arXiv subjects

Derui Sheng

Publications and source records attributed to Derui Sheng.

At least 19 recordsLinked to original sources

arXiv2026

A regularized truncated finite element method for degenerate parabolic stochastic PDE on non-compact graph

We study the numerical approximation of a class of degenerate parabolic stochastic partial differential equations on non-compact metric graphs, which naturally arise in the asymptotic analysis of Hamiltonian flows under small noise perturbations. The numerical discretization of these equations faces several challenges, including the non-compactness of the graph, the degeneracy of the differential operator near vertices, and the non-symmetry of the associated bilinear form. To address these issues, we propose a multi-step numerical strategy combining graph truncation, localized coefficient regularization, and finite element spatial discretization. By incorporating localization techniques, tightness arguments, and resolvent estimates, we establish the strong convergence of the proposed scheme in a weighted $L^2$-space. Our results provide a systematic methodology that is potentially extensible to more general non-compact graphs and degenerate operators.

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arXiv2026

A scalar auxiliary variable-based semi-implicit scheme for stochastic Cahn--Hilliard equation

In this paper, we present a novel semi-implicit numerical scheme for the stochastic Cahn--Hilliard equation driven by multiplicative noise. By reformulating the original equation into an equivalent stochastic scalar auxiliary variable (SSAV) system, our method enables an efficient and stable treatment of polynomial nonlinearities in a semi-implicit fashion. In order to accurately capture the impact of stochastic perturbations, we carefully incorporate It\^o correction terms into the SSAV approximation. Leveraging the smoothing properties of the underlying semigroup and the $H^{-1}$-dissipative structure of the nonlinear term, we establish the optimal strong convergence order of one-half for the proposed scheme in the trace-class noise case. Moreover, we show that the modified SAV energy asymptotically preserves the energy evolution law. Finally, numerical experiments are provided to validate the theoretical results and to explore the influence of noise near the sharp-interface limit.

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arXiv2025

Large and moderate deviation principles for stochastic partial differential equation on graph

In this paper, we study large and moderate deviation principles for stochastic partial differential equations (SPDEs) on metric graphs and their associated multiscale models via the weak convergence approach, providing a refined characterization of the probabilities of rare events. Several challenges unique to the graph setting are encountered, including operator degeneracy near vertices and the lack of compactness on non-compact graphs. To address these difficulties, we introduce novel weighted Sobolev spaces on graphs, and prove compact embedding results specifically adapted to the degeneracy structure. Our analysis is particularly applicable to SPDEs on graphs arising as limits of stochastic reaction-diffusion systems on narrow domains and from fast-flow asymptotics of stochastic incompressible fluids, yielding new deviation results for these models.

math.PR↗

arXiv2025

A dynamic domain semi-Lagrangian method for stochastic Vlasov equations

We propose a dynamic domain semi-Lagrangian method for stochastic Vlasov equations driven by transport noises, which arise in plasma physics and astrophysics. This method combines the volume-preserving property of stochastic characteristics with a dynamic domain adaptation strategy and a reconstruction procedure. It offers a substantial reduction in computational costs compared to the traditional semi-Lagrangian techniques for stochastic problems. Furthermore, we present the first-order convergence analysis of the proposed method, partially addressing the conjecture in the work [C.-E. Br\'{e}hier and D. Cohen, J. Comput. Dyn., 2024] on the convergence order of numerical methods for stochastic Vlasov equations. Several numerical tests are provided to show good performance of the proposed method.

math.NA↗

arXiv2024

Asymptotic-preserving approximations for stochastic incompressible viscous fluids and SPDEs on graph

The long-term dynamics of particles involved in an incompressible flow with a small viscosity ($\epsilon>0$) and slow chemical reactions, is depicted by a class of stochastic reaction-diffusion-advection (RDA) equations with a fast advection term of magnitude $1/\epsilon$. It has been shown in [7] the fast advection asymptotics of stochastic RDA equation in $\mathbb{R}^2$ can be characterized through a stochastic partial differential equation (SPDE) on the graph associated with certain Hamiltonian. To simulate such fast advection asymptotics, we introduce and study an asymptotic-preserving (AP) exponential Euler approximation for the multiscale stochastic RDA equation. There are three key ingredients in proving asymptotic-preserving property of the proposed approximation. First, a strong error estimate, which depends on $1/\epsilon$ linearly, is obtained via a variational argument. Second, we prove the consistency of exponential Euler approximations on the fast advection asymptotics between the original problem and the SPDE on graph. Last, a graph weighted space is introduced to quantify the approximation error for SPDE on graph, which avoids the possible singularity near the vertices. Numerical experiments are carried out to support the theoretical results.

