Section: New Results

Probabilistic numerical methods, stochastic modelling and applications

Participants : Mireille Bossy, Nicolas Champagnat, Quentin Cormier, Madalina Deaconu, Olivier Faugeras, Coralie Fritsch, Pascal Helson, Antoine Lejay, Radu Maftei, Victor Martin Lac, Hector Olivero-Quinteros, Paolo Pigato, Denis Talay, Etienne Tanré, Milica Tomašević, Denis Villemonais.

Published works and preprints

• M. Bossy, R. Maftei and Jean-François Jabir (National Research University Higher School of Economics, Moscow) propose and analyze the convergence of a time-discretization scheme for the motion of a particle when its instantaneous velocity is drifted by the known velocity of the carrying flow, and when the motion is taking into account the collision event with a boundary wall. We propose a symetrized version of the Euler scheme and prove a convergence of order one for the weak error. The regularity analysis of the associated Kolmogorov PDE is obtained by mixed variational and stochastic flow techniques for PDE problem with specular condition [46].

• N. Champagnat and B. Henry (IECL) studied a probabilistic approach for the Hamilton-Jacobi limit of non-local reaction-diffusion models of adaptive dynamics when mutations are small. They used a Feynman-Kac interpretation of the partial differential equation and large deviation estimates to obtain a variational characterization of the limit. They also studied in detail the case of finite phenotype space with exponentially rare mutations, where they were able to obtain uniqueness of the limit [48].

• N. Champagnat and P.-E. Jabin (Univ. Maryland) completed the study of the functional spaces in the article [18], devoted to the study of strong existence and pathwise uniqueness for stochastic differential equations (SDE) with rough coefficients, typically in Sobolev spaces.

• N. Champagnat and D. Villemonais consider, for general absorbed Markov processes, the notion of quasi-stationary distributions (QSD), which is a stationary distribution conditionally on non-absorbtion, and the associated $Q$-process, defined as the original Markov process conditioned to never be absorbed. They prove that, under the conditions of [5], in addition to the uniform exponential convergence of the conditional distributions to a unique QSD and the uniform exponential ergodicity of the $Q$-process, one also has the uniform convergence of the law of the process contionned to survival up to time $T$, when $T\to +\infty$. This allows them to obtain conditional ergodic theorems [22].

• N. Champagnat and D. Villemonais obtained criteria based on Lyapunov functions allowing to check the conditions of [5] which characterize the exponential uniform convergence in total variation of conditional distributions of an absorbed Markov process to a unique quasi-stationary distribution [50]. Among the various applications they give, they prove that these conditions apply to any logistic Feller diffusions in any dimension conditionned to the non extinction of all its coordinates. This question was left partly open since the first work of Cattiaux and Méléard on this topic  [64].

• N. Champagnat and D. Villemonais obtained general conditions based on Foster-Lyapunov criteria ensuring the exponential convergence in total variation of the conditional distributions of an absorbed Markov process to a quasi-stationary distribution (QSD), with a speed that can depend on the initial distribution. In particular, these results provide a non-trivial subset of the domain of attraction of the minimal QSD of an absorbed process in cases where there is not uniqueness of a QSD. Similar results were only known for the very specific branching models. They also show how these criteria can be checked for a wide range of Markov processes in discrete or continuous time and in discrete or continuous state spaces. In all these cases, they improve significantly the best known results. A particularly remarkable result is the existence of a principal eigenfunction for the generator of elliptic diffusion processes aborbed at the boundary of an open domain without any regularity assumption on the boundary of this domain [49].

• During his internship supervised by E. Tanré and R. Veltz (MathNeuro Inria team), Q. Cormier studied numerically and theoretically a model of spiking neurons in interactions [51]. This model generalizes classical integrate and fire models: the neurons no more spike after hitting a deterministic threshold but spikes with a rate given as a function of the membrane potential (see e.g. [65]). He showed existence and uniqueness of the corresponding limit equation, and was able to extend those results in the case of excitatory and inhibitory neurons. He is now studying the long time behavior of the model, as part of his thesis.

