arXiv subjects

Tony Lelièvre

Publications and source records attributed to Tony Lelièvre.

At least 19 recordsLinked to original sources

arXiv2026

Quantitative low-temperature spectral asymptotics for reversible diffusions in temperature-dependent domains

We derive novel low-temperature asymptotics for the spectrum of the infinitesimal generator of the overdamped Langevin dynamics. The novelty is that this operator is endowed with homogeneous Dirichlet conditions at the boundary of a domain which depends on the temperature. From the point of view of stochastic processes, this gives information on the long-time behavior of the diffusion conditioned on non-absorption at the boundary, in the so-called quasistationary regime. Our results provide precise estimates of the spectral gap and principal eigenvalue, extending the Eyring-Kramers formula. The phenomenology is richer than in the case of a fixed boundary and gives new insight into the sensitivity of the spectrum with respect to the shape of the domain near critical points of the energy function. Our work is motivated by--and is relevant to--the problem of finding optimal hyperparameters for accelerated molecular dynamics algorithms.

math.AP↗

arXiv2026

The narrow escape problem in arbitrary dimension

The narrow escape problem is a prototypical example for studying entropic metastability, motivated by the analysis of biological and chemical systems. The problem concerns the determination of the exit time and position of a Brownian particle trapped in a domain with a reflecting boundary pierced by narrow holes. Our goal is to investigate this problem in a general domain in any dimension (greater than or equal to two), using the quasi-stationary distribution approach to metastability. In particular, we derive first order asymptotic expansions with error bounds of the mean exit time and the law of the exit position in the limit where the hole sizes tend to zero. Our analytical predictions are illustrated by numerical simulations, using dedicated Monte Carlo techniques.

math.AP↗

arXiv2026

A spectral approach to the narrow escape problem in two-dimensional domains

We study the law of the exit time and exit point of a Brownian motion in a two-dimensional domain with reflecting boundary conditions, except on small disjoint exit windows through which the stochastic process can escape the domain. In the limit of infinitely small exit windows, it is natural to assume that the process starts from the quasi-stationary distribution. In this setting, we obtain a precise description of the exit event.

math.AP↗

arXiv2026

Approximating committor functions: Objective functions and training data sampling

Many molecular dynamics simulations aim at studying transitions between two states (from reactants to products). In this context, the committor function (which gives for a given molecular configuration the probability to reach the product state before the reactant state) is a pivotal quantity, in particular because it is the optimal importance function for rare event simulation methods such as importance sampling or splitting techniques. These methods are used to sample the reactive path ensemble, and estimate for example the transition rate. However, learning such a function is generally a challenging task due to the high dimensionality of the configuration space. In this work, after reviewing the existing methodologies to construct approximate committor functions, a new loss function based on the application of Itō's formula is proposed to learn the committor function with a minimization procedure on the parameters of a neural network. After comparing this novel approach to existing procedures on the Müller--Brown potential, we introduce a coupling strategy with the Adaptive Multilevel Splitting method to better approximate the committor function using a better sampling of the reactive trajectories. This methodology in which the committor function is iteratively learned only requires initially the knowledge of the reactant and product states.

physics.chem-ph↗

arXiv2026

Efficient Monte Carlo sampling of metastable systems using non-local collective variable updates

Monte Carlo simulations are widely used to simulate complex molecular systems, but standard approaches suffer from metastability. Lately, the use of non-local proposal updates in a collective-variable (CV) space has been proposed in several works. Here, we generalize these approaches and explicitly spell out an algorithm for non-linear CVs and underdamped Langevin dynamics. We prove reversibility of the resulting scheme and demonstrate its performance on several numerical examples, observing a substantial performance increase compared to methods based on overdamped Langevin dynamics as considered previously. Advances in generative machine-learning-based proposal samplers now enable efficient sampling in CV spaces of intermediate dimensionality (tens to hundreds of variables), and our results extend their applicability toward more realistic molecular systems.

cond-mat.stat-mech↗

arXiv2026

Shape optimization of metastable states

The definition of metastable states is an ubiquitous task in the design and analysis of molecular simulations, and is a crucial input in a variety of acceleration methods for the sampling of long configurational trajectories. Although standard definitions based on local energy minimization procedures can sometimes be used, these definitions are typically suboptimal, or entirely inadequate when entropic effects are significant, or when the lowest energy barriers are quickly overcome by thermal fluctuations. In this work, we propose an approach to the definition of metastable states, based on the shape-optimization of a local separation of timescale metric directly linked to the efficiency of a family of accelerated molecular dynamics algorithms. To realize this approach, we derive analytic expressions for shape-variations of Dirichlet eigenvalues for a class of operators associated with reversible elliptic diffusions, and use them to construct a local ascent algorithm, explicitly treating the case of multiple eigenvalues. We propose two methods to make our method tractable in high-dimensional systems: one based on dynamical coarse-graining, the other on recently obtained low-temperature shape-sensitive spectral asymptotics. We validate our method on a benchmark biomolecular system, showcasing a significant improvement over conventional definitions of metastable states.

