDEs, Preprint
Open source code available as an Insight Toolkit module :
Anisotropic diffusion in ITK, 2015 (module description), Interactive Figures
Adaptive and Anisotropic Finite Element Approximation
Video réalisée par EADS sur mes travaux
Mesh adaption procedures for finite elements approximation allow to adapt locally the resolution, by local refinement around the places of strong variation of the function. The use of anisotropic triangles allows to improve the efficiency of the procedure by introducing long and thin triangles fitting in particular the directions of the possible curves of discontinuity.
It is therefore natural to try to characterize, an produce optimal meshes, in the sense of the compromise between approximation error and complexity of the mesh, for a given function.
In the case of smooth functions, we have derived optimal error estimates and propose a caracterisation of the corresponding meshes (Article). We have also elaborated a hierarchical algorithm that produces such anisotropic triangulations, and which optimality has been proved for certain classes of functions.
Sharp asymptotic interpolation error:
-M., Optimally Adapted Meshes for Finite Elements of Arbitrary Order and W1p Norms, Numer Math, 2011
-Y. Babenko, T. Leskevich, M., Sharp asymptotics of the L p approximation error for interpolation on block partitions, Numerische Mathematik, 2011
-M., Optimal meshes for finite elements of arbitrary order, Constructive Approximation, 2010
Greedy refinement algorithms:
-A. Cohen, N. Dyn, F. Hecht, M., Adaptive multiresolution analysis based on anisotropic triangulations, Mathematics of Computation, 2011
-A. Cohen, M., Greedy bisection generates optimally adapted triangulations, Mathematics of Computation, 2011
-A. Cohen, M., Adaptive and anisotropic piecewise polynomial approximation, Chapter of the book Multiscale, Nonlinear and Adaptive Approximation , 2009
Link with image analysis:
-A. Cohen, M., Anisotropic smoothness classes: from finite element approximation to image models, JMIV, 2010
Design of Riemannian metrics for mesh generation:
-M., The optimal aspect ratio for piecewise quadratic anisotropic finite element approximation, proceedings of the conference SampTA 2011
Other subjects in numerical analysis
-J. Bleyer, G. Carlier, V. Duval, M., G. Peyré, A Gamma-Convergence Result for the Upper Bound Limit Analysis of Plates, M2AN, 2015
-M., Non conforming vector finite elements for H (curl) intersected with H (div), Applied Mathematic Letters, 2011
Awards :
Prix de la meilleure thèse de la fondation d'entreprise EADS (2011), en Mathématiques et leurs interactions.
Prix solennel de la chancellerie des universités de Paris (2011), Perrissin-Pirasset/Schneider, en Mathématiques Fondamentales et Appliquées, pour ma thèse.
CNRS researcher in Applied Mathematics
Université Paris-Sud, CNRS, Université Paris-Saclay,
Laboratoire de Mathématiques d’Orsay, bureau 2F1,
jean-marie.mirebeau «at» math.u-psud.fr
Address :
Département de Mathématiques Bâtiment 307
Faculté des Sciences d'Orsay Université Paris-Sud
F-91405 Orsay Cedex
Determination of the optimal aspect ratio, for anisotropic mesh generation
Anisotropic refinement by bisection
(1) Geometry of numerical scheme stencils for anisotropic eikonal equations. (2) An example of front propagation.
Three kinds of consumers in a principal agent model.
(Left) Original image. (Right) Effect of anisotropic diffusion.
Receiving the 8th Popov Prize, with Albert Cohen (left, former PhD advisor), Pencho Petrushev (right, chairman of the selection comittee).