CE40 - Mathématiques

Metastability for nonlinear processes – METANOLIN

Metastability for nonlinear processes

The project deals with questions linked to metastability for nonlinear stochastic processes: the so-called self-stabilizing diffusion (which is a McKean-Vlasov diffusion in which the law of the process intervenes in the drift through convolution), the self-interacting diffusio, (where the occupation measure intervenes in the drift) and Keller-Segel type diffusions (with space-time interaction).

To prove Kramers'type law for nonlinear diffusions

The project aims in providing Kramers'type law concerning the exit-time of nonlinear diffusions and their associated system of particles when available. More precisely, we are interested in self-stabilizing diffusions (probabilistic interpretation of the granular media equation), in self-interacting diffusions (which model polymers) and so-called Keller-Segel (or memorial McKean-Vlasov) diffusions which combine the first two types of interaction.

The main idea is to use a concave interacting potential instead of a convex one. Then, the interaction force is repulsive and we have a better exploration of the phase space.

A first work has been done by Paul-Eric Chaudru de Raynal, Pierre Monmarché, Milica Tomasevic and Julian Tugaut and also by Hong Duong (exterior to the ANR project). We obtained exit-time for overdamped and underdamped nonlinear diffusions with the three types of interaction mentioned previously. The main interest is that the method is robust with respect to the assumptions.
Here, the confining potential is convex and the interactive force is repulsive. We thus get a reduction of the exit-time which leads somehow to a reduction of the metastability (albeit there is no real metastability since V is convex). The paper will be submitted tomorrow.

A second work is betweenAshot Aleksian, Aline Kurtzmann and Julian Tugaut and also Pierre Del Moral (not a member of ANR project). We have obtained similar results to self-interacting diffusions than those obtained in self-stabilizing case/. Here, the confining potential and the interacting potential are convexes. This paper is about to be submitted. We also have been able to obtain results in non-convex case albeit nothing has been written yet.

A third work is between Julian Tugaut with non-members of ANR project (Daniel Adams, Gonçalo dos Reis, Romain Ravaille and William Salkeld). It deals with very general McKean-Vlasov diffusion in compact phase space albeit with convexity assumptions. This paper is under minor revision for Stochastic Processes and their Applications

A fourth work (which was not planned initially) is between Julian Tugaut and Jean-François Jabir (not a member of METANOLIN) and concerns the first time that two nonlinear diffusions (and their associated system of particles) collide. This preliminary work deals with simple cases. It is about to be submitted.

The main focus is the extension to the real non-convex case.

Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: uniform estimates in a compact soft case, Lucas Journel and Pierre Monmarché. arxiv.org/pdf/1910.05060.pdf

Large Deviations and Exit-times for reflected McKean-Vlasov equations with self-stabilizing terms and superlinear drifts, Daniel Adams, Gonçalo dos Reis, Romain Ravaille, William Salkeld, Julian Tugaut. arxiv.org/pdf/2005.10057.pdf

Reducing exit-times of diffusions with repulsive interactions, Paul-Eric Chaudru de Raynal, Hong Duong, Pierre Monmarché, Milica Tomasevic and Julian Tugaut.

The project deals with metastability problems for non-linear stochastic processes: the so-called self-stabilizing diffusion (which is a McKean-Vlasov diffusion in which the law of the process intervenes in the drift by a convolution), the self-interacting diffusion (which is a diffusion in which the occupation measure intervenes in the drift) and some Keller-Segel type diffusions (with interactions in space and time).

The main question is the following: how can we improve the convergence of algorithms like the simulated annealing one for the optimization of a given cost function ? Indeed, we already know that the rate of convergence of a stochastic gradient descent method is linked to the exit-time from a local minimum and therefore to the metastability of the associated Kolmogorov diffusion.

The theory of Freidlin and Wentzell allows us to quantify this exit-time for linear diffusions. Indeed, we know that the expectation of the exit-time behaves like the exponential of the height of the well divided by the temperature. As the temperature must be small to capture the geometry of the potential, this time is huge.

We look at non-linear diffusions in order to have a better exploration of the phase space by adding a repulsion. We already have some first encouraging results in the framework where the interacting potentials are convex. These results state that the exit-time, in the setting of the convexity and under some assumptions, behaves similarly except that the height of the well is larger. Consequently, the exit-time is increased. This strongly suggests that a judicious concave framework will drastically reduce the exit-time by reduction of the height of the well. The direct consequence is to significantly improve the convergence of classical algorithms.

In this project, we are interested in non-linear diffusions which are hydrodynamical limits of mean-field systems of particles in which the interacting potentials are repulsive. Thus, the particles of the system tend to repel themselves and intuitively to better explore the phase space. We also aim to look at the particles system itself in view of the numerical simulations. Indeed, the strength of the McKean-Vlasov diffusion is its ability to be easy to simulate with system of particles.

We are also interested in non-linear diffusions in which the past intervenes in the drift through the occupation measure. The strength (that we plan to use) of this type of diffusion is the loss of spatial interaction between the trajectories. Nevertheless, in view of numerical simulations, we will probably have to restrict to polynomial interacting potentials.

Finally, we are also interested in a mixture of spatial interaction and time interaction. This corresponds to a Keller-Segel type diffusion where the associated system of particles loses the Markov property. In a last step, we aim the interactions of this kind of diffusion to be singular so that there is a strong repulsion between the particles and their pasts.

Project coordination

Julian Tugaut (Institut Camille Jordan)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.


UJM/ICJ Institut Camille Jordan

Help of the ANR 87,696 euros
Beginning and duration of the scientific project: September 2019 - 48 Months

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