arXiv subjects

Pierre Monmarché

Publications and source records attributed to Pierre Monmarché.

At least 37 records · Page 2Linked to original sources

arXiv2025

Logarithmic Sobolev inequalities for non-equilibrium steady states

We consider two methods to establish log-Sobolev inequalities for the invariant measure of a diffusion process when its density is not explicit and the curvature is not positive everywhere. In the first approach, based on the Holley-Stroock and Aida-Shigekawa perturbation arguments [J. Stat. Phys., 46(5-6):1159-1194, 1987, J. Funct. Anal., 126(2):448-475, 1994], the control on the (non-explicit) perturbation is obtained by stochastic control methods, following the comparison technique introduced by Conforti [Ann. Appl. Probab., 33(6A):4608-4644, 2023]. The second method combines the Wasserstein-2 contraction method, used in [Ann. Henri Lebesgue, 6:941-973, 2023] to prove a Poincaré inequality in some non-equilibrium cases, with Wang's hypercontractivity results [Potential Anal., 53(3):1123-1144, 2020].

math.PR↗

arXiv2025

Velocity Jumps for Molecular Dynamics

We introduce the Velocity Jumps approach, denoted as JUMP, a new class of Molecular dynamics integrators, replacing the Langevin dynamics by a hybrid model combining a classical Langevin diffusion and a piecewise deterministic Markov process, where the expensive computation of long-range pairwise interactions is replaced by a resampling of the velocities at random times. This framework allows for an acceleration in the simulation speed while preserving sampling and dynamical properties such as the diffusion constant. It can also be integrated in classical multi-timestep methods, pushing further the computational speedup, while avoiding some of the resonance issues of the latter thanks to the random nature of jumps. The JUMP, JUMP-RESPA and JUMP-RESPA1 integrators have been implemented in the GPU-accelerated version of the Tinker-HP package and are shown to provide significantly enhanced performances compared to their BAOAB, BAOAB-RESPA and BAOAB-RESPA1 counterparts respectively.

physics.comp-ph↗

arXiv2025

Long-time propagation of chaos and exit times for metastable mean-field particle systems

Systems of stochastic particles evolving in a multi-well energy landscape and attracted to their barycenter is the prototypical example of mean-field process undergoing phase transitions: at low temperature, the corresponding mean-field deterministic limit has several stationary solutions, and the empirical measure of the particle system is then expected to be a metastable process in the space of probability measures, exhibiting rare transitions between the vicinity of these stationary solutions. We show two results in this direction: first, the exit time from such metastable domains occurs at time exponentially large with the number of particles and follows approximately an exponential distribution; second, up to the expected exit time, the joint law of particles remain close to the law of independent non-linear McKean-Vlasov processes.

math.PR↗

arXiv2025

Transient subtraction: A control variate method for computing transport coefficients

In molecular dynamics, transport coefficients measure the sensitivity of the invariant probability measure of the stochastic dynamics at hand with respect to some perturbation. They are typically computed using either the linear response of nonequilibrium dynamics, or the Green--Kubo formula. The estimators for both approaches have large variances, which motivates the study of variance reduction techniques for computing transport coefficients. We present an alternative approach, called the \emph{transient subtraction technique} (inspired by early work by Ciccotti and Jaccucci in 1975), which amounts to simulating a transient dynamics started off equilibrium and relaxing towards the equilibrium state, from which we subtract a sensibly coupled equilibrium trajectory, resulting in an estimator with smaller variance. We present the mathematical formulation of the transient subtraction technique, give error estimates on the bias and variance of the associated estimator, and demonstrate the relevance of the method through numerical illustrations for various systems.

math.NA↗

arXiv2025

Accuracy of the Ensemble Kalman Filter in the Near-Linear Setting

The filtering distribution captures the statistics of the state of a dynamical system from partial and noisy observations. Classical particle filters provably approximate this distribution in quite general settings; however they behave poorly for high dimensional problems, suffering weight collapse. This issue is circumvented by the ensemble Kalman filter which is an equal-weight interacting particle system. However, this finite particle system is only proven to approximate the true filter in the linear Gaussian case. In practice, however, it is applied in much broader settings; as a result, establishing its approximation properties more generally is important. There has been recent progress in the theoretical analysis of the algorithm, establishing stability and error estimates in non-Gaussian settings, but the assumptions on the dynamics and observation models rule out the unbounded vector fields that arise in practice and the analysis applies only to the mean field limit of the ensemble Kalman filter. The present work establishes error bounds between the filtering distribution and the finite particle ensemble Kalman filter when the dynamics and observation vector fields may be unbounded, allowing linear growth.

