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An Antithetic Multilevel Monte Carlo-Milstein Scheme for Stochastic Partial Differential Equations with non-commutative noise

Abdul-Lateef Haji-Ali, Andreas Stein

Abstract

We present a novel multilevel Monte Carlo approach for estimating quantities of interest for stochastic partial differential equations (SPDEs). Drawing inspiration from [Giles and Szpruch: Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation, Annals of Appl. Prob., 2014], we extend the antithetic Milstein scheme for finite-dimensional stochastic differential equations to Hilbert space-valued SPDEs. Our method has the advantages of both Euler and Milstein discretizations, as it is easy to implement and does not involve intractable Lévy area terms. Moreover, the antithetic correction in our method leads to the same variance decay in a MLMC algorithm as the standard Milstein method, resulting in significantly lower computational complexity than a corresponding MLMC Euler scheme. Our approach is applicable to a broader range of non-linear diffusion coefficients and does not require any commutative properties. The key component of our MLMC algorithm is a truncated Milstein-type time stepping scheme for SPDEs, which accelerates the rate of variance decay in the MLMC method when combined with an antithetic coupling on the fine scales. We combine the truncated Milstein scheme with appropriate spatial discretizations and noise approximations on all scales to obtain a fully discrete scheme and show that the antithetic coupling does not introduce an additional bias.

An Antithetic Multilevel Monte Carlo-Milstein Scheme for Stochastic Partial Differential Equations with non-commutative noise

Abstract

We present a novel multilevel Monte Carlo approach for estimating quantities of interest for stochastic partial differential equations (SPDEs). Drawing inspiration from [Giles and Szpruch: Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation, Annals of Appl. Prob., 2014], we extend the antithetic Milstein scheme for finite-dimensional stochastic differential equations to Hilbert space-valued SPDEs. Our method has the advantages of both Euler and Milstein discretizations, as it is easy to implement and does not involve intractable Lévy area terms. Moreover, the antithetic correction in our method leads to the same variance decay in a MLMC algorithm as the standard Milstein method, resulting in significantly lower computational complexity than a corresponding MLMC Euler scheme. Our approach is applicable to a broader range of non-linear diffusion coefficients and does not require any commutative properties. The key component of our MLMC algorithm is a truncated Milstein-type time stepping scheme for SPDEs, which accelerates the rate of variance decay in the MLMC method when combined with an antithetic coupling on the fine scales. We combine the truncated Milstein scheme with appropriate spatial discretizations and noise approximations on all scales to obtain a fully discrete scheme and show that the antithetic coupling does not introduce an additional bias.
Paper Structure (21 sections, 14 theorems, 139 equations, 2 figures)

This paper contains 21 sections, 14 theorems, 139 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Preliminaries
  5. Basic Notation
  6. Hilbert-Schmidt Operators and RKHS
  7. Martingales on Hilbert Spaces
  8. Wiener Process on a Hilbert Space
  9. Stochastic Partial Differential Equations
  10. Pathwise Approximations
  11. Spatial Discretization
  12. Noise Approximation
  13. Time Stepping
  14. Antithetic Multilevel Monte Carlo-Milstein Scheme
  15. Multilevel Monte Carlo Approximation
  16. ...and 6 more sections

Key Result

Theorem 2.7

Under Assumption ass:SPDE, there exists a unique mild solution $X\in\mathcal{X}_\mathbb{T}$ to eq:spde, such that for all $p\in (0, 8]$ and $\kappa\in[0, \alpha)$ it holds that

Figures (2)

  • Figure 1: Estimates of the $L^{2}(\Omega ; H)$ difference $\max_{m} \|Y_{m}^{N,K} - Y_{m}^{\lceil \sqrt{2} N \rceil, K}\|_{L^{2}(\Omega; H)}$ for the numerical example in Section \ref{['sec:numerics']} and when using the Galerkin method for different number of terms, $N$, in the spatial approximation. The estimates were obtained using Monte Carlo sampling with at least 4000 samples. The dashed reference lines are $\propto N^{-\min(1+s, 2)/d}$ and verify the assumed convergence rates in Assumption \ref{['ass:approximation']}.
  • Figure 2: Results for numerical example in Section \ref{['sec:numerics']} and $M,N$ and $K$ as in \ref{['eq:balance']}. (left) Shows the left-hand sides of \ref{['eq:var-decay']}, the variance for the antithetic estimator, and \ref{['eq:var-decay-EM']}, the variance for the "Standard" truncated Milstein estimator without the antithetic correction, for the smoothness parameter $s=3d/4$. (right) Shows the relative variance decay between the two estimators, i.e., $\max_{m}\mathbb{E}\left(\left\|\overline Y_m-Y_m^c\right\|_H^2\right) / \max_{m}\mathbb{E}\left(\left\|Y^{f}_m-Y_m^c\right\|_H^2\right) =\mathcal{O}(M^{-\min(s, 1)})$, for different smoothness parameters $s$. The variance estimates were obtained using Monte Carlo sampling with at least 4000 samples.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7
  • Proposition 3.1
  • proof
  • Example 3.3
  • Theorem 3.4
  • ...and 28 more