Non-asymptotic Error Analysis of Explicit Modified Euler Methods for Superlinear and Non-contractive SODEs

2025/09/10 by Zhihui Liu, Liu, Zhihui, Xiaojie Wang +5 · 2 citations
Engineering · Physics and Astronomy · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Model Reduction and Neural Networks #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2509.08410

Abstract

A family of explicit modified Euler methods (MEMs) is constructed for long-time approximations of super-linear SODEs driven by multiplicative noise. The proposed schemes can preserve the same Lyapunov structure as the continuous problems. Under a non-contractive condition, we establish a non-asymptotic error bound between the law of the numerical approximation and the target distribution in Wasserstein-1 (W₁) distance through a time-independent weak convergence rate for the proposed schemes. As a by-product of this weak error estimate, we obtain an O(τ|ln τ|) convergence rate between the exact and numerical invariant measures.

Citations

• Geometric Ergodicity and Strong Error Estimates for Tamed Schemes of Super-linear SODEs
• Strong convergence rates for long-time approximations of SDEs with non-globally Lipschitz continuous coefficients
• Taming under isoperimetry
• Unadjusted Langevin Algorithms for SDEs with Hoelder Drift
• Non-asymptotic convergence bounds for modified tamed unadjusted Langevin algorithm in non-convex setting
• The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients
• Unadjusted Langevin algorithm with multiplicative noise: Total variation\n and Wasserstein bounds
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• Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift
• Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature
• Approximation of the invariant measure with an Euler scheme for\n Stochastic PDE's driven by Space-Time White Noise

Cited by

• Discretization, Uniform-in-Time Estimations and Approximation of Invariant Measures for Nonlinear Stochastic Differential Equations with Non-Uniform Dissipativity
• Ergodic Estimates of One-Step Numerical Approximations for Superlinear SODEs

Related

• Ergodic Estimates of One-Step Numerical Approximations for Superlinear SODEs
• Non-asymptotic convergence bounds of modified EM schemes for non-dissipative SDEs
• Geometric Ergodicity and Strong Error Estimates for Tamed Schemes of Super-linear SODEs
• Uniform-in-Time Weak and Ergodic Error Estimates of a Nonlinearity-Explicit Full Discretization for Superlinear SPDEs Driven by Multiplicative Noise
• Strong convergence rates for long-time approximations of SDEs with non-globally Lipschitz continuous coefficients