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arXiv2023

Error analysis of numerical methods on graded meshes for stochastic Volterra equations

This paper presents the error analysis of numerical methods on graded meshes for stochastic Volterra equations with weakly singular kernels. We first prove a novel regularity estimate for the exact solution via analyzing the associated convolution structure. This reveals that the exact solution exhibits an initial singularity in the sense that its H\"older continuous exponent on any neighborhood of $t=0$ is lower than that on every compact subset of $(0,T]$. Motivated by the initial singularity, we then construct the Euler--Maruyama method, fast Euler--Maruyama method, and Milstein method based on graded meshes. By establishing their pointwise-in-time error estimates, we give the grading exponents of meshes to attain the optimal uniform-in-time convergence orders, where the convergence orders improve those of the uniform mesh case. Numerical experiments are finally reported to confirm the sharpness of theoretical findings.

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arXiv2023

Asymptotics of large deviations of finite difference method for stochastic Cahn--Hilliard equation

In this work, we establish the Freidlin--Wentzell large deviations principle (LDP) of the stochastic Cahn--Hilliard equation with small noise, which implies the one-point LDP. Further, we give the one-point LDP of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation. Our main result is the convergence of the one-point large deviations rate function (LDRF) of the spatial FDM, which is about the asymptotical limit of a parametric variational problem. The main idea for proving the convergence of the LDRF of the spatial FDM is via the $\Gamma$-convergence of objective functions, which relies on the qualitative analysis of skeleton equations of the original equation and the numerical method. In order to overcome the difficulty that the drift coefficient is not one-side Lipschitz, we use the equivalent characterization of the skeleton equation of the spatial FDM and the discrete interpolation inequality to obtain the uniform boundedness of the solution to the underlying skeleton equation. This plays an important role in deriving the $\Gamma$-convergence of objective functions.

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arXiv2023

Superiority Of Symplectic Methods For Stochastic Hamiltonian System Via Asymptotic Error Distribution

The superiority of symplectic methods for stochastic Hamiltonian systems has been widely recognized, yet the probabilistic mechanism behind this superiority remains incompletely understood. This paper studies the superiority of symplectic methods from the perspective of the asymptotic error distribution, i.e., the limit distribution of normalized error. Focusing on stochastic Hamiltonian systems driven by additive noise, we obtain the asymptotic limit of the normalized error distribution of the $\theta$ method $(\theta\in[0,1])$ that is symplectic if and only if $\theta=\frac12$. By establishing upper bounds for the second-order moment of the asymptotic error distribution, we show that the midpoint method minimizes the error constant of the $\theta$ method for a large time horizon $T$. Furthermore, we take the linear stochastic oscillator as a test equation and investigate exact asymptotic error constants of several symplectic and non-symplectic methods. Our result suggests that in the long-time computation, the probability that the error deviates from zero decays exponentially faster for the symplectic methods than that for the non-symplectic ones.

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arXiv2022

Scheme-Induced Minimum Action Methods for SPDEs: A Variational Framework for Convergence Analysis

The minimum action method (MAM) is an important numerical tool for computing the most probable transition paths and the leading exponential rates of rare-event probabilities for stochastic systems with small noise. To the best of our knowledge, no rigorous convergence analysis is available for fully discrete MAMs for stochastic partial differential equations (SPDEs). This paper establishes the convergence of the constrained minima of fully discrete scheme-induced action functionals for terminal observations of scalar semilinear SPDEs. A main difficulty is that a consistent discrete approximation of an admissible control generally fails to satisfy the discrete terminal constraint exactly. To overcome this difficulty, we construct an asymptotically negligible correction that restores the constraint, and incorporate this argument into an abstract variational framework. We then propose a criterion that reduces the assumptions of this framework to verifiable conditions involving the continuous and discrete Green functions. Applying this criterion to fully discrete schemes for a stochastic wave equation and a fourth-order parabolic equation, we prove convergence of the constrained minima along arbitrary spatial and temporal refinement sequences, without a mesh-ratio condition.

