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, Antoine Lejay, Sylvain Maire, Sebastian Niklitschek Soto, Nicolas Perrin, Denis Talay, Etienne Tanré, Denis Villemonais, Laurent Violeau.

Published works and preprints

• In collaboration with P.-E. Jabin (University of Maryland), J.-F. Jabir and J. Fontbona (CMM and Universidad de Chile, Santiago de Chile), M. Bossy 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 to an incompressible Lagrangian stochastic model in periodic domain. The paper is in positive revision for publication in Communications in Partial Differential Equations [33] . http://hal.inria.fr/hal-00691712

• 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 [13] , [14] 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 , http://hal.inria.fr/inria-00616765 . In collaboration with Mathieu Richard (Ecole Polytechnique, Palaiseau), they also extended some of these results to the case of splitting trees with mutations occuring at birth of individuals [15] , http://hal.inria.fr/hal-00736036 .

• N. Champagnat obtained with P. Diaconis (Stanford Univ.) and L. Miclo (Univ. Toulouse 3) the full spectral decomposition of the transition matrix of two-dimensional Markov chains ${(X_n,Y_n)}_{n\ge 0}$ in ${\mathbb{Z}}_{+}^2$, without immigration or mutation, which are neutral in the sense that ${(X_n+Y_n)}_{n\ge 0}$ is a Markov process. Because of the specific form of the eigenvectors, they were also able to characterize all the Dirichlet eigenvectors in subdomains of ${\mathbb{Z}}_{+}^2$ of the form $\{(i,j)\in {\mathbb{Z}}_{+}^2:i+j\ge d\}$ for all $d\ge 0$. As an application, they could determine the quasi-stationary and quasi-limiting distributions of such processes [12] , http://hal.inria.fr/hal-00672938 .

• N. Champagnat studied with F. Campillo (EPI Modemic , Inria Sophia Antipolis — Méditerrannée) individual based models of clonal plants where plants interact through the network formed by the rizhomes or stolons linking plants. In the limit of large population, they obtained a PDE governing the dynamics of population densities in space [11] , http://hal.inria.fr/hal-00723209 .

• M. Deaconu and S. Herrmann introduced a new method for the simulation of the hitting times of nonlinear boundaries for Bessel processes. This method combines the method of images and the random walk on spheres method. They construct the so called walk on moving spheres algorithm. This approach can be applied for the hitting time of a given level for the Cox-Ingersoll-Ross process and thus be used in models coming from finance and neuroscience [17] , http://hal.inria.fr/hal-00636056/en . This work is part of the ANR MANDy project.

• J. Inglis and E. Tanré studied with F. Delarue and S. Rubenthaler (Univ. Nice – Sophia Antipolis) the global solvability of a networked system of integrate-and-fire neurons proposed in the neuroscience literature. In the mean-field limit the equation resembles a McKean-Vlasov equation, but is highly non-standard and previous attempts at rigorous analysis were not satisfactory. They here bridge this gap, and shed light on a surprisingly complicated problem [35] , http://hal.inria.fr/hal-00747565 .

• A. Lejay continued his long term investigations on probabilistic interpretations and Monte Carlo simulations of interfaces conditions, such as ones arising in discontinuous media. With G. Pichot (IRISA, Rennes), he has developed a series of tests and benchmarks regarding one-dimensional Monte Carlo methods, such as the ones proposed in [19] , http://hal.inria.fr/hal-00649170 . He has also developed a new family of stochastic diffusion processes, called the snapping out Brownian motion, in order to take into account an interface condition where the concentration of the fluid is proportional to its gradient. Finally, A. Lejay and S. Maire also proposed new methods and tested a few ones to deal with the locally isotropic case for multidimensional problems [18] , http://hal.inria.fr/hal-00689581 .

• With A. Kohatsu-Higa (Ristumeikan University) and K. Yasuda (Hosei university), A. Lejay has continued his work [25] on the simulation of SDE with a discontinuous drift. http://hal.inria.fr/hal-00670123

• With L. Coutin (University of Toulouse), A. Lejay has developed an appropriate framework to deal with linear rough differential equations, extending some results (Magnus formula, Dyson series...) to this case. Using theses properties, they have studied the sensitivites of solutions of rough differential equations with respect to the signal, the vector field or the starting point. They have provided new results such as the Hölder continuity of the derivative of the so called Itô map which transforms a rough path to the solution of a rough differential equation [34] . http://hal.inria.fr/hal-00722900

• 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

• S. Maire and E. Tanré have generalised the spectral methods for elliptic PDEs developed in [42] , [43] 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 [40] . By taking into account these additional properties, they still obtained a spectral matrix having a condition number converging to one [36] . http://hal.inria.fr/hal-00677529

• C. Graham (Ecole Polytechnique) and D. Talay wrote the first volume [27] of their series of books published by Springer on the Foundations of Stochastic Simulations. They started to write the second volume.