math.ST↗

arXiv2025

L2 geometric ergodicity for the kinetic Langevin process with non-equilibrium steady states

In non-equilibrium statistical physics models, the invariant measure $μ$ of the process does not have an explicit density. In particular the adjoint $L^*$ in $L^2(μ)$ of the generator $L$ is unknown and many classical techniques fail in this situation. An important progress has been made in [5] where functional inequalities are obtained for non-explicit steady states of kinetic equations under rather general conditions. However in [5] in the kinetic case the geometric ergodicity is only deduced from the functional inequalities for the case with conservative forces, corresponding to explicit steady states. In this note we obtain $L^2$ convergence rates in the non-equilibrium case.

math.AP↗

arXiv2024

Non-asymptotic entropic bounds for non-linear kinetic Langevin sampler with second-order splitting scheme

The problem of sampling according to the probability distribution minimizing a given free energy, using interacting particles unadjusted kinetic Langevin Monte Carlo, is addressed. In this setting, three sources of error arise, related to three parameters: the number of particles $N$, the discretization step size $h$, and the length of the trajectory $n$. The main result of the present work is a quantitative estimate of strong convergence in relative entropy, implying non-asymptotic bounds for the quadratic risk of Monte Carlo estimators for bounded observables. The numerical discretization scheme considered here is a second-order splitting method, as commonly used in practice. In addition to $N,h,n$, the dependency in the ambient dimension $d$ of the problem is also made explicit, under suitable conditions. The main results are proven under general conditions (regularity, moments, log-Sobolev inequality), for which tractable conditions are then provided. In particular, a Lyapunov analysis is conducted under more general conditions than previous works; the nonlinearity may not be small and it may not be convex along linear interpolations between measures.

math.PR↗

arXiv2024

Uniform convergence of the Fleming-Viot process in a hard killing metastable case

We study the long-time convergence of a Fleming-Viot process, in the case where the underlying process is a metastable diffusion killed when it reaches some level set. Through a coupling argument, we establish the long-time convergence of the Fleming-Viot process toward some stationary measure at an exponential rate independent of $N$, the size of the system, as well as uniform in time propagation of chaos estimates.

math.PR↗

arXiv2024

Uniform log-Sobolev inequalities for mean field particles beyond flat-convexity

In the nice recent work [48], S. Wang established uniform log-Sobolev inequalities for mean field particles when the energy is flat convex. In this note we comment how to extend his proof to some semi-convex energies provided the curvature lower-bound is not too negative. It is not clear that this could be obtained simply by applying a posteriori a perturbation argument to Wang's result. Rather, we follow his proof and, at steps where the convexity is used, we notice that the uniform conditional or local functional inequalities assumed give some room to allow for a bit of concavity. In particular, this allows to recover other previous results on non-convex systems at high temperature or weak coupling, and to consider situations which mix flat-convexity and weak coupling.

math.AP↗

arXiv2024

Time-uniform log-Sobolev inequalities and applications to propagation of chaos

Time-uniform log-Sobolev inequalities (LSI) satisfied by solutions of semi-linear mean-field equations have recently appeared to be a key tool to obtain time-uniform propagation of chaos estimates. This work addresses the more general settings of time-inhomogeneous Fokker-Planck equations. Time-uniform LSI are obtained in two cases, either with the bounded-Lipschitz perturbation argument with respect to a reference measure, or with a coupling approach at high temperature. These arguments are then applied to mean-field equations, where, on the one hand, sharp marginal propagation of chaos estimates are obtained in smooth cases and, on the other hand, time-uniform global propagation of chaos is shown in the case of vortex interactions with quadratic confinement potential on the whole space. In this second case, an important point is to establish global gradient and Hessian estimates, which is of independent interest. We prove these bounds in the more general situation of non-attractive logarithmic and Riesz singular interactions.

math.PR↗

arXiv2024

Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case

Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with stochastic gradients under possibly nonconvex conditions. The algorithms considered include splitting schemes of kinetic Langevin diffusion commonly used in molecular dynamics simulations. To accommodate the degenerate noise structure corresponding to inertia existing in the chain, a characteristically discrete-in-time coupling and contraction proof is devised. As consequence, quantitative Gaussian concentration bounds are provided for empirical averages. Convergence in Wasserstein 2-distance and total variation are also given, together with numerical bias estimates.