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arXiv2022

Hitting properties of generalized fractional kinetic equation with time-fractional noise

This paper studies hitting properties for the system of generalized fractional kinetic equations driven by Gaussian noise fractional in time and white or colored in space. We derive the mean square modulus of continuity and some second order properties of the solution. These are applied to deduce lower and upper bounds for probabilities that the path process hits bounded Borel sets in terms of the $\mathfrak{g}_q$-capacity and $g_q$-Hausdorff measure, respectively, which yield the critical dimension for hitting points. Further, based on some fine analysis of the harmonizable representation for the solution, we prove that all points are polar in the critical dimension. This provides strong evidence for the conjecture raised in Hinojosa-Calleja and Sanz-Sol\'e [Stoch PDE: Anal Comp (2022). https://doi.org/10.1007/s40072-021-00234-6].

math.PR↗

arXiv2022

Mittag--Leffler Euler integrator and large deviations for stochastic space-time fractional diffusion equations

Stochastic space-time fractional diffusion equations often appear in the modeling of the heat propagation in non-homogeneous medium. In this paper, we firstly investigate the Mittag--Leffler Euler integrator of a class of stochastic space-time fractional diffusion equations, whose super-convergence order is obtained by developing a helpful decomposition way for the time-fractional integral. Here, the developed decomposition way is the key to dealing with the singularity of the solution operator. Moreover, we study the Freidlin--Wentzell type large deviation principles of the underlying equation and its Mittag--Leffler Euler integrator based on the weak convergence approach. In particular, we prove that the large deviation rate function of the Mittag--Leffler Euler integrator $\Gamma$-converges to that of the underlying equation.

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arXiv2022

Convergence analysis of a finite difference method for stochastic Cahn--Hilliard equation

This paper presents the convergence analysis of the spatial finite difference method (FDM) for the stochastic Cahn--Hilliard equation with Lipschitz nonlinearity and multiplicative noise. Based on fine estimates of the discrete Green function, we prove that both the spatial semi-discrete numerical solution and its Malliavin derivative have strong convergence order $1$. Further, by showing the negative moment estimates of the exact solution, we obtain that the density of the spatial semi-discrete numerical solution converges in $L^1(\mathbb R)$ to the exact one. Finally, we apply an exponential Euler method to discretize the spatial semi-discrete numerical solution in time and show that the temporal strong convergence order is nearly $\frac38$, where a difficulty we overcome is to derive the optimal H\"older continuity of the spatial semi-discrete numerical solution.

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arXiv2022

Strong error analysis of Euler methods for overdamped generalized Langevin equations with fractional noise: Nonlinear case

This paper considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the fractional Brownian motion and the singular kernel, which is overcome by means of the Malliavin calculus and fine estimates of several multiple singular integrals. Consequently, these two methods are proved to be strongly convergent with order nearly $\min\{2(H+\alpha-1), \alpha\}$, where $H \in (1/2,1)$ and $\alpha\in(1-H,1)$ respectively characterize the singularity levels of fractional noises and singular kernels in the underlying equation. This result improves the existing convergence order $H+\alpha-1$ of Euler methods for the nonlinear case, and gives a positive answer to the open problem raised in [4]. As an application of the theoretical findings, we further investigate the complexity of the multilevel Monte Carlo simulation based on the fast Euler method, which turns out to behave better performance than the standard Monte Carlo simulation when computing the expectation of functionals of the considered equation.

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arXiv2021

Influences of Numerical Discretizations on Hitting Probabilities for Linear Stochastic Parabolic System

This paper investigates the influences of standard numerical discretizations on hitting probabilities for linear stochastic parabolic system driven by space-time white noises. We establish lower and upper bounds for hitting probabilities of the associated numerical solutions of both temporal and spatial semi-discretizations in terms of Bessel-Riesz capacity and Hausdorff measure, respectively. Moreover, the critical dimensions of both temporal and spatial semi-discretizations turn out to be half of those of the exact solution. This reveals that for a large class of Borel sets $A$, the probability of the event that the paths of the numerical solution hit $A$ cannot converge to that of the exact solution.

