Binjie Li
A discretization of an optimal control problem of a stochastic parabolic equation driven by multiplicative noise is analyzed. The state equation is discretized by the continuous piecewise linear element method in space and by the backward Euler scheme in time. The convergence rate $ O(τ^{1/2} + h^2) $ is rigorously derived.
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A priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Time-Optimal Control Problems
本文研究了由线性抛物偏微分方程 governing 且受点约束控制的最优时间控制问题的空间时间有限元离散方法，基于二阶充分最优性条件获得了控制变量的最优先验误差估计。
Error estimates for space-time discretization of parabolic time-optimal control problems with bang-bang controls
本文推导了时间最优控制问题空间-时间离散化的先验误差估计，探讨了在不同假设下最优时间与控制变量的收敛性，特别研究了bang-bang控制的情况，并通过数值例子验证结果。
A non-uniform discretization of stochastic heat equations with multiplicative noise on the unit sphere
本文研究了单位球面上具有乘性噪声的随机热方程的离散化方法，采用谱方法处理空间离散化并截断Wiener过程，结合非均匀步长的隐式欧拉方案进行时间离散化，并通过地球表面温度数据进行数值实验。
This paper studies the convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation. The filtration is general, and the spatial semi-discretization uses the standard continuous piecewise linear element method. Firstly, higher regularity of the solution to the continuous equation is derived. Secondly, the first-order spatial accuracy is derived for the spatial semi-discretization. Thirdly, an application of the theoretical result to a stochastic linear quadratic control problem is presented.
This article deals with a simultaneous abstract evolution equation. This includes a parabolic-hyperbolic phase-field system as an example which consists of a parabolic equation for the relative temperature coupled with a semilinear damped wave equation for the order parameter. Although a time discretization of an initial value problem for an abstract evolution equation has been studied, time discretizations of initial value problems for simultaneous abstract evolution equations seem to be not studied yet. In this paper we focus on a time discretization of a simultaneous abstract evolution equa…
We analyze Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space-time discretization methods introduced in [IMA J. Numer. Anal., 33(1) (2013), pp. 242-260] by R. Andreev and in [Comput. Methods Appl. Math., 15(4) (2015), pp. 551-566] by O. Steinbach.
We consider a minimal residual discretization of a simultaneous space-time variational formulation of parabolic evolution equations. Under the usual `LBB' stability condition on pairs of trial- and test spaces we show quasi-optimality of the numerical approximations without assuming symmetry of the spatial part of the differential operator. Under a stronger LBB condition we show error estimates in an energy-norm which are independent of this spatial differential operator.
In this paper, a higher-order time-discretization scheme is proposed, where the iterates approximate the solution of the stochastic semilinear wave equation driven by multiplicative noise with general drift and diffusion. We employ a variational method for its error analysis and prove an improved convergence order of 3/2 for the approximates of the solution. The core of the analysis is Holder continuity in time and moment bounds for the solutions of the continuous and the discrete problem. Computational experiments are also presented.
Full Discretization of Stochastic Semilinear Schrödinger equation driven by multiplicative Wiener noise
本文针对乘性Wiener噪声驱动的随机半线性薛定谔方程，提出有限元与随机三角法结合的全离散格式，推导时空强收敛速率，并通过数值实验验证理论结果。
Full moment error estimates in strong norms for numerical approximations of stochastic Navier-Stokes equations with multiplicative noise, Part I: time discretization
本文针对带乘性噪声的随机Navier-Stokes方程的时间离散化，提出新框架以推导速度和压力在强范数下的最优阶全矩误差估计，核心方法包括指数稳定性估计、离散随机Gronwall不等式和自举论证。
Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise
本文分析带乘性噪声的三维随机Allen-Cahn方程的全离散化（Euler时间格式+有限元空间方法），推导一般空间L^q范数下的路径一致收敛速率并实验验证，贡献在于提出非线性随机抛物方程有限元全离散路径一致收敛的技术。
Dynamical low-rank tensor approximations to high-dimensional parabolic problems: existence and convergence of spatial discretizations
本文针对高维抛物问题提出动态低秩张量逼近框架，证明解的存在性、数据扰动稳定性及空间离散化收敛性，并兼容 tensor train 与分层张量等格式。
We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(Ω))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuška's theorem, we prove unique solvability in the continuous case. Furthermore, we est…
Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces
本文研究了在演化曲面上的准线性抛物型偏微分方程的全离散化收敛性，通过空间离散的正则性结果和时间积分器的稳定性结果，得出全离散问题的收敛性结论。
Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations
本文研究了拟线性抛物型方程的全隐式和线性隐式后向差分公式（BDF）方法，结合最大抛物正则性和能量估计，推导出时间离散近似解及其梯度在最大范数和能量范数下的最优阶误差界。
A discretization for the nonlinear parabolic evolution equation of fractional order in space
本文提出分数阶空间非线性抛物型方程的数值离散化方法，采用泛函分析设定定义分数导数，通过数值实验验证方法有效性并提出猜想。
A second-order correction method for loosely coupled discretizations applied to parabolic-parabolic interface problems
本文针对抛物-抛物界面问题，基于 Robin-Robin 分裂方法构造松耦合预测-校正格式，并证明校正步误差达到时间步长的二阶收敛，数值实验验证了理论分析。
Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints
本文研究了带有线性约束的抛物型问题的时间离散化方法，探讨了显式和隐式处理方式对变分公式的影响，并推导了误差界及超收敛结果。
Runge-Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity
本文研究了隐式Runge-Kutta方法对非线性抛物方程的时间半离散化，得出了在有限时间区间内W^{1,∞}范数下的误差界，并在改进的逼近阶下得到抛物能量范数的误差界，利用离散最大抛物正则性理论。
