Section: New Results

Probabilistic numerical methods, stochastic modelling and applications

Participants : Mireille Bossy, Nicolas Champagnat, Madalina Deaconu, Coralie Fritsch, Pascal Helson, Benoît Henry, Kouadio Jean Claude Kouaho, Antoine Lejay, Radu Maftei, Sylvain Maire, Paolo Pigato, Alexandre Richard, Denis Talay, Etienne Tanré, Milica Tomasevic, Denis Villemonais.

• M. Bossy with H. Quinteros (UChile) studied the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of one dimensional CEV type processes. They prove a strong rate of convergence of order one, recovering the classical result of Milstein for SDEs with smooth diffusion coefficient. In contrast with other recent results, the proof does not relies on Lamperti transformation, and it can be applied to a wide class of drift functions. Some numerical experiments and comparison with various other schemes complement the theoretical analysis that also applies for the simple projected Milstein scheme with same convergence rate ([14] accepted for publication in Bernoulli Journal).

• M. Bossy, R. Maftei, J.-P. Minier and C. Profeta worked on numerically determining the rate of convergence of the weak error for the discretised Langevin system with specular reflection conditions. The article [29] presents a discretisation scheme and offers a conjecture for the rate of convergence of the bias produced. Numerically, these conjectures are confirmed for the specular reflection scheme but also for the absorption scheme, which models perfect agglomeration. The scheme numerically follows a linear decrease. The Richardson-Romberg extrapolation is also presented with a quadratic decrease.

• M. Bossy, A. Rousseau (Lemon Inria team), JĖspina, JṀorice and C. Paris (Inria Chile) studied the computation of the wind circulation around mills, using a Lagrangian stochastic approach. They present the SDM numerical method and numerical experiments in the case of non rotating and rotating actuator disc models in [13]. First, for validation purpose they compare some numerical experiments against wind tunnel measurements. Second, they perform numerical experiments at the atmospheric scale and present some features of the numerical method, in particular the computation of the probability distribution of the wind in the wake zone, as a byproduct of the fluid particle model and the associated PDF method.

• Together with M. Baar and A. Bovier (Univ. Bonn), N. Champagnat studied the adaptive dynamics of populations under the assumptions of large population, rare and small mutations [11]. In this work, the three limits are taken simultaneously, contrary to the classical approach, where the limits of large population and rare mutations are taken first, and next the limit of small mutations  [57]. We therefore obtain the precise range of parameters under which these limits can be taken, and provide explicit biological conditions for which our approximation is valid.

• 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 QSD of the processes controled by Markov controls [36].

• N. Champagnat and C. Fritsch worked with F. Campillo (Inria Sophia-Antipolis, Lemon team) on the links between a branching process and an integro-differential equation of a growth-fragmentation-death model [15]. They proved that the two representations of the model lead to the same criteria of invasion of a population in a given environment. They also studied the variations of the principal eigenvalue (resp. the survival probability) of an integro-differential equation (resp. branching process) of growth-fragmentation models with respect to an environmental parameter in [35].

• 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, degammad as the original Markov process conditioned to never be absorbed. They prove that, under the conditions of [17], in addition to the uniform exponential convergence of 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 [41].

• 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.

• Using a new method to compute the expectation of an integral with respect to a random measure, N. Champagnat and B. Henry obtained explicit formulas for the moments of the frequency spectrum in the general branching processes known as Splitting Trees, with neutral mutations and under the infinitely-many alleles model [16]. This allows them to obtain a law of large numbers for the frequency spectrum in the limit of large time.

• N. Champagnat and D. Villemonais obtained criteria for existence, uniqueness and exponential convergence in total variation to QSD for discrete population processes with unbounded absorption rate, using a non-linear Lyapunov criterion [38]. For logistic multidimensional birth and death processes absorbed when one coordinate gets extinct, they show that their criterion covers cases stronger intra-spectific competition than inter-specific competition.

• N. Champagnat and D. Villemonais extended their work [17] to general penalized processes, including time-inhomogeneous Markov processes with absorption and Markov processes in varying environments [40]. Their method allows to improve significantly the former results of  [58], [59].

• M. Deaconu worked with L. Beznea and O. Lupaşcu (Bucharest, Romania) and analyzed the description of rupture phenomena like avalanches, by using fragmentation models. The main physical properties of the model are deeply involved in this study. They obtained new results on a stochastic equation of fragmentation and branching processes related to avalanches [12].