math.PR↗

arXiv2024

Impacts of Tempo and Mode of Environmental Fluctuations on Population Growth: Slow- and Fast-Limit Approximations of Lyapunov Exponents for Periodic and Random Environments

We examine to what extent the tempo and mode of environmental fluctuations matter for the growth of structured populations. The models are switching, linear ordinary differential equations $x'(t)=A(σ(ωt))x(t)$ where $x(t)=(x_1(t),\dots,x_d(t))$ corresponds to the population densities in the $d$ individual states, $σ(t)$ is a piece-wise constant function representing the fluctuations in the environmental states $1,\dots,N$, $ω$ is the frequency of the environmental fluctuations, and $A(1),\dots,A(n)$ are Metzler matrices. $σ(t)$ can either be a periodic function or correspond to a continuous-time Markov chain. Under suitable conditions, there is a Lyapunov exponent $Λ(ω)$ such that $\lim_{t\to\infty} \frac{1}{t}\log\sum_i x_i(t)=Λ(ω)$ for all non-negative, non-zero initial conditions $x(0)$ (with probability one in the random case). For both forms of switching, we derive analytical first-order and second-order approximations of $Λ(ω)$ in the limits of slow ($ω\to 0$) and fast ($ω\to\infty$) environmental fluctuations. When the order of switching and the average switching times are equal, we show that the first-order approximations of $Λ(ω)$ are equivalent in the slow-switching limit, but not in the fast-switching limit. We illustrate our results with applications to stage-structured and spatially-structured models. When dispersal rates are symmetric, the first order approximations suggest that population growth rates increase with the frequency of switching -- consistent with earlier work on periodic switching. In the absence of dispersal symmetry, we demonstrate that $Λ(ω)$ can be non-monotonic in $ω$. In conclusion, our results show how population growth rates depend on the tempo ($ω$) and mode (random versus deterministic) of the environmental fluctuations.

q-bio.PE↗

arXiv2024

HMC and underdamped Langevin united in the unadjusted convex smooth case

We consider a family of unadjusted generalized HMC samplers, which includes standard position HMC samplers and discretizations of the underdamped Langevin process. A detailed analysis and optimization of the parameters is conducted in the Gaussian case, which shows an improvement from $1/κ$ to $1/\sqrtκ$ for the convergence rate in terms of the condition number $κ$ by using partial velocity refreshment, with respect to classical full refreshments. A similar effect is observed empirically for two related algorithms, namely Metropolis-adjusted gHMC and kinetic piecewise-deterministic Markov processes. Then, a stochastic gradient version of the samplers is considered, for which dimension-free convergence rates are established for log-concave smooth targets over a large range of parameters, gathering in a unified framework previous results on position HMC and underdamped Langevin and extending them to HMC with inertia.

math.PR↗

arXiv2024

Lambda-ABF: Simplified, Portable, Accurate and Cost-effective Alchemical Free Energy Computations

We introduce an efficient and robust method to compute alchemical free energy differences, resulting from the application of multiple walker Adaptive Biasing Force (ABF) in conjunction with strongly damped Langevin $λ$-dynamics. Unbiased alchemical free energy surfaces are naturally recovered by Thermodynamic Integration (TI). No manual optimization of the $λ$ schedule is required as the sampling of the $λ$ variable is continuous and converges towards a uniform distribution. Free diffusion of $λ$ improves orthogonal relaxation compared to fixed $λ$ methods such as standard TI or Free Energy Perturbation (FEP). Furthermore, the multiple walker strategy provides coverage of orthogonal space in a generic way with minimal user input and negligible computational overhead. Of practical importance, no adiabatic decoupling between the alchemical and Cartesian degrees of freedom is assumed, ensuring unbiased estimates for a wide envelope of numerical parameters. We present two high-performance implementations of the method in production molecular dynamics engines, namely NAMD and Tinker-HP, through coupling with the Colvars open source library. These interfaces enable the combination of the rich feature sets of those packages. We demonstrate the correctness and efficiency of the approach on several real-world cases: from solvation free energies up to ligand-receptor binding (using a recently proposed binding restraint scheme) with both fixed-charge and polarizable models. We find that, for a chosen accuracy, the computational cost is strongly reduced compared to state-of-the-art fixed-lambda methods and that results within 1~kcal/mol of experimental value are recovered for the most complex system. The implementation is publicly available and readily usable by practitioners of current alchemical methods.

physics.chem-ph↗

arXiv2024

Second order quantitative bounds for unadjusted generalized Hamiltonian Monte Carlo

