On the random generation of Butcher trees

TL;DR

This paper develops a Monte Carlo framework to represent ODE solutions via random Butcher trees, providing an alternative to fixed-order truncations of the Butcher series and avoiding random branching times. The core idea is to sample labelled Butcher trees with sizes drawn from a distribution $p_n$ and express the solution as an expectation of $F(\mathcal T)(x_0)$ scaled by the tree size and $p_{|\mathcal T|}$. The authors extend the approach to semilinear ODEs using Poisson-distributed tree sizes and a continuous-time Markov generator $A$, yielding an exponential Butcher-like series and enabling connections to semilinear PDEs. They present both theoretical moment bounds and practical Monte Carlo implementations, along with numerical experiments and multidimensional code to illustrate variance behavior and computational tradeoffs compared to traditional truncations. The work provides a probabilistic numerical tool that complements deterministic Runge-Kutta methods and has potential applications in nonlinear PDE representations via branching-tree constructions.

Abstract

The main goal of this paper is to provide an algorithm for the random sampling of Butcher trees and the probabilistic numerical solution of ordinary differential equations (ODEs). This approach complements and simplifies a recent approach to the probabilistic representation of ODE solutions, by removing the need to generate random branching times. The random sampling of trees is compared to the finite order truncation of Butcher series in numerical experiments.

Figures (4)

• Figure 1: Generation of Butcher trees.
• Figure 2: Comparisons of \ref{['Butcher']}vs.\ref{['random-Butcher']}.
• Figure 3: Second moment lower bound.
• Figure 4: Numerical solution of \ref{['e1']} by the Monte Carlo method \ref{['random-Butcher']}.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 4
• Proposition 2