• M. Deaconu and S. Herrmann continued and completed the study of the simulation of hitting times of given boundaries for Bessel processes. These problems are of great interest in many application fields, such as finance and neurosciences. In a previous work, the authors introduced a new method for the simulation of hitting times for Bessel processes with integer dimension. The method was based mainly on explicit formula for the distribution of the hitting times and on the connexion between the Bessel process and the Euclidean norm of the Brownian motion. The method does not apply for a non-integer dimension. In this new work they consider the simulation of the hitting time of Bessel processes with non integer dimension and provide a new algorithm by using the additivity property of the laws of squared Bessel processes. Each simulation step is splitted in two parts: one is using the integer dimension case and the other one exhibits hitting time of a Bessel process starting from zero [20].

• M. Deaconu and S. Herrmann studied the Initial-Boundary Value Problem for the heat equation and solved it by using a new algorithm based on a random walk on heat balls [44]. Even if it represents a sophisticated and challenging 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 new mean value formula for the heat equation. The convergence results and numerical examples allow to emphasize the efficiency and accuracy of the algorithm.

• M. Deaconu, B. Dumortier and E. Vincent (EPI Multispeech are working with the Venathec SAS on the acoustic control of wind farms. They constructed a new approach to control wind farms based on real-time source separation. They expressed the problem as a non-linear knapsack problem and solve it using an efficient branch-and-bound algorithm that converges asymptotically to the global optimum. The algorithm is initialised with a greedy heuristic that iteratively downgrades the turbines with the best acoustical to electricity loss ratio. The solution is then regammad using a depth-first search strategy and a bounding stage based on a continuous relaxation problem solved with an adapted gradient algorithm. The results are evaluated using data from 28 real wind farms [46].

• C. Fritsch and B. Cloez (INRA, Montpellier) proved central limit theorems for chemostat models in finite and infinite dimensions in [42]. From these theorems, they obtianed gaussian approximations of individual-based models and made a numerical analysis for the model in finite dimension in order to discuss the validity of these approximations in different contexts.

• Together with R. Azaïs (Bigs Inria team) and A. Genadot (Univ. Bordeaux), B. Henry studied an estimation problem for a forest of size-constrained Galton-Watson trees [31]. Using the asymptotic behavior of the Harris contour process, they constructed estimators for the inverse standard deviation of the birth distribution. In addition to the theoretical convergence results obtained in this work, they used the method to study the evolution of Wikipedia webpages in order, for instance, to detect vandalism.

• In [49], B. Henry showed a central limit theorem for the population counting process of a supercritical Splitting Tree in the limit of large time. Thanks to the results of [16], he also obtained a central limit theorem for the frequency spectrum of Splitting Trees with neutral mutations and under the infinitely-many alleles model.

• In collaboration with Laure Coutin, A. Lejay have studied the sensitivity of solution of rough differential equations with respect to their parameters using a Banach space version of the implicit function theorem. This result unifies and extends all the similar results on the subject [43].

• A. Lejay have studied the parametric estimation of the bias coefficient of skew random walk, as a toy model for the problem of estimation of the parameter of the Skew Brownian motion [50].

• P. Pigato has continued with V. Bally (Univ. Marne-la-Vallée) and L. Caramellino (Univ. Roma Tor Vergata) his PhD work on the regularity of diffusions under Hörmander-type conditions [32], [33].

• A. Richard and D. Talay ended their work on the sensitivity of the first hitting time of fractional SDEs, when $H>\frac{1}{2}$[54]. This study is being completed by the rough case $H\in (\frac{1}{4},\frac{1}{2}]$. In relation to fractional SDEs, another short work on accurate Gaussian-like upper bounds on density of one-dimensional fractional SDEs is almost finished.

• In [21], S. Herrmann and E. Tanré propose a new algorithm to simulate the first hitting times of a deterministic continuous function by a one-dimensional Brownian motion. They give explicit rate of convergence of the algorithm.

• E. Tanré and Pierre Guiraud (Univ. of Valparaiso) have studied the synchronization in a model of neural network with noise. 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. [48]

• V. Reutenauer and E. Tanré have worked on extensions of the exact simulation algorithm introduced by Beskos et al.  [56]. They propose an unbiased algorithm to approximate the two first derivatives with respect to the initial condition $x$ of quantities with the form $𝔼\Psi \left(X_T^x\right)$, where $X$ is a one-dimensional diffusion process and $\Psi$ any test-function. They also propose an efficient modification of Beskos et al. algorithm. [53]

• During his internship supervised by E. Tanré, A. Papic worked on multi scales generator of Markov processes. He presents a method to approximate such processes with an application in neuroscience for noisy Hodgkin-Huxley model [52].

• D. Villemonais worked with P. Del Moral (Univ. Sydney) on the conditional ergodicity of time inhomogeneous diffusion processes [45]. They proved that, conditionally on non extinction, an elliptic time-inhomogeneous diffusion process forgets its initial distribution exponentially fast. An interacting particle scheme to numerically approximate the conditional distribution is also provided.