This paper provides a convergence analysis for generalized Hamiltonian Monte Carlo samplers, a family of Markov Chain Monte Carlo methods based on leapfrog integration of Hamiltonian dynamics and kinetic Langevin diffusion, that encompasses the unadjusted Hamiltonian Monte Carlo method. Assuming that the target distribution $π$ satisfies a log-Sobolev inequality and mild conditions on the corresponding potential function, we establish quantitative bounds on the relative entropy of the iterates defined by the algorithm, with respect to $π$. Our approach is based on a perturbative and discrete version of the modified entropy method developed to establish hypocoercivity for the continuous-time kinetic Langevin process. As a corollary of our main result, we are able to derive complexity bounds for the class of algorithms at hand. In particular, we show that the total number of iterations to achieve a target accuracy $\varepsilon >0$ is of order $d/\varepsilon^{1/4}$, where $d$ is the dimension of the problem. This result can be further improved in the case of weakly interacting mean field potentials, for which we find a total number of iterations of order $(d/\varepsilon)^{1/4}$.

math.PR↗

arXiv2024

Wasserstein contraction for the stochastic Morris-Lecar neuron model

Neuron models have attracted a lot of attention recently, both in mathematics and neuroscience. We are interested in studying long-time and large-population emerging properties in a simplified toy model. From a mathematical perspective, this amounts to study the long-time behaviour of a degenerate reflected diffusion process. Using coupling arguments, the flow is proven to be a contraction of the Wasserstein distance for long times, which implies the exponential relaxation toward a (non-explicit) unique globally attractive equilibrium distribution. The result is extended to a McKean-Vlasov type non-linear variation of the model, when the mean-field interaction is sufficiently small. The ergodicity of the process results from a combination of deterministic contraction properties and local diffusion, the noise being sufficient to drive the system away from non-contractive domains.

math.PR↗

arXiv2023

Discrete sticky couplings of functional autoregressive processes

In this paper, we provide bounds in Wasserstein and total variation distances between the distributions of the successive iterates of two functional autoregressive processes with isotropic Gaussian noise of the form $Y_{k+1} = \mathrm{T}_γ(Y_k) + \sqrt{γσ^2} Z_{k+1}$ and $\tilde{Y}_{k+1} = \tilde{\mathrm{T}}_γ(\tilde{Y}_k) + \sqrt{γσ^2} \tilde{Z}_{k+1}$. More precisely, we give non-asymptotic bounds on $ρ(\mathcal{L}(Y_{k}),\mathcal{L}(\tilde{Y}_k))$, where $ρ$ is an appropriate weighted Wasserstein distance or a $V$-distance, uniformly in the parameter $γ$, and on $ρ(π_γ,\tildeπ_γ)$, where $π_γ$ and $\tildeπ_γ$ are the respective stationary measures of the two processes. The class of considered processes encompasses the Euler-Maruyama discretization of Langevin diffusions and its variants. The bounds we derive are of order $γ$ as $γ\to 0$. To obtain our results, we rely on the construction of a discrete sticky Markov chain $(W_k^{(γ)})_{k \in \mathbb{N}}$ which bounds the distance between an appropriate coupling of the two processes. We then establish stability and quantitative convergence results for this process uniformly on $γ$. In addition, we show that it converges in distribution to the continuous sticky process studied in previous work. Finally, we apply our result to Bayesian inference of ODE parameters and numerically illustrate them on two particular problems.

math.PR↗

arXiv2023

A note on a Vlasov-Fokker-Planck equation with non-symmetric interaction

In the recent [3], Cesbron and Herda study a Vlasov-Fokker-Planck (VFP) equation with non-symmetric interaction, introduced in physics to model the distribution of electrons in a synchrotron particle accelerator. We make four remarks in view of their work: first, it is noticed in [3] that the free energy classically considered for the (symmetric) VFP equation is not a Lyapunov function in the non-symmetric case, and we will show however that this is still the case for a suitable definition of the free energy (with no explicit expression in general). Second, when the interaction is sufficiently small (in $W^{1,\infty}$), it is proven in [3] that the equation has a unique stationary solution which is locally attractive; in this spirit, we will see that, when the interaction force is Lipschitz with a sufficiently small constant, the convergence is global. Third, we also briefly discuss the mean-field interacting particle system corresponding to the VFP equation which, interestingly, is a non-equilibrium Langevin process. Finally, we will see that, in the small interaction regime, a suitable (explicit) non-linear Fisher information is contracted at constant rate, similarly to the situation of Wasserstein gradient flows for convex functionals (although here the dynamics is not a gradient flow).

math.PR↗