physics.comp-ph↗

arXiv2026

Convergence rates for an Adaptive Biasing Potential scheme from a Wasserstein optimization perspective

Free-energy-based adaptive biasing methods, such as Metadynamics, the Adaptive Biasing Force (ABF) and their variants, are enhanced sampling algorithms widely used in molecular simulations. Although their efficiency has been empirically acknowledged for decades, providing theoretical insights via a quantitative convergence analysis is a difficult problem, in particular for the kinetic Langevin diffusion, which is non-reversible and hypocoercive. We obtain the first exponential convergence result for such a process, in an idealized setting where the dynamics can be associated with a mean-field non-linear flow on the space of probability measures. A key of the analysis is the interpretation of the (idealized) algorithm as the gradient descent of a suitable functional over the space of probability distributions.

math.PR↗

arXiv2025

Improving sampling by modifying the effective diffusion

Markov chain Monte Carlo samplers based on discretizations of (overdamped) Langevin dynamics are commonly used in the Bayesian inference and computational statistical physics literature to estimate high-dimensional integrals. One can introduce a non-constant diffusion matrix to precondition these dynamics, and recent works have optimized it in order to improve the rate of convergence to stationarity by overcoming entropic and energy barriers. However, the introduced methodologies to compute these optimal diffusions are generally not suited to high-dimensional settings, as they rely on costly optimization procedures. In this work, we propose to optimize over a class of diffusion matrices, based on one-dimensional collective variables (CVs), to help the dynamics explore the latent space defined by the CV. The form of the diffusion matrix is chosen in order to obtain an efficient effective diffusion in the latent space. We describe how this class of diffusion matrices can be constructed and learned during the simulation. We provide implementations of the Metropolis--Adjusted Langevin Algorithm and Riemann Manifold (Generalized) Hamiltonian Monte Carlo algorithms, and discuss numerical optimizations in the case when the CV depends only on a few degrees of freedom of the system. We illustrate the efficiency gains by computing mean transition durations between two metastable states of a dimer in a solvent.

math.NA↗

arXiv2025

Optimizing the diffusion coefficient of overdamped Langevin dynamics

Overdamped Langevin dynamics are reversible stochastic differential equations which are commonly used to sample probability measures in high-dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian inference. By varying the diffusion coefficient, there are in fact infinitely many overdamped Langevin dynamics which are reversible with respect to the target probability measure at hand. This suggests to optimize the diffusion coefficient in order to increase the convergence rate of the dynamics, as measured by the spectral gap of the generator associated with the stochastic differential equation. We analytically study this problem here, obtaining in particular necessary conditions on the optimal diffusion coefficient. We also derive an explicit expression of the optimal diffusion in some appropriate homogenized limit. Numerical results, both relying on discretizations of the spectral gap problem and Monte Carlo simulations of the stochastic dynamics, demonstrate the increased quality of the sampling arising from an appropriate choice of the diffusion coefficient.

math.NA↗

arXiv2024

Sampling metastable systems using collective variables and Jarzynski-Crooks paths

We consider the problem of sampling a high dimensional multimodal target probability measure. We assume that a good proposal kernel to move only a subset of the degrees of freedoms (also known as collective variables) is known a priori. This proposal kernel can for example be built using normalizing flows. We show how to extend the move from the collective variable space to the full space and how to implement an accept-reject step in order to get a reversible chain with respect to a target probability measure. The accept-reject step does not require to know the marginal of the original measure in the collective variable (namely to know the free energy). The obtained algorithm admits several variants, some of them being very close to methods which have been proposed previously in the literature. We show how the obtained acceptance ratio can be expressed in terms of the work which appears in the Jarzynski-Crooks equality, at least for some variants. Numerical illustrations demonstrate the efficiency of the approach on various simple test cases, and allow us to compare the variants of the algorithm.

cond-mat.stat-mech↗

arXiv2024

Quasi-stationary distribution for kinetic SDEs with low regularity coefficients

We consider kinetic SDEs with low regularity coefficients in the setting recently introduced in [6]. For the solutions to such equations, we first prove a Harnack inequality. Using the abstract approach of [5], this inequality then allows us to prove, under a Lyapunov condition, the existence and uniqueness (in a suitable class of measures) of a quasi-stationary distribution in cylindrical domains of the phase space. We finally exhibit two settings in which the Lyapunov condition holds: general kinetic SDEs in domains which are bounded in position, and Langevin processes with a non-conservative force and a suitable growth condition on the force.