• D. Villemonais wrote with S. Méléard (École Polytechnique) a survey on quasi-stationary distributions and $Q$-processes for stochastic models of population dynamics. This survey also contains a detailed numerical study of the behaviour of classical models with extinction [23] . http://hal.inria.fr/hal-00653834

• D. Villemonais worked on the empirical distribution of Fleming-Viot type particle systems. Using couplings with reflected diffusion processes, he proved the uniform tightness of such empirical distributions and deduced the non-degeneracy of the law of diffusion processes conditioned not to hit a boundary [39] . http://hal.inria.fr/hal-00681601

• D. Villemonais proved in [38] a general approximation method for Markov processes conditioned not be killed. The method is based on a mean field interacting particles system which is easy to simulate. The study also details the particular case of time/environment dependent diffusion processes. http://hal.archives-ouvertes.fr/hal-00598085

Other works in progress

• N. Champagnat and D. Villemonais obtained criterions for existence and uniqueness of quasi-stationary distributions and $Q$-processes for general absorbed Markov processes. A quasi-stationary distribution is a stationary distribution conditionnally on non-absorbtion, and the $Q$-process is defined as the original Markov process conditionned to never be absorbed. The criterion that they obtain also ensures exponential convergence of the conditionned $t$-marginal of the process conditionned not to be absorbed at time $t$ to the quasi-stationary distribution and the exponential ergodicity of the $Q$-process. This work is currently being written.

• N. Champagnat and D. Villemonais work on time-reversal of absorbed processes, which allow to characterize the path to extinction in extinct populations which are known to be non-extinct at some time in the past. They plan to apply these results on practical ecological situations.

• 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 finite-horizon optimal control problem on a branching-diffusion process. In addition, he is working on modelling of a pH-mediated cancer treament.

• M. Deaconu and S. Herrmann continue the study of the hitting times for Bessel processes in the situation of noninteger dimensions and also in the application of this method to the simulation of the Brownian hitting time,

• M. Deaconu starts a collaboration with L. Beznea (Simion Stoilow Institute of Mathematics of the Romanian Academy) on coagulation-fragmentation models and their connection with branching processes.

• M. Deaconu studies in collaboration with F. Nobile and F. Tesei (EPFL) a pollution model by using hitting times of stochastic processes.

• S. Herrmann and E. Tanré worked on a scheme to construct an efficient algorithm to simulate 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 neuroscience. This work is part of the ANR MANDy project.

• S. Larnier joined the team in September as a post-doctoral researcher and began working with A. Lejay on data assimilation in order to predict the ocean wave energy from the knowledge of near-shore incoming waves. They started a collaboration on video data with R. Almar (LEGOS, Toulouse) and R. Cienfuegos (Pontificia Universidad Católica de Chile).

• S. Maire works with M. Simon (Mainz Univ.) on electrical impedance tomography problems using new Monte Carlo schemes that deal with Robin and transmission boundary conditions.

• S. Maire develops with I. Dimov (Bulgarian academy of sciences) a Monte Carlo method called the walk on equations to solve linear systems of algebraic equations.

• S. Niklitschek has continued his PhD. work under the supervision of D. Talay. They were able to extended their first work in which they gave a probabilistic interpretation of a parabolic equation with discontinuous drift and proved the weak rate of convergence of the Euler method using the accurate pointwise estimates obtained for the derivatives of the solution, to the case in which both drift and diffusion coefficients are discontinuous. Both results are consistent with each other, and also with the results obtained by M. Martinez and D. Talay in [22] .

• N. Perrin continued his PhD. on stochastic methods in molecular dynamics under the supervision of M. Bossy, N. Champagnat and D. Talay. This year, he studied a stochastic interpretation of parabolic PDEs with divergence form operators involved in the Poisson-Boltzmann PDE of molecular dynamics, and the associated numerical Monte Carlo method. He also continued his study of 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.

• P. Guiraud (University of Valparaiso) and E. Tanré study the effect of noise in the phenomenon of spontaneous synchronisation in a network of full connected integrate-and-fire neurons. They detail cases in which the phenomenon of synchronization persists in a noisy environment, cases in which noise permits to accelerate synchronization, and cases in which noise permits to observe synchronization while noiseless model does not have synchronization.

• P. Orio (Centro Interdisciplinario de Neurociencia de Valparaiso) and E. Tanré work on the comparison of global properties of the solution of mathematical models and the associated measurements obtained by experiments.

• 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 (Moise team, Inria Sophia Antipolis – Méditerranée, Montpellier). Laurent studied this year the rate of convergence of the Nadaraya-Watson conditional estimator for “linear” kinetic processes. He is currently working on the rate of convergence of the particle approximation of kinetic conditional McKean-Vlasov 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.

• D. Talay is working with J. Bion-Nadal (Ecole Polytechnique) on applications of risk measures to the calibration of stochastic models, with N. Touzi (Ecole Polytechnique) on the stochastic control of stochastic differential equations with weighted local times, and with O. Bardou (GDF) on Edgeworth expansions for the Central Limit Theorem for Brownian martingales whose integrands depend on ergodic diffusion processes.