• M. Deaconu and S. Herrmann studied the simulation of the hitting time of some given boundaries for Bessel processes. These problems are of great interest in many application fields as finance and neurosciences. More precisely they obtained recently a new method for the simulation of hitting times for Bessel processes with a non integer dimension. The main idea is to consider the simulation of the hitting time of Bessel processes with integer dimension and provide a new algorithm by using the additivity property of the laws of squared Bessel processes [26].

• M. Deaconu and S. Herrmann studied the Initial-Boundary Value Problem for the heat equation [25]. They construct an algorithm based on a random walk on heat balls in order to approximate the solution. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to solve the Dirichlet problem for Laplace's equation, its implementation is rather easy. The definition of the random walk is based on a particular mean value formula for the heat equation and they obtained also a probabilistic formulation of this formula. They proved convergence results for this algorithm and illustrate them by numerical examples.

• M. Deaconu, S. Herrmann and S. Maire [27] introduced a new method for the simulation of the exit time and position of a $\delta$-dimensional Brownian motion from a domain. The main interest of this method is that it avoids splitting time schemes as well as inversion of complicated series. The idea is to use the connexion between the $\delta$-dimensional Bessel process and the $\delta$-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time.

• M. Deaconu, B. Dumortier and E. Vincent (EPI Multispeech) are working with the Venathec SAS on the acoustic control of wind farms. Wind turbine noise is often annoying for humans living in close proximity to a wind farm. Reliably estimating the intensity of wind turbine noise is a necessary step towards quantifying and reducing annoyance, but it is challenging because of the overlap with background noise sources. Current approaches involve measurements with on/off turbine cycles and acoustic simulations, which are expensive and unreliable. This raises the problem of separating the noise of wind turbines from that of background noise sources and coping with the uncertainties associated with the source separation output. In their work they propose to assist a black-box source separation system with a model of wind turbine noise emission and propagation in a recursive Bayesian estimation framework. This new approach is validated on real data with simulated uncertainties using different nonlinear Kalman filters [38].

• M. Deaconu is working with L. Beznea and O. Lupaşcu (Bucharest, Romania) on the stochastic interpretation of rupture phenomena. They constructed a stochastic differential equation and a branching process for the fragmentation model. The main physical model involved in their study is the avalanche one and their model includes physical properties of the phenomenon. They introduced a new numerical algorithm issued from this study, which captures the fractal property of the avalanche [43].

• C. Fritsch, F. Campillo (Inria Sophia-Antipolis, MATHNEURO team) and O. Ovaskainen (Univ. Helsinki) proposed a numerical approach to determine mutant invasion fitness and evolutionary singular strategies using branching processes and integro-differential models in [31]. They illustrate this method with a mass-structured individual-based chemostat model.

• P. Helson, E. Tanré and R. Veltz (MathNeuro Inria team), have numerically and theoretically studied a model of spiking neurons in interaction with stochastic plasticity. A slow-fast analysis enabled to split the dynamic in two inhomogeneous Markov chains: one models the slow variable, the other one the fast variable. The jump rates of the slow chain is governed by the invariant distribution of the fast one. In his PhD thesis, P. Helson has proved existence and uniqueness of solution. Simple conditions for the slow variable to be recurrent and transient are given [53].

• A. Lejay, L. Lenôtre (CMAP, École Polytechnique) and G. Pichot (Inria Paris, SERENA team) have continued their work on the simulation of processes on discontinuous media [56]. A new Monte Carlo scheme, called the exponential timestepping scheme and based on closed form expression of the resolvent, is being studied.

• A. Lejay, E. Mordecki (U. de la República, Uruguay) and S. Torres (U. de Valparaíso, Chile) have continuous their work on the estimation of the parameter of the Skew Brownian motion [57].

• A. Lejay and P. Pigato have studied the estimation of the parameter of the Oscillating Brownian motion, which is a solution of a stochatic differential equation whose diffusivity takes two values [35].

• A. Lejay have given an alternative proof of the Girsanov theorem which is based on semigroups [39].

