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, Émilie Soret, Denis Talay, Etienne Tanré, Milica Tomasevic, Denis Villemonais.

Published works and preprints

• M. Bossy, J. Fontbona (Universidad de Chile, Chile) and H. Olivero-Quinteros (CIMFAV, Valparaíso, Chile) analysed mathematical model for the collective behavior of a fully connected network of finitely many neurons. They obtained that the whole system synchronize, up to some error controlled by the channels noise level. The associated nonlinear McKean-Vlasov equation concentrates, as time goes to infinity, around the dynamics of a single Hodgkin-Huxley neuron with a chemical neurotransmitter channel [42].

• M. Bossy, A. Dupré, P. Drobinski, L. Violeau and C. Briard (Zephyr ENR) obtained advances in stochastic Lagrangian approach for atmospheric boundary layer simulation, on the analysis of an optimal rate of convergence for the particle approximation method, and on validation case with the simulation of a Zephyr ENR wind farm site of six turbines [36].

• M. Di Iorio (Marine Energy Research and Innovation Center, Santiago, Chile), M.  Bossy, C. Mokrani (Marine Energy Research and Innovation Center, Santiago, Chile), and A. Rousseau obtained advances in stochastic Lagrangian approaches for the simulation of hydrokinetic turbines immersed in complex topography [33], [50].

• Together with M. Andrade-Restrepo (Univ. Paris Diderot) and R. Ferrière (Univ. Arizona and École Normale Supérieure), N. Champagnat studied deterministic and stochastic spatial eco-evolutionary dynamics along environmental gradients. This work focuses on numerical and analytical analysis of the clustering phenomenon in the population, and on the patterns of invasion fronts [40].

• N. Champagnat and J. Claisse (Ecole Polytechnique) studied the ergodic and infinite horizon controls of discrete population dynamics with almost sure extinction in finite time. This can either correspond to control problems in favor of survival or of extinction, depending on the cost function. They have proved that these two problems are related to the quasi-stationary distribution of the processes controled by Markov controls [16].

• N. Champagnat and B. Henry (Univ. Lille 1) 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 [17].

• N. Champagnat and D. Villemonais solved a general conjecture on the Fleming-Viot particle systems approximating quasi-stationary distributions (QSD): in cases where several quasi-stationary distributions exist, it is expected that the stationary distribution of the Fleming-Viot processes approaches a particular QSD, called minimal QSD. They proved that this holds true for general absorbed Markov processes with soft obstacles [48].

• N. Champagnat, K. Coulibaly-Pasquier (Univ. Lorraine) and D. Villemonais obtained general criteria for existence, uniqueness and exponential convergence in total variation to QSD for multi-dimensional diffusions in a domain absorbed at its boundary [37]. These results improve and simplify the existing results and methods.

• N. Champagnat and D. Villemonais obtained contraction properties in total variation of general penalized processes, including time-inhomogeneous Markov processes with absorption and Markov processes in varying environments [20]. Their method allows to improve significantly the former results of  [62], [63].

• N. Champagnat and D. Villemonais studied with R. Schott (Univ. Lorraine) models of deadlocks in distributed systems. They use the approach developped recently by the first two authors to study quasi-stationary distributions in order to characterize and compute numerically the asymptotic behaviour of the deadlock time and the behaviour of the system before deadlock, both for discrete and for diffusion models [47].

• A. Lejay and A. Brault have followed their work on rough flow, which provides an unified framework to deal with the theory of rough paths from the points of view of flows. In particular, they have shown existence of flows even when the associated rough differential equations have multiple solutions [44], [45].

• A. Lejay and P. Pigato have provided an estimator of the diffusion and drift coefficients when they are discontinuous at a threshold. These estimators have been applied to financial data and exhibit leverage as well as mean-reversion effects on S&P 500 stocks' prices [57], [30]

• A. Lejay, L. Lenôtre and G. Pichot have proposed a new Monte Carlo method based on random exponential time steps to deal with discontinuous diffusions coefficients and drift [35], [56]

• A. Lejay, S. Haraketi and E. Haoula have shown how to construct a diffusion on the Sierpinski gasket lifted to the Heisenberg group [53].

• J. Bion-Nadal (Ecole Polytechnique) and D. Talay have pursued their work on 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 and can be expressed in terms of the solution to a stochastic control problem, which allows one to deduce a priori estimates or to obtain numerical evaluations: cf. [41]. This solution is now shown to exist and be smooth even in cases where the infinitesimal generators of the considered diffusion processes are not strongly elliptic.

  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? The objective being to select a model minimizing the above distance to a target model, the construction and analysis of an efficient stochastic algorithm are being in progress.

• In [60] D. Talay and M. Tomasevic have developed and analysed a new type of stochastic interpretation of the one-dimensional 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. In [12] M. Tomasevic has proven that the two-dimensional parabolic-parabolic Keller-Segel system in the whole Euclidean space and the corresponding McKean-Vlasov stochastic differential equation are well-posed under some explicit conditions on the parameters of the model.

• D. Talay and M. Tomasevic are studying the well-posedness and the propagation of chaos of the particle system related to the two-dimensional parabolic-parabolic Keller-Segel system. The singularity of the interaction kernel being more critical than in the one-dimensional case, the preceding analysis [26] cannot be extended and a fully new methodology needs to be developed.

• V. Martin Lac, D. Talay and M. Tomasevic have worked on theoretical and algorithmic questions related to the simulation of the Keller–Segel particle systems. A preliminary version of a library has been developed.

