Section: New Results

Probabilistic numerical methods, stochastic modelling and applications

Participants : Mireille Bossy, Nicolas Champagnat, Julia Charrier, Julien Claisse, Madalina Deaconu, Samuel Herrmann, James Inglis, Pierre-Emmanuel Jabin, Antoine Lejay, Sylvain Maire, Sebastian Niklitschek Soto, Nicolas Perrin, Denis Talay, Etienne Tanré, Laurent Violeau.

Published works and preprints

• M. Bossy in collaboration with J.-F. Jabir (Univ. Chile) proved the well posedness of the confined Lagrangian models, in association with no-permeability boundary conditions.

  When the confining domain is a hyperplane, they proved the strong existence of the trace of the density of particles following the kinetic stochastic equation of a simplified McKean Vlasov Lagrangian model in [12] , http://hal.inria.fr/inria-00515481/en .

  When the confining domain $𝒟$ is bounded with smooth boundary, they constructed a confined primitive of Brownian motion in $𝒟$ and characterized the solution to the corresponding martingale problem by showing that the time marginal density is the unique solution to a mild equation with specular condition. This key step allowed them to finish the construction in the non linear case, using previous work on Vlasov-Fokker-Plank PDE with specular boundary condition. Two papers are being written.

• In collaboration with J.-F. Jabir and J. Fontbona (CMM and Universidad de Chile, Santiago de Chile), M. Bossy and P.-E. Jabin have studied the link between the Lagrangian version of divergence free constraint (and the uniform density constraint), with an additional potential term, in the Lagrangian equation, having some similarity with the role of the Eulerian pressure term. They obtained the local existence of analytical solutions for an incompressible Lagrangian stochastic model in periodic domain. An article is currently being writen.

• N. Champagnat worked with A. Lambert (Univ. Paris 6) on splitting trees with Poissonian mutations. Assuming that each mutation is neutral and gives a new type in the population, they obtained in [15] explicit expressions for the expected number of types carried by a fixed number of individuals living in the population at time $t$. In [31] , they also obtained large time convergence results on the sizes of the largest families and the ages of the oldest families in the population. http://hal.inria.fr/inria-00515481/en , http://hal.inria.fr/inria-00616765/en .

• N. Champagnat and P.-E. Jabin studied the limit of some population dynamics models under the assumption that the time scale for mutations is much larger than the time scale for reproduction. They are able to provide the first full characterization of the corresponding limit equation [14] , http://hal.inria.fr/inria-00488979/en .

• M. Deaconu and S. Herrmann developed a new method for the simulation of the hitting time of nonlinear boundaries for Bessel processes. This method is based on a walk on moving spheres algorithm and can be applied for the hitting time of a given level for the Cox-Ingersoll-Ross process [32] , http://hal.inria.fr/hal-00636056/en . This work is part of the ANR MANDy project.

• S. Herrmann and E. Tanré worked on a scheme to construct an efficient algorithm to similate the first hitting time of curves by a one dimensional Brownian motion. They apply the result to estimate the spiking time of leaky integrate fire models in neurosciences. This work is part of the ANR MANDy project.

• P.-E. Jabin and F. Ben Belgacem (Univ. of Monastir, Tunisia) have studied a new class of models which have seen considerable development in applications for biosciences (flocking, chemotaxis, pedestrian flows...). These models include some non linear corrections to classical linear continuity equations. In [30] , they introduce new, critical regularity estimates to obtain well posedness. http://www2.cscamm.umd.edu/~jabin/transportlcs2.pdf .

• P.-E. Jabin and M. Hauray (Aix-Marseille Université) have studied the mean field limit for systems of many interacting particles. It is the only result able to deal with singular forces and physically realistic initial configurations [33] , http://hal.inria.fr/hal-00609453/en .

• P.-E. Jabin and A. Nouri (Aix-Marseille Université) studied a highly singular kinetic equation in dimension 1. This equation is obtained as a quasi-neutral limit in plasma physics. In [18] , they were able to prove well posedness in short time of analytic solutions. http://dx.doi.org/10.1016/j.crma.2011.03.024 .

• P.-E. Jabin and G. Raoul (Cambridge University) prove the convergence to a unique stable equilibrium for a wide class of competitive models in population dynamics [19] , http://dx.doi.org/10.1007/s00285-010-0370-8 .

• P.-E. Jabin and J. Calvo (Universidad de Granada) investigate the long time asymptotics of a new class of models for interacting particles inspired from various phenomena in the biosciences. In this model, when two particles collide they may coalesce and then completely stop moving [13] , http://hal.inria.fr/hal-00601969/en .

• In collaboration with G. Pichot (INRIA Rennes Bretagne Atlantique), A. Lejay has developed a new Monte Carlo methods for discontinuous media that relies on the simulation of the Skew Brownian motion [22] , [35] , http://hal.inria.fr/hal-00642194/en , http://hal.inria.fr/hal-00649170/en .

• A. Lejay developped a new method for the simulation of a stochastic process in a layered media using the properties of the Brownian path [20] , http://hal.inria.fr/inria-00583127/en .

