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Convergence of the micro-macro Parareal Method for a Linear Scale-Separated Ornstein-Uhlenbeck SDE: extended version

Ignace Bossuyt, Giovanni Samaey, Stefan Vandewalle

TL;DR

This work develops and validates a micro-macro Parareal framework for linear, scale-separated OU SDEs, combining a high-dimensional fine propagator with a reduced coarse propagator. By deriving moment-based ODEs and leveraging averaging, the authors establish convergence results for mean and covariance across iterations, including affine (inhomogeneous) extensions of classical Parareal theory. They prove that, for iterations k ≥ 1, the inhomogeneity does not affect error propagation in the covariance analysis, while the zeroth iteration carries the inhomogeneity impact. Numerical experiments confirm the predicted convergence behavior, showing faster convergence for smaller time-scale separation parameter ε. The results provide a principled basis for parallel-in-time simulation of multiscale SDEs and highlight avenues for extension to higher dimensions and nonlinear settings.

Abstract

Time-parallel methods can reduce the wall clock time required for the accurate numerical solution of differential equations by parallelizing across the time-dimension. In this paper, we present and test the convergence behavior of a multiscale, micro-macro version of a Parareal method for stochastic differential equations (SDEs). In our method, the fine propagator of the SDE is based on a high-dimensional slow-fast microscopic model; the coarse propagator is based on a model-reduced version of the latter, that captures the low-dimensional, effective dynamics at the slow time scales. We investigate how the model error of the approximate model influences the convergence of the micro-macro Parareal algorithm and we support our analysis with numerical experiments. This is an extended and corrected version of [Domain Decomposition Methods in Science and Engineering XXVII. DD 2022, vol 149 (2024), pp. 69-76, Bossuyt, I., Vandewalle, S., Samaey, G.].

Convergence of the micro-macro Parareal Method for a Linear Scale-Separated Ornstein-Uhlenbeck SDE: extended version

TL;DR

This work develops and validates a micro-macro Parareal framework for linear, scale-separated OU SDEs, combining a high-dimensional fine propagator with a reduced coarse propagator. By deriving moment-based ODEs and leveraging averaging, the authors establish convergence results for mean and covariance across iterations, including affine (inhomogeneous) extensions of classical Parareal theory. They prove that, for iterations k ≥ 1, the inhomogeneity does not affect error propagation in the covariance analysis, while the zeroth iteration carries the inhomogeneity impact. Numerical experiments confirm the predicted convergence behavior, showing faster convergence for smaller time-scale separation parameter ε. The results provide a principled basis for parallel-in-time simulation of multiscale SDEs and highlight avenues for extension to higher dimensions and nonlinear settings.

Abstract

Time-parallel methods can reduce the wall clock time required for the accurate numerical solution of differential equations by parallelizing across the time-dimension. In this paper, we present and test the convergence behavior of a multiscale, micro-macro version of a Parareal method for stochastic differential equations (SDEs). In our method, the fine propagator of the SDE is based on a high-dimensional slow-fast microscopic model; the coarse propagator is based on a model-reduced version of the latter, that captures the low-dimensional, effective dynamics at the slow time scales. We investigate how the model error of the approximate model influences the convergence of the micro-macro Parareal algorithm and we support our analysis with numerical experiments. This is an extended and corrected version of [Domain Decomposition Methods in Science and Engineering XXVII. DD 2022, vol 149 (2024), pp. 69-76, Bossuyt, I., Vandewalle, S., Samaey, G.].

Paper Structure

This paper contains 21 sections, 12 theorems, 75 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and motivation
  2. Convergence of the micro-macro Parareal Algorithm for a linear multiscale ODE
  3. Linear multiscale ODE system and its properties
  4. Convergence of micro-macro Parareal for a linear multiscale test ODE
  5. Convergence of micro-macro Parareal for a linear multiscale test ODE with another lifting operator
  6. Generalization of convergence theory of Parareal for linear ODEs to affine ODEs
  7. Stochastic differential equation model problem
  8. Derivation of a reduced model for multiscale SDEs
  9. Moments of the Ornstein-Uhlenbeck process and its reduced model
  10. Using micro-macro Parareal for the model problem
  11. Overview of assumptions on model parameters
  12. Convergence of micro-macro Parareal for the multiscale Ornstein-Uhlenbeck SDE model problem
  13. Properties of the homogeneous part of the ODE describing the covariance
  14. Properties of the non-homogeneous ODE describing the covariance
  15. Convergence of micro-macro Parareal for the inhomogeneous ODE system describing the covariance
  16. ...and 6 more sections

Key Result

lemma 1

For the initial-value problem test_system_Samaey_2013 and its reduced model definition_lambda, there exists $\epsilon_0 \in (0,1)$, and a constant $C > 0$, independent of $\epsilon$, such that, for all $\epsilon < \epsilon_0$, where $t^{\mathrm{BL}}_{\epsilon}$ is the length of a boundary layer in time of the order of $\epsilon$:

Figures (2)

  • Figure 1: As a function of $\epsilon$, we plotted (left) the real part of the eigenvalues of $A_{\Sigma}$ and (right) the value of $\mu_{\Sigma}$.
  • Figure 2: Error as function of time-scale separation parameter $\epsilon$. We used $\infty$-norm over time (only considering coarse discretization points) and the 2-norm for the micro error. Top left: macro error on mean, Top right: micro error on mean, Bottom left: macro error on variance, Bottom right; micro error on variance. We used a numerical solver to discretize the moment equations \ref{['OU_mean_reduced']} - \ref{['OU_variance_full']} with a very stringent tolerance, so that the effect of numerical discretization errors can be neglected.

Theorems & Definitions (27)

  • lemma 1: Properties of the multiscale system \ref{['test_system_Samaey_2013']} and its reduced model \ref{['definition_lambda']}
  • proof
  • proof
  • lemma 2: Convergence of micro-macro Parareal for homogeneous linear multiscale ODEs \ref{['test_system_Samaey_2013']}-\ref{['definition_lambda']}
  • proof
  • lemma 3: Convergence of micro-macro Parareal with lifting based on the initial condition in the zeroth iteration for homogeneous linear multiscale ODEs
  • proof
  • lemma 4: Error propagation property of micro-macro Parareal for nonhomogenous linear ODEs with constant coefficients
  • remark 1: Alternative derivation of reduced model for evolution of slow variance
  • lemma 5: Inverse of $B_{\Sigma}$
  • ...and 17 more