Monte-Carlo/Moments micro-macro Parareal method for unimodal and bimodal scalar McKean-Vlasov SDEs

TL;DR

A micro-macro parallel-in-time Parareal method for scalar McKean-Vlasov stochastic differential equations (SDEs) and shows that convergence typically takes place in a low number of iterations, depending on the quality of the ODE predictor.

Abstract

We propose a micro-macro parallel-in-time Parareal method for scalar McKean-Vlasov stochastic differential equations (SDEs). In the algorithm, the fine Parareal propagator is a Monte Carlo simulation of an ensemble of particles, while an approximate ordinary differential equation (ODE) description of the mean and the variance of the particle distribution is used as a coarse Parareal propagator to achieve speedup. We analyse the convergence behaviour of our method for a linear problem and provide numerical experiments indicating the parallel weak scaling of the algorithm on a set of examples. We show, with numerical experiments, that convergence typically takes place in a low number of iterations, depending on the quality of the ODE predictor. For bimodal SDEs, we avoid quality deterioration of the coarse predictor (compared to unimodal SDEs) through the usage of multiple ODEs, each describing the mean and variance of the particle distribution in locally unimodal regions of the phase space. The benefit of the proposed algorithm can be viewed through two lenses: (i) through the parallel-in-time lens, speedup is obtained through the use of a very cheap coarse integrator (an ODE moment model), and (ii) through the moment models lens, accuracy is iteratively gained through the use of parallel machinery as a corrector. In contrast to the isolated use of a moment model, the proposed method (iteratively) converges to the true distribution generated by the SDE.

Figures (18)

• Figure 1: Illustration of a potential pitfall with particle transformations. The regions of attraction are separated by a separatrix. The green distributions are the locally unimodal distributions in each region of attraction before matching. The cyan ones result after application of the matching operator, but this changes the particle fractions.
• Figure 1: Illustration of the convergence bound for a simple test problem. The reference solution equals the sequential solution using the fine propagator $\mathcal{F}_n$ with $\beta= 0$. (Blue) $\beta = 10^{-2}$, (orange) $\beta= 10^{-6}$ and (green) $\beta = 0$.
• Figure 1: Modified geometric Brownian motion: (left) approximation of a nonlinear QoI using two approximation methods, (middle) MC-moments Parareal convergence of the weak error for the QoI with three variants, (right) histogram at the beginning and at the end of the time window.
• Figure 1: Overview of test systems for the bimodal SDE: evolution of particle fractions in function of time.
• Figure 2: Convergence of (the error on) the mean and variance in the Parareal approximation of the Ornstein-Uhlenbeck SDE. We also plot the minimum of the linear and superlinear convergence bounds from \ref{['lemma_what_happens_with_noise']} with and without statistical noise included.

Theorems & Definitions (33)

• Remark 1: About expectations
• Lemma 2.1: Exactness of the Taylor-based moment model for linear SDEs
• Remark 2: Computation of the regions of attraction
• Proof 1
• Remark 3: micro-macro Parareal revisited
• Lemma 3.3: Consistency of the operators $\mathcal{S}$ and $\mathcal{T}$
• Definition 3.4: MC-moments Parareal for unimodal scalar McKean-Vlasov SDEs
• Definition 3.5: MC-moments Parareal for multimodal scalar McKean-Vlasov SDEs
• Proof 2
• Remark 4: About the update of the particle fractions