• S. Maire and C. Prissette (Univ. du Sud – Toulon – Var) have developed in [21] a stochastic algorithm to solve Sudoku puzzles using estimation of distribution coupled with restart techniques. http://hal.inria.fr/inria-00591852_v1/

• S. Maire and E. Tanré have generalised the spectral methods for elliptic PDEs developed in [39] , [40] to the case of pure Neumann boundary conditions. Some additional difficulties occur because the stochastic representation of the solutions is defined only up to an additive constant and as a limit involving local time approximations [38] . By taking into account these additional properties, they still obtained a spectral matrix having a condition number converging to one.

• D. Talay and E. Tanré, in collaboration with F. Delarue and S. Rubenthaler (Univ. Nice – Sophia Antipolis), have given a precise approximation of the interspike intervals for the LIF model, describing the activity of a single neuron. This work is part of the ANR MANDy project (see Section  7.1.1 ).

• D. Talay, in collaboration with M. Martinez (Univ. Paris-Est), achieved to develop their stochastic approach for one-dimensional transmission parabolic problems. Owing to their stochastic representation of the solutions, they obtained accurate pointwise estimates for the derivatives of these solutions, from which they got accurate convergence rate estimates in the weak sense for a numerically effective discretization scheme of stochastic differential equations with weighted local times which are related to elliptic partial differential operators under divergence form with a discontinuous coefficient [36] , http://hal.inria.fr/inria-00607967/en .

Other works in progress

• N. Champagnat studies in collaboration with S. Méléard (Ecole Polytechnique, Palaiseau) adaptive dynamics and evolutionary branching in individual-based models of populations competing for resources, similar to those involved in chemostat systems of ODEs.

• N. Champagnat studies in collaboration with A. Lambert the process of the time to the most recent common ancestor in a family of subcritical branching processes whose genealogy is given by splitting trees.

• J. Charrier joined the team in September as a post-doctoral researcher and began working with M. Bossy and D. Talay on the long time behaviour of stochastic particules systems in McKean-Vlasov interaction.

• J. Claisse continued his PhD. under the supervision of N. Champagnat and D. Talay on stochastic control of population dynamics. He completed a finite-horizon and an infinite-horizon optimal control problem on a birth-death process. He is currently working on a birth-death process whose parameters depend on a controlled ordinary differential equation. In addition, he is working on applications of branching processes in biology and optimal control theory, and more specifically in cancer therapy.

• M. Deaconu and S. Herrmann continue the study of the hitting time for Bessel processes in the situation of noninteger dimensions.

• J. Inglis joined the team in October 2011 as a post-doctoral researcher (ANR MANDy), and began working with E. Tanré, D. Talay, F. Delarue (University of Nice) and S. Rubenthaler (University of Nice) on problems related to the rigorous justification of mean field models used in neuroscience.

• J. Inglis, E. Tanré and M. Tejo (PUC, Chile) started a collaboration on the numerical simulation of spiking times of neurons described by some new stochastic models related to the Hodgkin-Huxley equation. This work is a part of Anestoch associated team.

• A. Lejay and S. Maire study some new Monte Carlo methods for multi-dimensional discontinuous media.

• In collaboration with J.-R. Li (INRIA Rocquencourt & Neurospin), A. Lejay studies some probabilistic representation for interface condition arising in diffusion Magnetic Resonance Imaging.

• In collaboration with G. Pichot and J. Erhel (INRIA Rennes Bretagne Atlantique), A. Lejay studies Monte Carlo methods for discontinuous media as well as benchmarks and test on existing methods.

• With L. Coutin (Univ. Toulouse), A. Lejay studies some perturbation results for solutions of Rough Differential Equations.

• S. Maire develops with C. de Luigi (Univ. du Sud – Toulon – Var) and Jerôme Lelong (IMAG, Grenoble) resolution algorithms for the price of various european options in high dimension by coupling an adaptive deterministic integration algorithm and Principal Component Analysis tools.

• S. Niklitschek continued his PhD. under the supervision of D. Talay on discretized stochastic differential equations related to one-dimensional partial differential equations of parabolic type involving a discontinuous drift coefficient. He obtained accurate pointwise estimates for the derivatives of these solutions, from which he gets convergence rate estimates in the weak sense of the stochastic discretization scheme. Now he is working on the extension of these results to the multi-dimensional setup.

• N. Perrin continued his PhD. on stochastic methods in molecular dynamics under the supervision of M. Bossy, N. Champagnat and D. Talay. He is studying a method due to P. Malliavin (French Academy of Science) based on the Fourier analysis of covariance matrices with delay in order to identify the fast and slow components of a molecular dynamics and to construct simplified projected dynamics. He also studied probabilistic interpretation of the nonlinear Poisson-Boltzmann equation in Molecular Dynamics with BSDEs [37] , http://hal.inria.fr/hal-00648180/en .

• L. Violeau continued his PhD. on Stochastic Lagrangian Models and Applications to Downscaling in Fluid Dynamics under the supervision of M. Bossy and A. Rousseau (Mere team, INRIA Sophia Antipolis – Méditerranée, Montpellier). He studied the convergence in law of a sequence of penalized processes to the so called reflected langevin process in a convex domain. He is currently working on the rate of convergence of the particle approximation of conditional McKean stochastic models.

• P-E. Jabin and D. Talay continue to develop their innovating approach, which combines stochastic analysis and PDE analysis, for the time varying Hamilton-Jacobi-Bellman-McKean-Vlasov equations of the Lasry and Lions mean-field stochastic control theory.