• D. Villemonais worked with his Research Project student William Oçafrain (École des Mines de Nancy) on an original mean-field particle system [51]. 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.

• M. Bossy and R. Maftei are working on determining the rate of convergence of the weak error of a discretised scheme for the Langevin system with specular boundary reflection on the position. The velocity process allows for a bounded and smooth drift. In order to determine the optimal rate of convergence, the regularity of the associated PDE is required and also regularity results for the derivative of flow of the process w.r.t. the initial conditions.

• N. Champagnat and B. Henry are studying limits of small mutations in Lokta-Volterra type PDEs of population dynamics using probabilistic representations and large deviations.

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

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

• M. Deaconu, O. Lupaşcu and L. Beznea (Bucharest, Romania) worked on the numerical scheme for the simulation of an avalanche by using the fragmentation model. This work will be submitted soon.

• M. Deaconu, B. Dumortier and E. Vincent (EPI Multispeech ) work on handling uncertainties in the model of acoustic control of wind farms they develop, in order to design a stochastic algorithm based on filtering methods. They will submit another article to IEEE transaction on sustainable energy.

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

• 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.

• B. Cloez (INRA Montpellier) and B. Henry started a work on the asymptotic behavior of splitting trees in random environment. In addition, they begin the study of scaling limits of splitting trees in varying environment.

• Together with Ernesto Mordecki (Universidad de la República, Uruguay) and Soledad Torres (Universidad de Valparaíso), A. Lejay is working on the estimation of the parameter of the Skew Brownian motion.

• A. Lejay, and P. Pigato are working on the estimation of the parameters of diffusions with discontinuous coefficients, with application to financial data.

• Together with Laure Coutin and Antoine Brault (Université Toulouse 3), A. Lejay is studying application of the Trotter-Kato theorem in the context of rough differential equations, in order to solve some Stochastic Partial Differential Equations.

• A. Lejay and H. Mardones are working on a Monte Carlo simulation of the Navier-Stokes equations which is based on a novel probabilistic representation due to F. Delbaen et al.  [60].

• In a research visit to Chile, P. Pigato has worked with R. Rebolledo and S. Torres on the estimation of parameters of diffusions from the occupation time and the local time of the process.

• Together with Laure Coutin and Antoine Brault (Université Toulouse 3), A. Lejay is studying application of the Trotter-Kato theorem in the context of rough differential equations, in order to solve some Stochastic Partial Differential Equations.

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

• In collaboration with J. Bion-Nadal (CNRS and École Polytechnique) D. Talay ended the first paper on an innovating calibration method for stochastic models belonging to a family of solutions to martingale problems. The methodology involves the introduction of a new Wasserstein-type distance and stochastic control problems. The manuscript is being finished.

• Motivated by the study of systems of non-linear PDE's by stochastic methods, M. Tomasevic and D. Talay studied a system of differential equations interacting through a singular kernel, depending on all the past of the solutions. They have proved the existence of a solution in the space of Lipschitz functions in short time interval and performed numerical simulations. In the same time, they studied a non-linear stochastic differential equation whose drift is given as a convolution of a singular kernel with the unknown one dimensional time marginals both in time and space. Combining probabilistic and PDE techniques, they are currently finishing the proof of the existence and uniqueness of a weak solution up to an arbitrary finite time horizon. Properties of the corresponding particle system (well-posedness and propagation of chaos) are also studied.

• A. Richard and E. Tanré's work with Patricio Orio (CINV, Chile) on the modelling and measurement of long-range dependence in neuronal spike trains is almost completed. They exhibit evidence of memory effect in genuine neuronal data and compared their fractional integrate-and-fire model with the existing Markovian models. A. Richard and E. Tanré are still working on the convergence of the statistical estimator that measures this phenomenon.

• A. Richard, E. Tanré are working with S. 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 $H>\frac{1}{2}$. In addition, 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 case and are working on the question of uniqueness and on the relaxation of ellipticity.

• During his internship supervised by E. Tanré and Romain Veltz (Mathneuro team), Pascal Helson studied numerically and theoretically a model of spiking neurons in interaction with plasticity. He showed that a simple model without plasticity could reproduce biological phenomena such as oscillations. In order to add plasticity, he enabled synaptic weights to evolve in a probabilistic way, in agreement with biological laws. He is now studying the convergence of this model and the existence of separable time scales, which is part of his thesis.

• D. Villemonais started a collaboration with Camille Coron (Univ. Paris Sud) and Sylvie Méléard (École Polytechnioque) on the question of simultaneous/non-simultaneous extinction of traits in a structured population

• D. Villemonais currently works on the computation of lower bounds for the Wasserstein curvature of interacting particle systems.

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