In this paper, we analyse a proximal method based on the idea of forward-backward splitting for sampling from distributions with densities that are not necessarily smooth. In particular, we study the non-asymptotic properties of the Euler-Maruyama discretization of the Langevin equation, where the forward-backward envelope is used to deal with the non-smooth part of the dynamics. An advantage of this envelope, when compared to widely-used Moreu-Yoshida one and the MYULA algorithm, is that it maintains the MAP estimator of the original non-smooth distribution. We also study a number of numerical experiments that corroborate that support our theoretical findings.
This paper considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the fractional Brownian motion and the singular kernel, which is overcome by means of the Malliavin calculus and fine estimates of several multiple singular integrals. Consequently, these two methods are proved to be strongly convergent with order nearly $\min\{2(H+α-1), α\}$, where $H \in (1/2,1)$ and $α\in(1-H,1)$ respectively characterize the singularity levels o…
The Strictly Correlated Electrons (SCE) limit of the Levy-Lieb functional in Density Functional Theory (DFT) gives rise to a symmetric multi-marginal optimal transport problem with Coulomb cost, where the number of marginal laws is equal to the number of electrons in the system, which can be very large in relevant applications. In this work, we design a numerical method, built upon constrained overdamped Langevin processes to solve Moment Constrained Optimal Transport (MCOT) relaxations (introduced in A. Alfonsi, R. Coyaud, V. Ehrlacher and D. Lombardi, Math. Comp. 90, 2021, 689--737) of symme…
We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution $π$ on $\mathbb{R}^d$, based on (overdamped) Langevin diffusions. This method inspired by \cite{mainPPlangevin} and \cite{giles_szpruch_invariant} relies on a multilevel occupation measure, $i.e.$ on an appropriate combination of $R$ occupation measures of (constant-step) Euler schemes with respective steps $γ_r = γ_0 2^{-r}$, $r=0,\ldots,R$. We first state a quantitative result under general assumptions which guarantees an \textit{$\varepsilon$-approximation} (in a $L^2$-sense) with a cost of …
We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of non-degeneracy and er…