• H. Olivero (Inria, now University of Valparaiso, Chile) and D. Talay have constructed and analysed an hypothesis test which helps to detect when the probability distribution of complex stochastic simulations has an heavy tail and thus possibly an infinite variance. This issue is notably important when simulating particle systems with complex and singular McKean-Vlasov interaction kernels whick make it extremely difficult to get a priori estimates on the probability laws of the mean-field limit, the related particle system, and their numerical approximations. In such situations the standard limit theorems do not lead to effective tests. In the simple case of independent and identically distributed sequences the procedure developed this year and its convergence analysis are based on deep tools coming from the statistics of semimartingales.

• V. Martin Lac, H. Olivero-Quinteros and D. Talay have worked on theoretical and algorithmic questions related to the simulation of large particle systems under singular interactions and to critical numerical issues related to the simulation of independent random variables with heavy tails. A preliminary version of a library has been developed.

• 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 have ended their work on a mean-field game and shown the convergence of the joint density function of the controlled particle system. The construction of the limit has required the construction of suitable Sobolev spaces on sets of probability measures on Polish spaces.

• E. Tanré and Pierre Guiraud (Univ. of Valparaíso) have worked on the synchronization in a model of network of noisy biological neurons. Using a large deviation principle, they prove the stability of the synchronized state under stochastic perturbations. They also give a lower bound on the probability of synchronization for networks which are not initially synchronized. This bound shows the robustness of the emergence of synchronization in presence of small stochastic perturbations [25].

• E. Tanré, P. Grazieschi (Univ. Warwick), M. Leocata (Univ. Pisa), C. Mascart (Univ. Côte d'Azur), J. Chevallier (Univ. of Grenoble) and F. Delarue (Univ. Côte d'Azur) have extended the previous work [9] to sparse networks of interacting neurons. They have obtained a precise description of the limit behavior of the mean field limit according to the probability of (random) interactions between two individual LIF neurons [52].

• E. Tanré has worked with Nicolas Fournier (Sorbonne Université) 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 membrane 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. They prove the existence and uniqueness of a heuristically derived mean-field limit of the system when $n\to \infty$ [51].

• 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 [31].

• Q. Cormier and E. Tanré studied with Romain Veltz (team MathNeuro ) the long time behavior of a McKean-Vlasov SDE modeling a large assembly of neurons. A convergence to the unique (in this case) invariant measure is obtained assuming that the interactions between the neurons are weak enough. The key quantity in this model is the “firing rate”: it gives the average number of jumps per unit of times of the solution of the SDE. They derive a non-linear Voltera equation satisfied by this rate. They used methods from integral equation to control finely the long time behavior of this firing rate [49].

• D. Villemonais collaborates with the Gerontology Service of CHRU Nancy on statistics of time evolution of telomere lengths in human blood cells. This is a collaboration with Anne Gégout Petit (IECL, Inria Bigs ), Simon Toupance (CHRU Nancy), Eliane Albuisson (CHRU Nancy), Athanasios Benetos (CHRU Nancy), Daphnée Germain (Ecole des Mines de Nancy). They proposed in [32] a telomeric signature for human beings, stable along age evolution. Lionel Lenôtre works as a post-doc on this topic within the project GEENAGE of LUE.

• D. Villemonais studied with C. Coron (Univ. Paris-Saclay) and S. Méléard (École Polytechnique) the extinction probability before fixation for multi-dimensional models of Wright-Fisher type with mutations [21].

• In collaboration with E. Horton and A. Kyprianou (University of Bath), D. Villemonais studied the large-time asymptotic behaviour of the neutron transport equation in a three-dimensional domain [55]. This work is motivated by the simulation of the flow of particles in a nuclear tank.

• D. Villemonais studied with C. Mailler (University of Bath) the asymptotic behaviour of generalized measure-valued Polya urn models taking values in non-compact sets, using techniques from the theory of stochastic algorithms [58].

Other works in progress

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

• N. Champagnat and D. Villemonais are working with M. Benaïm (Univ. Neuchatel) on the convergence of stochastic algorithms to the quasi-stationary distribution of diffusion processes absorbed at the boundary of a domain.

• N. Champagnat is working with S. Méléard (École Polytechnique) and C. Tran Viet (Univ. Lille 1) on evolutionary models of bacteria with horizontal transfer. They study a scaling of parameters taking into account the influence of negligible but non-extinct populations, allowing to study specific phenomena observed in these models (re-emergence of traits, cyclic evolutionary dynamics and evolutionary suicide).

• Q. Cormier is investigating new methods to explore the long time behavior of the McKean-Vlasov SDE of [49], to go beyond the weak interactions case. The long time behavior of such McKean-Vlasov equations can be intricate as there can be multiple invariant measures or stable oscillations of the law of the process. The objective of this work is to develop (numerical and theoretical) methods to check the local stability of a given invariant measure of this non-linear SDE.

• C. Fritsch is working with A. Gégout-Petit (Univ. Lorraine and EPI Bigs ), B. Marçais (INRA, Nancy) and M. Grosdidier (INRA, Avignon) on a statistical analysis of a Chalara Fraxinea model [34].

• C. Fritsch is working with Marianne Clausel (Univ. Lorraine) and Julien Trombini (Two-I) on the modeling of emotions spreading in a crowd.

• A. Lejay and A. Brault (U. Paris Descartes) continue their work to extend the framework of rough flows.

• O. Faugeras (MathNeuro Inria Research Team), É. Soret (joint postdoc with MathNeuro Inria Research Team) and É. Tanré are working on Mean-Field description of thermodynamics limits of large population of neurons with random interactions. They study the asymptotic behaviour for an asymmetric neuronal dynamics in a network of linear Hopfield neurons. They obtain the convergence in law of each component to a Gaussian process. The limit object is not a Markov process.

• P. Helson, E. Tanré and R. Veltz (MathNeuro Inria team), are working on a neural network model of memory. The aim is to propose a new retrieval criterion and its mathematical analysis.

• E. Tanré has worked 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 where 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.