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Alvin Tse

Publications and source records attributed to Alvin Tse.

6 recordsLinked to original sources
arXiv2021

Central limit theorem over non-linear functionals of empirical measures with applications to the mean-field fluctuation of interacting particle systems

In this work, a generalised version of the central limit theorem is proposed for nonlinear functionals of the empirical measure of i.i.d. random variables, provided that the functional satisfies some regularity assumptions for the associated linear functional derivative. This generalisation can be applied to Monte-Carlo methods, even when there is a nonlinear dependence on the measure component. We use this result to deal with the contribution of the initialisation in the convergence of the fluctuations between the empirical measure of interacting diffusion and their mean-field limiting measure (as the number of particles goes to infinity), when the dependence on measure is nonlinear. A complementary contribution related to the time evolution is treated using the master equation, a parabolic PDE involving L-derivatives with respect to the measure component, which is a stronger notion of derivative that is nonetheless related to the linear functional derivative.

math.PR↗
arXiv2021

Uniform in time weak propagation of chaos on the torus

We address the long time behaviour of weakly interacting diffusive particle systems on the d-dimensional torus. Our main result is to show that, under certain regularity conditions, the weak error between the empirical distribution of the particle system and the theoretical law of the limiting process (governed by a McKean-Vlasov stochastic differential equation) is of the order O(1/N), uniform in time on [0, infinity), where N is the number of particles in the interacting diffusion. This comprises general interaction terms with a small enough mean-field dependence together with interactions terms driven by an H-stable potential. Our approach relies on a systematic analysis of the long-time behaviour of the derivatives of the semigroup generated by the McKean-Vlasov SDE, which may be explicitly computed through the linearised Fokker-Planck equation. Ergodic estimates for the latter hence play a key role in our analysis. We believe that this strategy is flexible enough to cover a wider broad of situations. To wit, we succeed in adapting it to the super-critical Kuramoto model, for which the corresponding McKean-Vlasov equation has several invariant measures.

math.AP↗
arXiv2021

Antithetic multilevel particle system sampling method for McKean-Vlasov SDEs

Let $μ\in \mathcal{P}_2(\mathbb R^d)$, where $\mathcal{P}_2(\mathbb R^d)$ denotes the space of square integrable probability measures, and consider a Borel-measurable function $Φ:\mathcal P_2(\mathbb R^d)\rightarrow \mathbb R $. IIn this paper we develop Antithetic Monte Carlo estimator (A-MLMC) for $Φ(μ)$, which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws $μ^N = \frac1N \sum_{i=1}^{N}δ_{X_i}$, where a) $(X_i)_{i=1}^N$ is a sequence of i.i.d samples from $μ$ or b) $(X_i)_{i=1}^N$ is a system of interacting particles (diffusions) corresponding to a McKean-Vlasov stochastic differential equation (McKV-SDE). Each case requires a separate analysis. For a mean-field particle system, we also consider the empirical law induced by its Euler discretisation which gives a fully implementable algorithm. As by-products of our analysis, we establish a dimension-independent rate of uniform \textit{strong propagation of chaos}, as well as an $L^2$ estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.

math.PR↗
arXiv2019

Iterative Particle Approximation for McKean-Vlasov SDEs with application to Multilevel Monte Carlo estimation

The mean field limits of systems of interacting diffusions (also called stochastic interacting particle systems (SIPS)) have been intensively studied since McKean \cite{mckean1966class}. The interacting diffusions pave a way to probabilistic representations for many important nonlinear/nonlocal PDEs, but provide a great challenge for Monte Carlo simulations. This is due to the nonlinear dependence of the bias on the statistical error arising through the approximation of the law of the process. This and the fact that particles/diffusions are not independent render classical variance reduction techniques not directly applicable and consequently make simulations of interacting diffusions prohibitive. In this article, we provide an alternative iterative particle representation, inspired by the fixed point argument by Sznitman \cite{sznitman1991topics}. This new representation has the same mean field limit as the classical SIPS. However, unlike classical SIPS, it also allows decomposing the statistical error and the approximation bias. We develop a general framework to study integrability and regularity properties of the iterated particle system. Moreover, we establish its weak convergence to the McKean-Vlasov SDEs (MVSDEs). One of the immediate advantages of iterative particle system is that it can be combined with the Multilevel Monte Carlo (MLMC) approach for the simulation of MVSDEs. We proved that the MLMC approach reduces the computational complexity of calculating expectations by an order of magnitude. Another perspective on this work is that we analyse the error of nested Multilevel Monte Carlo estimators, which is of independent interest. Furthermore, we work with state dependent functionals, unlike scalar outputs which are common in literature on MLMC. The error analysis is carried out in uniform, and what seems to be new, weighted norms.

math.PR↗
arXiv2019

Weak quantitative propagation of chaos via differential calculus on the space of measures

Consider the metric space $(\mathcal{P}_2(\mathbb{R}^d),W_2)$ of square integrable laws on $\mathbb{R}^d$ with the topology induced by the 2-Wasserstein distance $W_2$. Let $Φ: \mathcal{P}_2( \mathbb{R}^d) \to \mathbb{R}$ be a function and $μ_N$ be the empirical measure of a sample of $N$ random variables distributed as $μ$. The main result of this paper is to show that under suitable regularity conditions, we have \[ |Φ(μ) - \mathbb{E}Φ(μ_N)|= \sum_{j=1}^{k-1}\frac{C_j}{N^j} + O(\frac{1}{N^k}), \] for some positive constants $C_1, \ldots, C_{k-1}$ that do not depend on $N$, where $k$ corresponds to the degree of smoothness. We distinguish two cases: a) $μ_N$ is the empirical measure of $N$-samples from $μ$; b) $μ$ is a marginal law of McKean-Vlasov stochastic differential equation in which case $μ_N$ is an empirical law of marginal laws of the corresponding particle system. The first case is studied using functional derivatives on the space of measures. The second case relies on an Itô-type formula for the flow of probability measures and is intimately connected to PDEs on the space of measures, called the master equation in the literature of mean-field games. We state the general regularity conditions required for each case and analyse the regularity in the case of functionals of the laws of McKean-Vlasov SDEs. Ultimately, this work reveals quantitative estimates of propagation of chaos for interacting particle systems. Furthermore, we are able to provide weak propagation of chaos estimates for ensembles of interacting particles and show that these may have some remarkable properties.

math.PR↗