On the randomized Euler schemes for ODEs under inexact information

TL;DR

This paper analyzes the errors, stability, and optimality of randomized Euler schemes for solving ODEs when only noisy, inexact information about the right-hand side is available, considering both global and local Lipschitz conditions.

Contribution

It provides a comprehensive error analysis and stability assessment of randomized Euler methods under inexact data, extending understanding of their performance in noisy environments.

Findings

Error bounds for randomized Euler schemes with noisy data

Stability conditions for explicit and implicit methods

Optimality results for algorithm performance

Abstract

We analyse errors of randomized explicit and implicit Euler schemes for approximate solving of ordinary differential equations (ODEs). We consider classes of ODEs for which the right-hand side functions satisfy Lipschitz condition globally or only locally. Moreover, we assume that only inexact discrete information, corrupted by some noise, about the right-hand side function is available. Optimality and stability of explicit and implicit randomized Euler algorithms are also investigated.