arXiv subjects

Yubin Yan

Publications and source records attributed to Yubin Yan.

6 recordsLinked to original sources

arXiv2026

Drift-Randomized Milstein-Galerkin Finite Element Method for Semilinear Stochastic Evolution Equations

Kruse and Wu [Math. Comp. 88 (2019) 2793--2825] proposed a fully discrete randomized Galerkin finite element method for semilinear stochastic evolution equations (SEEs) driven by additive noise and showed that this method attains a temporal strong convergence rate exceeding order $\frac{1}{2}$ without imposing any differentiability assumptions on the drift nonlinearity. They further discussed a potential extension of the randomized method to SEEs with multiplicative noise and introduced the so-called drift-randomized Milstein-Galerkin finite element fully discrete scheme, but without providing a corresponding strong convergence analysis. This paper aims to fill this gap by rigorously analyzing the strong convergence behavior of the drift-randomized Milstein-Galerkin finite element scheme. By avoiding the use of differentiability assumptions on the nonlinear drift term, we establish strong convergence rates in both space and time for the proposed method. The obtained temporal convergence rate is $O(\Delta t^{1-\varepsilon_0})$, where $\Delta t$ denotes the time step size and $\varepsilon_0$ is an arbitrarily small positive number. Numerical experiments are reported to validate the theoretical findings.

math.NA↗

arXiv2024

High-order BDF convolution quadrature for stochastic fractional evolution equations driven by integrated additive noise

The numerical analysis of stochastic time fractional evolution equations presents considerable challenges due to the limited regularity of the model caused by the nonlocal operator and the presence of noise. The existing time-stepping methods exhibit a significantly low order convergence rate. In this work, we introduce a smoothing technique and develop the novel high-order schemes for solving the linear stochastic fractional evolution equations driven by integrated additive noise. Our approach involves regularizing the additive noise through an $m$-fold integral-differential calculus, and discretizing the equation using the $k$-step BDF convolution quadrature. This novel method, which we refer to as the ID$m$-BDF$k$ method, is able to achieve higher-order convergence in solving the stochastic models. Our theoretical analysis reveals that the convergence rate of the ID$2$-BDF2 method is $O(τ^{α+ γ-1/2})$ for $1< α+ γ\leq 5/2$, and $O(τ^{2})$ for $5/2< α+ γ<3$, where $α\in (1, 2)$ and $γ\in (0, 1)$ denote the time fractional order and the order of the integrated noise, respectively. Furthermore, this convergence rate could be improved to $O(τ^{α+ γ-1/2})$ for any $α\in (1, 2)$ and $γ\in (0, 1)$, if we employ the ID$3$-BDF3 method. The argument could be easily extended to the subdiffusion model with $α\in (0, 1)$. Numerical examples are provided to support and complement the theoretical findings.

math.NA↗

arXiv2022

L1 scheme for solving an inverse problem subject to a fractional diffusion equation

This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle $< π/2 $, that is the resolvent set of this operator contains $ \{z\in\mathbb C\setminus\{0\}:\ |\operatorname{Arg} z|< θ\}$ for some $ π/2 < θ< π$. The relationship between the time fractional order $α\in (0, 1)$ and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as $ α$ approaches $ 1 $. Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results.

math.NA↗

arXiv2021

Correction of high-order $L_k$ approximation for subdiffusion

The subdiffusion equations with a Caputo fractional derivative of order $α\in (0,1)$ arise in a wide variety of practical problems, which is describing the transport processes, in the force-free limit, slower than Brownian diffusion. In this work, we derive the correction schemes of the Lagrange interpolation with degree $k$ ($k\leq 6$) convolution quadrature, called $L_k$ approximation, for the subdiffusion, which are easy to implement on variable grids. The key step of designing correction algorithm is to calculate the explicit form of the coefficients of $L_k$ approximation by the polylogarithm function or Bose-Einstein integral. To construct a $τ_8$ approximation of Bose-Einstein integral, the desired $(k+1-α)$th-order convergence rate can be proved for the correction $L_k$ scheme with nonsmooth data, which is higher than $k$th-order BDF$k$ method in [Jin, Li, and Zhou, SIAM J. Sci. Comput., 39 (2017), A3129--A3152; Shi and Chen, J. Sci. Comput., (2020) 85:28]. The numerical experiments with spectral method are given to illustrate theoretical results.

math.NA↗

arXiv2020

Finite-time blow-up of a non-local stochastic parabolic problem

The main aim of the current work is the study of the conditions under which (finite-time) blow-up of a non-local stochastic parabolic problem occurs. We first establish the existence and uniqueness of the local-in-time weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noise-induced blow-up. In the second part we first prove the $C^{1}$-spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf's type boundary value point lemma. The preceding results together with Kaplan's eigenfunction method are then employed to provide a (non-local) drift term induced blow-up result. In the last part of the paper, we present a method which provides an upper bound of the probability of (non-local) drift term induced blow-up.

math.AP↗

arXiv2018

Numerical Approximation of Stochastic Time-Fractional Diffusion

We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order $\alpha\in(0,1)$, and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order $\gamma \in[0,1]$ in the front). The numerical scheme approximates the model in space by the Galerkin method with continuous piecewise linear finite elements and in time by the classical Gr\"unwald-Letnikov method, and the noise by the $L^2$-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the deterministic counterpart. Numerical results are presented to support the theoretical findings.

math.NA↗