• In [61] D. Talay and M. Tomašević propose a new type of stochastic interpretation of the parabolic-parabolic Keller-Segel systems. It involves an original type of McKean-Vlasov interaction kernel. At the particle level, each particle interacts with all the past of each other particle. At the mean-field level studied here, the McKean-Vlasov limit process interacts with all the past time marginals of its probability distribution. They prove that the one-dimensional parabolic-parabolic Keller-Segel system in the whole Euclidean space and the corresponding McKean-Vlasov stochastic differential equation are well-posed for any values of the parameters of the model.

• In collaboration with Jean-François Jabir (National Research University Higher School of Economics, Moscow) D. Talay and M. Tomašević prove the well–posedness of an original singularly interacting stochastic particle system associated to the one-dimensional parabolic-parabolic Keller-Segel model. They also establish the propagation of chaos towards this model [55].

• In [44] J. Bion-Nadal (Ecole Polytechnique) and D. Talay have introduced a Wasserstein-type distance on the set of the probability distributions of strong solutions to stochastic differential equations. This new distance is defined by restricting the set of possible coupling measures. They proved that it may also be defined by means of the value function of a stochastic control problem whose Hamilton–Jacobi–Bellman equation has a smooth solution, which allows one to deduce a priori estimates or to obtain numerical evaluations. They have exhibited an optimal coupling measure and characterized it as a weak solution to an explicit stochastic differential equation, and they finally have described procedures to approximate this optimal coupling measure.

  A notable application concerns the following modeling issue: given an exact diffusion model, how to select a simplified diffusion model within a class of admissible models under the constraint that the probability distribution of the exact model is preserved as much as possible?

• E. Tanré has worked with Patricio Orio (CINV, Chile) and Alexandre Richard (Centrale-Supelec) on the modelling and measurement of long-range dependence in neuronal spike trains. They exhibit evidence of memory effect in genuine neuronal data and compared a fractional integrate-and-fire model with the existing Markovian models (paper in revision: [60]).

• D. Villemonais worked with his Research Project student William Oçafrain (École des Mines de Nancy) on an original mean-field particle system [36]. They proved that the mean-field particle system converges in full generality toward the distribution of a conditioned Markov process, with applications to the approximation of the quasi-stationary distribution of piecewise deterministic Markov processes.

• D. Villemonais, Camille Coron (Université Paris XI) and Sylvie Méléard (École Polytechnique) proved a criterion for the integrability of paths of one-dimensional diffusion processes in [52] from which we derive new insights on allelic fixation in several situations.

• D. Villemonais obtains a lower bound for the coarse Ricci curvature of continuous time pure jump Markov processes in [62], with an emphasis on interacting particle systems. In this preprint, several models are studied, with a detailed study of the herd behavior of a simple model of interacting agents. The lower bound is shown to be sharp for birth and death processes.

Other works in progress

• M. Bossy, J. Fontbona (Universidad de Chile, Chile) and H. Olivero-Quinteros are working in a model for a network of neurons interacting electrically and chemically in a mean field fashion. They have proved the synchronization of the network under suitable values for the parameters of the model and a concentration result for the mean field limit.

• N. Champagnat is working with P. Vallois (IECL and Inria Bigs team) and L. Vallat (CHRU Strasbourg) on the inference of dynamical gene networks from RNAseq and proteome data.

• N. Champagnat, C. Fritsch and S. Billiard (Univ. Lille) are working on food web modeling.

• N. Champagnat, C. Fritsch and D. Villemonais are working with A. Gégout-Petit, P. Vallois, A. Mueller-Gueudin (IECL and Inria Bigs team), A. Kurtzmann (IECL), A. Harlé, J.-L. Merlin (ICL and CRAN) and E. Pencreac'h (CHRU Strasbourg) within an ITMO Cancer project on modeling and parametric estimation of dynamical models of circulating tumor DNA (ctDNA) of tumor cells, divided into resistant and sensitive ctDNA depending on whether they hold mutations known to provide resistance to a given targeted therapy or not. The goal of the project is to predict sooner and more accurately the emergence of resistance to the targeted therapy in a patient's tumor, so that the patient's therapy can be modulated more efficiently.

• M. Deaconu and S. Herrmann are working on numerical approaches for hitting times of some general stochastic differential equations.

