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The random timestep Euler method and its continuous dynamics

Jonas Latz

TL;DR

The paper introduces stochastic Euler dynamics, a continuous-time Markov process that linear-interpolates a forward Euler step with exponentially distributed random timesteps, yielding a piecewise linear trajectory with a companion constant process. It establishes convergence to the ODE solution via infinitesimal-generator analysis, derives a local RMS truncation error of order $O(\varepsilon^2)$ for linear ODEs, and proves stability results using mean, second moment, and Foster–Lyapunov criteria, with explicit thresholds depending on the step parameter $h$ and spectrum of the linear part. It further extends the framework to second-order stochastic Euler dynamics and validates the theory through numerical experiments on a 1D linear problem and an underdamped oscillator, highlighting both the potential and limitations of higher-order stochastic schemes. The results provide a principled basis for randomized timestep methods in irregular and chaotic ODEs and offer directions for future work in random timestep Runge-Kutta methods and data-driven dynamics.

Abstract

ODE solvers with randomly sampled timestep sizes appear in the context of chaotic dynamical systems, differential equations with low regularity, and, implicitly, in stochastic optimisation. In this work, we propose and study the stochastic Euler dynamics - a continuous-time Markov process that is equivalent to a linear spline interpolation of a random timestep (forward) Euler method. We understand the stochastic Euler dynamics as a path-valued ansatz for the ODE solution that shall be approximated. We first obtain qualitative insights by studying deterministic Euler dynamics which we derive through a first order approximation to the infinitesimal generator of the stochastic Euler dynamics. Then we show convergence of the stochastic Euler dynamics to the ODE solution by studying the associated infinitesimal generators and by a novel local truncation error analysis. Next we prove stability by an immediate analysis of the random timestep Euler method and by deriving Foster-Lyapunov criteria for the stochastic Euler dynamics; the latter also yield bounds on the speed of convergence to stationarity. The paper ends with a discussion of second-order stochastic Euler dynamics and a series of numerical experiments that appear to verify our analytical results.

The random timestep Euler method and its continuous dynamics

TL;DR

The paper introduces stochastic Euler dynamics, a continuous-time Markov process that linear-interpolates a forward Euler step with exponentially distributed random timesteps, yielding a piecewise linear trajectory with a companion constant process. It establishes convergence to the ODE solution via infinitesimal-generator analysis, derives a local RMS truncation error of order for linear ODEs, and proves stability results using mean, second moment, and Foster–Lyapunov criteria, with explicit thresholds depending on the step parameter and spectrum of the linear part. It further extends the framework to second-order stochastic Euler dynamics and validates the theory through numerical experiments on a 1D linear problem and an underdamped oscillator, highlighting both the potential and limitations of higher-order stochastic schemes. The results provide a principled basis for randomized timestep methods in irregular and chaotic ODEs and offer directions for future work in random timestep Runge-Kutta methods and data-driven dynamics.

Abstract

ODE solvers with randomly sampled timestep sizes appear in the context of chaotic dynamical systems, differential equations with low regularity, and, implicitly, in stochastic optimisation. In this work, we propose and study the stochastic Euler dynamics - a continuous-time Markov process that is equivalent to a linear spline interpolation of a random timestep (forward) Euler method. We understand the stochastic Euler dynamics as a path-valued ansatz for the ODE solution that shall be approximated. We first obtain qualitative insights by studying deterministic Euler dynamics which we derive through a first order approximation to the infinitesimal generator of the stochastic Euler dynamics. Then we show convergence of the stochastic Euler dynamics to the ODE solution by studying the associated infinitesimal generators and by a novel local truncation error analysis. Next we prove stability by an immediate analysis of the random timestep Euler method and by deriving Foster-Lyapunov criteria for the stochastic Euler dynamics; the latter also yield bounds on the speed of convergence to stationarity. The paper ends with a discussion of second-order stochastic Euler dynamics and a series of numerical experiments that appear to verify our analytical results.
Paper Structure (22 sections, 11 theorems, 40 equations, 9 figures)

This paper contains 22 sections, 11 theorems, 40 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The random timestep Euler method and the stochastic Euler dynamics
  3. Contributions and outline
  4. Deterministic Euler dynamics
  5. Convergence and error analysis
  6. Uniform approximation for bounded $f$
  7. Uniform approximation for Lipschitz continuous $f$
  8. Local root mean square truncation errors
  9. Stability
  10. Stability of the random timestep Euler method
  11. Foster--Lyapunov criteria in the case of linear ODEs
  12. Second-order stochastic Euler dynamics
  13. Stability of the second order stochastic Euler dynamics jump chain
  14. Numerical experiments
  15. One-dimensional linear problem
  16. ...and 7 more sections

Key Result

Theorem 2.2

Let $A \in \mathbb{R}^{d \times d}$ and let the eigenvalues $\lambda_1,\ldots,\lambda_d \in \mathbb{C}$ of $A$ have negative real parts: $\Re(\lambda_1),\ldots, \Re(\lambda_d) < 0$. Then, the eigenvalues of $B$ have negative real parts, if $h > 0$ is chosen such that with $-\Re(\lambda_i)/0 := \infty$. Indeed, if $f(u) = Au$ and the assumption above holds, there is a $c >0$ such that the dynamica

Figures (9)

  • Figure 1.1: Realisations of the stochastic Euler dynamics considering the ODEs $u'=(1-u)u$ (left) and $u'=-u$ (right) with initial values $u_0 = 0.25$ and stepsize parameter $h= 0.8$.
  • Figure 2.1: Plots of the deterministic Euler dynamics regarding $u'=-u$, $u_0 = 1$, and $h \in \{0.625, 0.125, \ldots, 1\}$.
  • Figure 3.1: The trajectory of an underdamped harmonic oscillator $u_1' = u_2, u_2' = -u_1-u_2$ with initial value $u(0) = (1,0)^T$ and realisations of the corresponding stochastic Euler dynamics ${(V(t))_{t\geq 0}}$ for $h \in \{0.25, 0.125, 0.0625\}$. We show the ground truth ${(u(t))_{t\geq 0}}$ and one realisation of ${(V(t))_{t\geq 0}}$ for each of the stepsize parameters. The stochastic Euler dynamics approaches the ODE solution as $h$ decreases.
  • Figure 6.1: Distances between deterministic Euler dynamics $(w(t))_{t \geq 0}$ and ODE solution $(u(t))_{t \geq 0}$, as well as between deterministic Euler dynamics $(w(t))_{t \geq 0}$ and its companion process $(\overline{w}(t))_{t \geq 0}$ with respect to $h$ at time points $t \in \{0.01, 0.1, 1\}$. Here, $(u(t))_{t \geq 0}$ and $(w(t),\overline{w}(t))_{t \geq 0}$ correspond to the linear model ODE $u' = -u, u(0) = 1$.
  • Figure 6.2: Estimates of the local root mean square truncation error of the stochastic Euler dynamics and second-order stochastic Euler dynamics at multiple time points $\varepsilon > 0$ regarding the linear model problem $u' = -u, u(0) = 1$ for $h \in \{0.1, 1\}$ and $h = \varepsilon$. $\widehat{\mathrm{E}}$ denotes the sample mean and is computed using $10^5$ samples.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Example 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Lemma 4.1
  • ...and 2 more