On the properties of the exceptional set for the randomized Euler and Runge-Kutta schemes

TL;DR

This paper demonstrates that the probability of large errors in randomized ODE solvers decreases exponentially, providing confidence intervals and numerical validation for methods like randomized Euler and Runge-Kutta schemes.

Contribution

It establishes exponential decay of the exceptional set probability for a broad class of randomized ODE algorithms, including explicit, implicit, and two-stage Runge-Kutta schemes.

Findings

Exponential decay of exceptional set probability.

Designed confidence intervals for solutions of IVPs.

Numerical experiments validate theoretical results.

Abstract

We show that the probability of the exceptional set decays exponentially for a broad class of randomized algorithms approximating solutions of ODEs, admitting a certain error decomposition. This class includes randomized explicit and implicit Euler schemes, and the randomized two-stage Runge-Kutta scheme (under inexact information). We design a confidence interval for the exact solution of an IVP and perform numerical experiments to illustrate the theoretical results.