Showing papers by "Benjamin Jourdain published in 2018"

Convergence and Efficiency of Adaptive Importance Sampling Techniques with Partial Biasing

Abstract: We propose a new Monte Carlo method to efficiently sample a multimodal distribution (known up to a normalization constant). We consider a generalization of the discrete-time Self Healing Umbrella Sampling method, which can also be seen as a generalization of well-tempered metadynamics. The dynamics is based on an adaptive importance technique. The importance function relies on the weights (namely the relative probabilities) of disjoint sets which form a partition of the space. These weights are unknown but are learnt on the fly yielding an adaptive algorithm. In the context of computational statistical physics, the logarithm of these weights is, up to an additive constant, the free-energy, and the discrete valued function defining the partition is called the collective variable. The algorithm falls into the general class of Wang–Landau type methods, and is a generalization of the original Self Healing Umbrella Sampling method in two ways: (i) the updating strategy leads to a larger penalization strength of already visited sets in order to escape more quickly from metastable states, and (ii) the target distribution is biased using only a fraction of the free-energy, in order to increase the effective sample size and reduce the variance of importance sampling estimators. We prove the convergence of the algorithm and analyze numerically its efficiency on a toy example.

Computation of sensitivities for the invariant measure of a parameter dependent diffusion

Abstract: We consider the solution to a stochastic differential equation with a drift function which depends smoothly on some real parameter λ, and admitting a unique invariant measure for any value of λ around λ = 0. Our aim is to compute the derivative with respect to λ of averages with respect to the invariant measure, at λ = 0. We analyze a numerical method which consists in simulating the process at λ = 0 together with its derivative with respect to λ on long time horizon. We give sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to λ of the mean of an observable through Monte Carlo simulations.

Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean

Abstract: In this paper, we prove that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step h of a system of N interacting particles is O(1/N + h). We provide numerical experiments confirming this behaviour and showing that it extends to more general mean-field interaction and study the efficiency of the antithetic sampling technique on the same examples.

Squared quadratic Wasserstein distance : optimal couplings and Lions differentiability

Abstract: In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance $W^2_2(\mu, u)$ between two probability measures $\mu$ and $ u$ with finite second order moments on $\mathbb{R}^d$ is the composition of a martingale coupling with an optimal transport map ${\mathcal T}$. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between $\mu$ and ${\mathcal T}\#\mu$. Next, we give a direct proof that $\sigma\mapsto W_2^2(\sigma, u)$ is differentiable at $\mu$ in the Lions sense iff there is a unique optimal coupling between $\mu$ and $ u$ and this coupling is given by a map. It was known combining results by Ambrosio, Gigli and Savar\'e and Ambrosio and Gangbo that, under the latter condition, geometric differentiability holds. Moreover, the two notions of differentiability are equivalent according to the recent paper of Gangbo and Tudorascu. Besides, we give a self-contained probabilistic proof that mere Fr\'echet differentiability of a law invariant function $F$ on $L^2(\Omega,\mathbb{P};\mathbb{R}^d)$ is enough for the Fr\'echet differential at $X$ to be a measurable function of $X$.

Evolution of the Wasserstein distance between the marginals of two Markov processes

Abstract: In this paper, we are interested in the time derivative of the Wasserstein distance between the marginals of two Markov processes. As recalled in the introduction, the Kantorovich duality leads to a natural candidate for this derivative. Up to the sign, it is the sum of the integrals with respect to each of the two marginals of the corresponding generator applied to the corresponding Kantorovich potential. For pure jump processes with bounded intensity of jumps, we prove that the evolution of the Wasserstein distance is actually given by this candidate. In dimension one, we show that this remains true for Piecewise Deterministic Markov Processes.

Lifted and geometric differentiability of the squared quadratic Wasserstein distance

Abstract: In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance $W^2_2(\mu, u)$ between two probability measures $\mu$ and $ u$ with finite second order moments on $\mathbb{R}^d$ is the composition of a martingale coupling with an optimal transport map ${\mathcal T}$. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between $\mu$ and ${\mathcal T}\#\mu$. Next, we prove that $\sigma\mapsto W_2^2(\sigma, u)$ is differentiable at $\mu$ in the Lions~\cite{Lions} and the geometric senses iff there is a unique optimal coupling between $\mu$ and $ u$ and this coupling is given by a map. Besides, we give a self-contained proof that mere Frechet differentiability of a law invariant function $F$ on $L^2(\Omega,\mathbb{P};\mathbb{R}^d)$ is enough for the Frechet differential at $X$ to be a measurable function of $X$.

A new family of one dimensional martingale couplings

Abstract: In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures $\mu$ and $ u$ in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of $\mu$ and $ u$. It contains the inverse transform martingale coupling which is explicit in terms of the associated cumulative distribution functions. The integral of $\vert x-y\vert$ with respect to each of these couplings is smaller than twice the $W_1$ distance between $\mu$ and $ u$. When $\mu$ and $ u$ are in the decreasing (resp. increasing) convex order, the construction is generalised to exhibit super (resp. sub) martingale couplings.