[Paper Review] Exotic B-series and S-series: algebraic structures and order conditions for invariant measure sampling

Eugen Bronasco|arXiv (Cornell University)|Sep 22, 2022

Advanced Topics in Algebra4 citations

TL;DR

This paper introduces exotic B-series and S-series with a novel symmetry normalization to analyze order conditions for stochastic Runge-Kutta methods that sample the invariant measure of ergodic SDEs. By establishing algebraic relationships between Grossman-Larson algebras over exotic and grafted forests and their dual Connes-Kreimer coalgebras, the authors derive a multiplicative composition law that ensures certain order conditions are automatically satisfied, significantly simplifying the construction of high-order invariant measure integrators.

ABSTRACT

B-series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-series was introduced to study the order conditions for sampling the invariant measure of ergodic SDEs. Introducing a new symmetry normalization coefficient, we analyze the algebraic structures related to exotic B-series and S-series. Precisely, we prove the relationship between the Grossman-Larson algebras over exotic and grafted forests and the corresponding duals to the Connes-Kreimer coalgebras and use it to study the natural composition laws on exotic S-series. Applying this algebraic framework to the derivation of order conditions for a class of stochastic Runge-Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.

Motivation & Objective

• To develop a systematic algebraic framework for analyzing order conditions in stochastic integrators targeting the invariant measure of ergodic SDEs.
• To extend B-series and S-series to exotic and grafted forests in the stochastic context, particularly for additive noise SDEs.
• To introduce a new symmetry normalization coefficient to unify the treatment of exotic trees and forests.
• To establish a composition law for exotic S-series that enables automatic satisfaction of certain order conditions.
• To apply the framework to stochastic Runge-Kutta methods, deriving explicit order conditions up to order 3 for invariant measure sampling.

Proposed method

• The paper constructs exotic aromatic forests and defines exotic S-series using a new symmetry normalization coefficient σ(τ) analogous to the deterministic case.
• It establishes a duality between Grossman-Larson algebras over exotic and grafted forests and the Connes-Kreimer coalgebras, enabling the definition of natural composition laws.
• The authors use decorated aromatic forests to model vector fields arising in SDEs with additive noise, particularly in the overdamped Langevin equation.
• A multiplicative composition law is derived for exotic S-series, which ensures that certain order conditions are satisfied automatically when the method satisfies lower-order conditions.
• The framework is applied to stochastic Runge-Kutta methods, with order conditions derived via an algorithmic procedure and verified using explicit formulas.
• The paper provides a complete list of order conditions up to order 3, with marked conditions that are automatically satisfied due to the composition law.

Experimental results

Research questions

• RQ1 How can exotic B-series and S-series be algebraically structured to analyze order conditions for invariant measure sampling in ergodic SDEs?
• RQ2 What is the role of the new symmetry normalization coefficient in unifying the treatment of exotic and grafted forests in the stochastic setting?
• RQ3 How do the algebraic structures of Grossman-Larson and Connes-Kreimer relate in the context of exotic forests and their duals?
• RQ4 Which order conditions for stochastic Runge-Kutta methods are automatically satisfied due to a multiplicative composition law in the exotic S-series framework?
• RQ5 What explicit order conditions up to order 3 govern the convergence of stochastic integrators to the invariant measure?

Key findings

• The paper derives a multiplicative composition law for exotic S-series that ensures certain order conditions are automatically satisfied, reducing the number of conditions to verify.
• For order 3, the method identifies three conditions (marked with ∗ in Table 2) that are automatically satisfied due to the composition law, simplifying the construction of high-order integrators.
• The framework successfully generates all order conditions up to order 3 for stochastic Runge-Kutta methods targeting the invariant measure, with explicit formulas provided.
• The symmetry normalization coefficient σ(τ) is shown to be essential in maintaining consistency between the algebraic structure and the stochastic dynamics of the problem.
• The duality between Grossman-Larson algebras over exotic forests and Connes-Kreimer coalgebras is rigorously established, enabling the derivation of composition laws.
• The application to the overdamped Langevin equation confirms the framework's utility in analyzing ergodic SDEs with additive noise.