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Derivation of optimal stochastic Runge-Kutta methods with exotic and decorated Butcher series for the weak integration of stochastic dynamics

Adrien Busnot Laurent, Kristian Debrabant, Anne Kværnø

Abstract

The design of numerical integrators for solving stochastic dynamics with high weak order relies on tedious calculations and is subject to a high number of order conditions. The original approaches from the literature consider strong approximations and adapt them for the weak approximation by replacing the iterated stochastic integrals by appropriate random variables. The methods obtained this way are sub-optimal in their number of function evaluations and the analysis of order conditions is unnecessarily complicated. We provide in this paper a novel approach, relying on well-chosen sets of random Runge-Kutta coefficients, that greatly reduce the number of order conditions. The approach is successfully applied to the creation of a collection of new stochastic Runge-Kutta methods of second weak order with an optimal number of function evaluations and a smaller number of random variables. The efficiency of the new methods is confirmed with numerical experiments and a modern algebraic approach using Hopf algebras is provided for the derivation and the study of the order conditions.

Derivation of optimal stochastic Runge-Kutta methods with exotic and decorated Butcher series for the weak integration of stochastic dynamics

Abstract

The design of numerical integrators for solving stochastic dynamics with high weak order relies on tedious calculations and is subject to a high number of order conditions. The original approaches from the literature consider strong approximations and adapt them for the weak approximation by replacing the iterated stochastic integrals by appropriate random variables. The methods obtained this way are sub-optimal in their number of function evaluations and the analysis of order conditions is unnecessarily complicated. We provide in this paper a novel approach, relying on well-chosen sets of random Runge-Kutta coefficients, that greatly reduce the number of order conditions. The approach is successfully applied to the creation of a collection of new stochastic Runge-Kutta methods of second weak order with an optimal number of function evaluations and a smaller number of random variables. The efficiency of the new methods is confirmed with numerical experiments and a modern algebraic approach using Hopf algebras is provided for the derivation and the study of the order conditions.
Paper Structure (15 sections, 12 theorems, 76 equations, 4 figures, 1 table)

This paper contains 15 sections, 12 theorems, 76 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Reduced order conditions for stochastic Runge-Kutta methods
  3. General order conditions for weak order two
  4. Reduced order conditions for Itô SDEs
  5. Reduced order conditions for Stratonovich SDEs
  6. New second order stochastic Runge-Kutta methods
  7. Optimal stochastic Runge-Kutta methods
  8. Stochastic Runge-Kutta methods with high deterministic order
  9. Numerical applications
  10. High order analysis with decorated and exotic series
  11. Decorated and exotic Butcher forests
  12. Algebraic expansion of flows with decorated and exotic forests
  13. Decorated and exotic S-series for stochastic numerics
  14. Conclusion
  15. Implicit-explicit stochastic Runge-Kutta methods

Key Result

Proposition 1

Consider a one-step integrator that has a Taylor expansion equation:num_expansion that satisfies Assume further that the integrator has bounded moments of any order, Then, the method has global weak order $p$, that is, for $T>0$, for all $h\leq h_0$ small enough with $Nh=T$, for all test functions $\phi\in\mathcal{C}\xspace^\infty_P(\mathbb{R}\xspace^d)$ and initial conditions $X_0$, there exist

Figures (4)

  • Figure 1: Numerical results for \ref{['ex:KP']}, methods of order (2,2)
  • Figure 2: Numerical results for \ref{['ex:KP']}, methods of order (3,2)
  • Figure 3: Numerical results for \ref{['ex:DR']}, methods of order (2,2)
  • Figure 4: Numerical results for \ref{['ex:DR']}, methods of order (3,2)

Theorems & Definitions (28)

  • Proposition 1: Milstein85waoMilstein04snf
  • Theorem 2.1
  • Remark 1
  • Theorem 2.2
  • Remark 2
  • Theorem 2.3
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • ...and 18 more