Thesis

Randomised numerical schemes : generalisation and applications

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Awarding institution
  • University of Strathclyde
Date of award
  • 2026
Thesis identifier
  • T18094
Person Identifier (Local)
  • 202253409
Qualification Level
Qualification Name
Department, School or Faculty
Abstract
  • This thesis focuses on the theoretical analysis, numerical implementation, and practical applications of randomized numerical schemes for solving differential equations characterized by low regularity and temporal irregularities. Randomized schemes, which introduce stochastic perturbations into traditional deterministic numerical methods, can offer advantages in convergence analysis and error estimation in certain low-regularity settings, especially when dealing with non-smooth coefficient functions. The research is structured into three primary components. First, we study delay differential equations (DDEs) with Carathéodory-type coefficients that are measurable in time and continuous in the state variables, and establish existence and uniqueness under appropriate assumptions. We then propose a novel randomized two-stage Runge-Kutta scheme and establish convergence rates under strengthened low-regularity assumptions—Lipschitz in the state and Hölder in delay and time—extending the class of DDEs tractable beyond standard deterministic methods. Numerical experiments validate these theoretical results and demonstrate the scheme’s effectiveness. Second, the thesis explores stochastic delay differential equations (SDDEs) with drift coefficients satisfying Carathéodory-type conditions. By constructing and analysing a randomized Euler scheme tailored for these equations, we derive rigorous error bounds and confirm convergence properties theoretically. Implementation details and extensive numerical experiments illustrate the practical performance and robustness of the proposed scheme. Lastly, the thesis introduces a randomized numerical scheme within neural ordinary differential equation (NODE) frameworks for image denoising applications. A new deep learning model, NODE-ImgNet, is developed by embedding a randomized Euler solver within a NODE architecture. Evaluated on multiple image denoising benchmarks, the proposed model demonstrates competitive and, in many cases, superior empirical performance relative to several strong baseline methods. Overall, this thesis establishes randomized numerical methods as powerful tools for handling differential equations and data-driven models with limited regularity. The work not only provides rigorous theoretical insights but also broadens the practical applicability of numerical methods, opening pathways for future research into higher order randomized schemes and deeper connections with machine learning techniques.
Advisor / supervisor
  • Wu, Yue
Resource Type
DOI

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