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arXiv2021

Convergence analysis for minimum action methods coupled with a finite difference method

The minimum action method (MAM) is an effective approach to numerically solving minimums and minimizers of Freidlin--Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are $1/2$ and $1$, respectively. Our main result also reveals the convergence of the stochastic $\theta$-method for SDEs with small noise in terms of large deviations.

math.PR↗

arXiv2021

A splitting semi-implicit method for stochastic incompressible Euler equations on $\mathbb T^2$

The main difficulty in studying numerical method for stochastic evolution equations (SEEs) lies in the treatment of the time discretization (J. Printems. [ESAIM Math. Model. Numer. Anal. (2001)]). Although fruitful results on numerical approximations for SEEs have been developed, as far as we know, none of them include that of stochastic incompressible Euler equations. To bridge this gap, this paper proposes and analyses a splitting semi-implicit method in temporal direction for stochastic incompressible Euler equations on torus $\mathbb{T}^2$ driven by an additive noise. By a Galerkin approximation and the fixed point technique, we establish the unique solvability of the proposed method. Based on the regularity estimates of both exact and numerical solutions, we measure the error in $L^2(\mathbb{T}^2)$ and show that the pathwise convergence order is nearly $\frac{1}{2}$ and the convergence order in probability is almost $1$.

math.NA↗

arXiv2020

Numerically asymptotical preservation of the large deviations principles for invariant measures of Langevin equations

In this paper, we focus on two kinds of large deviations principles (LDPs) of the invariant measures of Langevin equations and their numerical methods, as the noise intensity $\epsilon\to 0$ and the dissipation intensity $\nu\to\infty$ respectively. First, by proving the weak LDP and the exponential tightness, we conclude that the invariant measure $\{\mu_{\nu,\epsilon}\}$ of the exact solution satisfies the LDPs as $\epsilon\to0$ and $\nu\to\infty$ respectively. Then, we study whether there exist numerical methods asymptotically preserving these two LDPs of $\{\mu_{\nu,\epsilon}\}$ in the sense that the rate functions of invariant measures of numerical methods converge pointwise to the rate function of $\{\mu_{\nu,\epsilon}\}$ as the step-size tends to zero. The answer is positive for the linear Langevin equation. For the small noise case, we show that a large class of numerical methods can asymptotically preserve the LDP of $\{\mu_{\nu,\epsilon}\}_{\epsilon>0}$ as $\epsilon\to0$. For the strong dissipation case, we study the stochastic $\theta$-method ($\theta\in[1/2,1]$) and show that only the midpoint scheme ($\theta=1/2$) can asymptotically preserve the LDP of $\{\mu_{\nu,\epsilon}\}_{\nu>0}$ as $\nu\to\infty$. These results indicate that in the linear case, the LDP as $\epsilon\to0$ and the LDP as $\nu\to\infty$ for the invariant measures of numerical methods have intrinsic differences: the common numerical methods can asymptotically preserve the LDP of $\{\mu_{\nu,\epsilon}\}_{\epsilon>0}$ as $\epsilon\to0$ while the asymptotical preservation of numerical methods for the LDP of $\{\mu_{\nu,\epsilon}\}_{\nu>0}$ as $\nu\to\infty$ depends on the choice of numerical methods. To the best of our knowledge, this is the first result of investigating the relationship between the LDPs of invariant measures of stochastic differential equations and those of their numerical methods.

math.NA↗

arXiv2020

Convergence of Density Approximations for Stochastic Heat Equation

This paper investigates the convergence of density approximations for stochastic heat equation in both uniform convergence topology and total variation distance. The convergence order of the densities in uniform convergence topology is shown to be exactly $1/2$ in the nonlinear case and nearly $1$ in the linear case. This result implies that the distributions of the approximations always converge to the distribution of the origin equation in total variation distance. As far as we know, this is the first result on the convergence of density approximations to the stochastic partial differential equation.

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