• M. Deaconu, O. Lupaşcu and L. Beznea (Bucharest, Romania) are working on the connexion between branching processes and partial differential equations in fluid mechanics.

• M. Deaconu, B. Dumortier and E. Vincent (EPI Multispeech) are working on handling uncertainties in the wind farms model in order to design a stochastic algorithm.

• M. Deaconu and R. Stoica (Université de Lorraine, Nancy) are working on the ABC Shadow algorithm and its possible generalizations.

• O. Faugeras, E. Soret and E. Tanré are working on Mean-Field descriptions or thermodynamics limits of large populations of neurons. They study a system of EDS which describes the evolution of membrane potential of each neuron over the time when the synaptic weights are random variables (not assumed to be independent).

• O. Faugeras, James Maclaurin (Univ. of Utah) and E. Tanré have worked on the asymptotic behavior of a model of neurons in interaction with correlated gaussian synaptic weights. They have obtained the limit equation as a singular non-linear SDE and a Large Deviation Principle for the law of the finite network.

• C. Fritsch is working with A. Gégout-Petit (Univ. Lorraine and sc Bigs team), B. Marçais (INRA, Nancy) and M. Grosdidier (INRA, Nancy) on a statistical analysis of a Chalara fraxinea model.

• P. Helson, E. Tanré and R. Veltz (MathNeuro Inria team) are working on a mathematical framework for plasticity models. The aim is to propose a `optimized' model of memory capacity and memory lifetime.

• A. Lejay, A. Brault (Univ. Toulouse) and L. Coutin (Univ. Toulouse) are working on a non linear generalization of the sewing lemma, which is the main technical tool in the theory of rough paths.

• V. Martin Lac, H. Olivero-Quinteros and D. Talay are working on theoretical and algorithmic questions related to the simulation of large particle systems under singular interactions and to the simulation of independent random variables with heavy tails.

• C. Graham (École Polytechnique) and D. Talay are ending and polishing the second volume of their series on Mathematical Foundation of Stochastic Simulation to be published by Springer.

• P-E. Jabin (University of Maryland) and D. Talay are working on a mean-field game and developing a new technique to analyse it.

• E. Tanré is working with Nicolas Fournier (Univ. Pierre et Marie Curie, Paris 6) and Romain Veltz (MathNeuro Inria team) on a network of spiking networks with propagation of spikes along the dendrites. Consider a large number $n$ of neurons randomly connected. When a neuron spikes at some rate depending on its electric potential, its potential is set to a minimum value $v_{min}$, and this makes start, after a small delay, two fronts on the dendrites of all the neurons to which it is connected. Fronts move at constant speed. When two fronts (on the dendrite of the same neuron) collide, they annihilate. When a front hits the soma of a neuron, its potential is increased by a small value $w_n$. Between jumps, the potentials of the neurons are assumed to drift in $[v_{min},\infty )$, according to some well-posed ODE. We prove the existence and uniqueness of a heuristically derived mean-field limit of the system when $n\to \infty$.

• E. Tanré is working with Alexandre Richard (Centrale-Supelec) and Soledad Torres (Universidad de Valparaíso, Chile) on a one-dimensional fractional SDE reflected on the line. The existence and uniqueness of this process is known in the case of the Hurst parameter $H$ of the noise (fBM) is larger than $0.5$. They have proved the existence of a penalization scheme (suited to numerical approximation) to approach this object. When $H\in (\frac{1}{4},\frac{1}{2})$, they have proved the existence in the elliptic.

• D. Villemonais works in collaboration with Éliane Albuisson (CHRU of Nancy), Athanase Benetos (CHRU of Nancy), Simon Toupance (CHRU of Nancy), Daphné Germain (École des Mines de Nancy), Anne Gégout-Petit (Inria Bigs team) and Sylvain Chabanet (École des Mines de Nancy). The aim of this collaboration is to conduct a statistical study of the time evolution of telomere's length in human cells.

• D. Villemonais started a collaboration with Cécile Mailler (University of Bath) with the aim of studying the almost sure convergence of measure valued Pólya urns models.