math.PR↗

arXiv2024

Condensation in Fleming--Viot particle systems with fast selection mechanism

We study the Fleming--Viot particle system in a discrete state space, in the regime of a fast selection mechanism, namely with killing rates which grow to infinity. This asymptotics creates a time scale separation which results in the formation of a condensate of all particles, which then evolves according to a continuous-time Markov chain with jump rates depending in a nontrivial way on both the underlying mutation dynamics and the relative speed of growth to infinity of the killing rate between neighbouring sites. We prove the convergence of the particle system and completely describe the dynamics of the condensate in the case where the number of particles is kept fixed, and partially in the case when the number of particles goes to infinity together with (but slower than) the minimal killing rate.

math.PR↗

arXiv2024

A spectral approach to the narrow escape problem in the disk

We study the narrow escape problem in the disk, which consists in identifying the first exit time and first exit point distribution of a Brownian particle from the ball in dimension 2, with reflecting boundary conditions except on small disjoint windows through which it can escape. This problem is motivated by practical questions arising in various scientific fields (in particular cellular biology and molecular dynamics). We apply the quasi-stationary distribution approach to metastability, which requires to study the eigenvalue problem for the Laplacian operator with Dirichlet boundary conditions on the small absorbing part of the boundary, and Neumann boundary conditions on the remaining reflecting part. We obtain rigorous asymptotic estimates of the first eigenvalue and of the normal derivative of the associated eigenfunction in the limit of infinitely small exit regions, which yield asymptotic estimates of the first exit time and first exit point distribution starting from the quasi-stationary distribution within the disk.

math.AP↗

arXiv2024

Analyzing multimodal probability measures with autoencoders

Finding collective variables to describe some important coarse-grained information on physical systems, in particular metastable states, remains a key issue in molecular dynamics. Recently, machine learning techniques have been intensively used to complement and possibly bypass expert knowledge in order to construct collective variables. Our focus here is on neural network approaches based on autoencoders. We study some relevant mathematical properties of the loss function considered for training autoencoders, and provide physical interpretations based on conditional variances and minimum energy paths. We also consider various extensions in order to better describe physical systems, by incorporating more information on transition states at saddle points, and/or allowing for multiple decoders in order to describe several transition paths. Our results are illustrated on toy two dimensional systems and on alanine dipeptide.

physics.chem-ph↗

arXiv2023

An Application of Reduced Basis Methods to Core Computation in APOLLO3

In the aim of reducing the computational cost of the resolution of parameter-dependent eigenvalue problems, a model order reduction (MOR) procedure is proposed. We focus on the case of non-self-adjoint generalized eigenvalue problems, such as the stationary multigroup neutron diffusion equations. The method lies in an approximation of the manifold of solutions using a Proper Orthogonal Decomposition approach. The numerical method is composed of two stages. In the offline stage, we build a reduced space which approximates the manifold. In the online stage, for any given new set of parameters, we solve a reduced problem on the reduced space within a much smaller computational time than the required time to solve the high-fidelity problem. This method is applied to core computations in the APOLLO3 code.

math.NA↗

arXiv2023

Using Witten Laplacians to locate index-1 saddle points

We introduce a new stochastic algorithm to locate the index-1 saddle points of a function $V:\mathbb R^d \to \mathbb R$, with $d$ possibly large. This algorithm can be seen as an equivalent of the stochastic gradient descent which is a natural stochastic process to locate local minima. It relies on two ingredients: (i) the concentration properties on index-1 saddle points of the first eigenmodes of the Witten Laplacian (associated with $V$) on $1$-forms and (ii) a probabilistic representation of a partial differential equation involving this differential operator. Numerical examples on simple molecular systems illustrate the efficacy of the proposed approach.

math.NA↗

arXiv2023

Unbiasing Hamiltonian Monte Carlo algorithms for a general Hamiltonian function

Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo method that allows to sample high dimensional probability measures. It relies on the integration of the Hamiltonian dynamics to propose a move which is then accepted or rejected thanks to a Metropolis procedure. Unbiased sampling is guaranteed by the preservation by the numerical integrators of two key properties of the Hamiltonian dynamics: volume-preservation and reversibility up to momentum reversal. For separable Hamiltonian functions, some standard explicit numerical schemes, such as the Störmer-Verlet integrator, satisfy these properties. However, for numerical or physical reasons, one may consider a Hamiltonian function which is nonseparable, in which case the standard numerical schemes which preserve the volume and satisfy reversibility up to momentum reversal are implicit. When implemented in practice, such implicit schemes may admit many solutions or none, especially when the timestep is too large. We show here how to enforce the numerical reversibility, and thus unbiasedness, of HMC schemes in this context by introducing a reversibility check. In addition, for some specific forms of the Hamiltonian function, we discuss the consistency of these HMC schemes with some Langevin dynamics, and show in particular that our algorithm yields an efficient discretization of the metropolized overdamped Langevin dynamics with position-dependent diffusion coefficients. Numerical results illustrate the relevance of the reversibility check on simple problems.

math.NA↗

arXiv2023

Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations

In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the eigenvalue and left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation.

math.NA↗