Weak convergence of Galerkin approximations of stochastic partial differential equations driven by additive Lévy noise

漂移随机化米尔斯坦-伽辽金有限元方法用于半线性随机演化方程

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

本文提出一种无需假设非线性漂移项可微的漂移随机化米尔斯坦-伽辽金有限元方法，实现空间和时间的强收敛性分析，并验证理论结果。

关于具有乘性噪声的半线性随机波动方程空间谱伽辽金近似弱收敛率的研究

Weak convergence rates for spatial spectral Galerkin approximations of semilinear stochastic wave equations with multiplicative noise

本文研究了具有乘性噪声的半线性随机波动方程的空间谱伽辽金近似的弱收敛率，通过Kolmogorov方程、Schatten范数的Hölder不等式和温和Itô公式等方法，建立了该类方程的精确弱收敛率结果，并作为特殊情况，证明了超几何安德森模型的精确弱收敛率。

带加性时空白噪声的四阶非线性随机偏微分方程的数值逼近

Numerical approximation of nonlinear fourth-order SPDEs with additive space-time white noise

针对$d=1,2,3$维环面上带加性时空白噪声的四阶非线性SPDE，采用空间谱Galerkin与时间Euler全离散格式，通过随机缝合技术证明其空间收敛阶近$2-\ rac{d}{2}$、时间收敛阶为$\ rac{6-d}{4}$。

具有全局部单调系数的随机偏微分方程最优控制的存在性

Existence of optimal controls for stochastic partial differential equations with fully local monotone coefficients

本文利用极小化序列与 Faedo–Galerkin 逼近，证明全单调系数受控 SPDE 的适定性及随机最优反馈控制存在性，并可推广至随机对流扩散等方程。

Weak convergence of the backward Euler method for stochastic Cahn--Hilliard equation with additive noise

We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen--Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin c…

半线性随机偏微分方程Milstein-伽辽金有限元方案的一致性与稳定性

Consistency and Stability of a Milstein-Galerkin Finite Element Scheme for Semilinear SPDE

本文提出了一种分析半线性随机偏微分方程数值方案误差的抽象概念，并通过证明Milstein-伽辽金有限元方案的强收敛性展示了其有效性。通过将Beyn与Kruse（DCDS B，2010）中的双稳概念推广到Hilbert空间中的半群框架，得出时空离散化的双侧误差估计。

Random attractors for a class of stochastic partial differential equations driven by general additive noise

The existence of random attractors for a large class of stochastic partial differential equations (SPDE) driven by general additive noise is established. The main results are applied to various types of SPDE, as e.g. stochastic reaction-diffusion equations, the stochastic $p$-Laplace equation and stochastic porous media equations. Besides classical Brownian motion, we also include space-time fractional Brownian Motion and space-time Lévy noise as admissible random perturbations. Moreover, cases where the attractor consists of a single point are considered and bounds for the speed of attraction…

A Galerkin approximation scheme for the mean correction in a mean-reversion stochastic differential equation

This paper is concerned with the following Markovian stochastic differential equation of mean-reversion type \[ dR_t= (θ+σα(R_t, t))R_t dt +σR_t dB_t \] with an initial value $R_0=r_0\in\mathbb{R}$, where $θ\in\mathbb{R}$ and $σ>0$ are constants, and the mean correction function $α:\mathbb{R}\times[0,\infty)\to α(x,t)\in\mathbb{R}$ is twice continuously differentiable in $x$ and continuously differentiable in $t$. We first derive that under the assumption of path independence of the density process of Girsanov transformation for the above stochastic differential equation, the mean correction f…

Chaos in stochastic 2d Galerkin-Navier-Stokes

We prove that all Galerkin truncations of the 2d stochastic Navier-Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies $N\geq 392$. By "chaotic" we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degene…

基于Gauss-Seidel和Jacobi算法的求解器和预条件子用于非对称随机格架方程系统

Solvers and precondtioners based on Gauss-Seidel and Jacobi algorithms for non-symmetric stochastic Galerkin system of equations

本文研究了基于Gauss-Seidel和Jacobi算法的求解器和预条件子，用于具有随机输入数据的偏微分方程的随机格架离散化。这些算法被设计成能够利用现有软件来减少计算工作量，并作为Krylov迭代方法的预条件子。通过求解稳态对流扩散方程，结果表明近似版本的Gauss-Seidel算法是求解非对称格架方程系统的良好预条件子。

Weak convergence rates for an explicit full-discretization of stochastic Allen-Cahn equation with additive noise

We discretize the stochastic Allen-Cahn equation with additive noise by means of a spectral Galerkin method in space and a tamed version of the exponential Euler method in time. The resulting error bounds are analyzed for the spatio-temporal full discretization in both strong and weak senses. Different from existing works, we develop a new and direct approach for the weak error analysis, which does not rely on the use of the associated Kolmogorov equation or Itô's formula and is therefore non-Markovian in nature. Such an approach thus has a potential to be applied to non-Markovian equations su…