In this paper we consider the numerical approximation of a general second order semi-linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element method for space discretization and the exponential Rosenbrock-Euler method for time discretization, we provide a rigorous convergence proof in space and time under only the standard Lipschitz condition of the nonlinear part for both smooth and nonsmooth initial solution. This is in contrast to very restrictive assumptions made in the literature, where the autho…
This paper is concerned with the approximation of linear and nonlinearinitial-boundary-value problems of pseudo-parabolic equations with Dirichlet boundary conditions. They are discretized in space by spectral Galerkin and collocation methods based on Legendre and Chebyshev polynomials. The time integration is carried out suitably with robust schemes attending to qualitative features such as stiffness and preservation of strong stability to simulate nonregular problems more correctly. The corresponding semidiscrete and fully discrete schemes are described and the performance of the methods is …
In this paper, we consider the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic ex…
A priori error estimates for stable generalized finite element discretization of parabolic interface optimal control problems
本文研究了控制作用于界面的抛物界面方程最优控制问题，提出基于稳定广义有限元、后向欧拉格式与变分离散的完全离散逼近方法，并证明了控制、状态及伴随状态的先验误差估计，数值算例验证了理论结果。
BDDC preconditioners for a space-time finite element discretization of parabolic problems
本文研究了利用BDDC方法求解由空间-时间有限元离散化得到的抛物型初始-边界值问题大规模线性代数方程组，将时间视为另一种空间坐标，采用不规则三角形空间-时间网格上的连续分段线性有限元，通过BDDC预处理GMRES方法求解空间-时间有限元Schur补方程，并通过数值研究展示了预处理子程序的鲁棒性。
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 probabil…
We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks, which have a single hidden layer and can be trained at a modest computational cost as compared with Deep Learning Neural Networks. Our approach addresses the time evolution by applying classical ODEs techniques and uses ELM-based collocation for solving the resulting stationary elliptic problems. In this framework, the $θ$-method and Backward Difference Formulae (BDF) techniques are investigated on some linear parabolic PDEs that are challeging problems for the stability and accuracy properties…
In this paper, we propose a new approach for the time-discretization of the incompressible stochastic Stokes equations with multiplicative noise. Our new strategy is based on the classical Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of at most $1$ for both velocity and pressure approximations. The proof is based on a new Hölder continuity estimate of the velocity solution. While the errors of the velocity approximation are estimated in the standard $L^2$- and $H^1$-norms, the pressure errors are car…
Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [7]. We improve the earlier time stepping algorithm based on this theory, and specifically address its stable and efficient implementation in the context of high-order methods. The considered methods include an L1-2 method and continuous collocation methods of arbitrary order, for which adaptive temporal meshes are shown to yield optimal convergence rates in the presen…
An exponential Integrator for finite volume discretization of nonlinear parabolic differential equation
针对一般二阶半线性抛物型偏微分方程，采用正则网格上的两点通量近似有限体积法与一阶指数时间差分（ETD1）进行时空离散，在非线性项局部Lipschitz假设下推导了L²估计，并通过数值模拟验证理论结果。
A nonnegativity-preserving finite element method for a class of parabolic SPDEs with multiplicative noise
本文提出一种非负有限元方法，用于求解具有乘性噪声的抛物型随机偏微分方程，该方法在非负初始条件下的数值解保持非负性，并通过数学分析和数值实验验证了其有效性。
We construct a space-time parallel method for solving parabolic partial differential equations by coupling the Parareal algorithm in time with overlapping domain decomposition in space. The goal is to obtain a discretization consisting of "local" problems that can be solved on parallel computers efficiently. However, this introduces significant sources of error that must be evaluated. Reformulating the original Parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint-based a posteriori error analysis to be performed. Through an app…
We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented c…
Optimal Order Space-Time Discretization Methods for the Nonlinear Stochastic Elastic Wave Equations with Multiplicative Noise
本文针对带乘性噪声的非线性随机弹性波方程，提出了基于非标准时间步进和标准有限元的最优阶半离散与全离散格式，通过高阶矩稳定性与改进的梯形求积误差估计证明了最优收敛速率，并通过数值实验验证了理论。
A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations
本文提出基于对偶的后验误差估计方法，用于非线性抛物方程的离散误差估计与自适应控制，通过IMEX时间步进方案和有限元方法进行空间离散，实现空间和时间误差分离以驱动自适应网格细化和时间步长选择。
An exponentially convergent discretization for space-time fractional parabolic equations using $hp$-FEM
针对时空分数阶抛物型问题，结合基于 sinc 求积的 Riesz-Dunford 积分离散与空间 $hp$-FEM，提出指数收敛格式，严格证明其在小时间及数据不兼容启动奇点下的鲁棒性。
Quasi-optimal time-space discretizations for a class of nonlinear parabolic PDEs
本文提出了一类非线性抛物型PDEs的准最优时间-空间离散化方法，通过引入额外变量构造等价系统，并利用Galerkin离散化与近似Uzawa算法求解，同时发展了后验条件以确保准最优性。
Some notes concerning preconditioning of linear parabolic optimal control problems
本文研究带Neumann边界条件的线性抛物型最优控制问题的条件数，证明约化问题条件数与离散水平无关，并基于已有工作提出全耦合方法的预处理器，其特征值分布也与离散水平无关，数值实验验证了结果与预处理器的有效性。
Strong convergence of numerical discretizations for semilinear stochastic evolution equations driven by multiplicative white noise
本文研究了系数比经典全局Lipschitz条件更一般的semilinear随机演化方程的数值离散化的强收敛性，提出了一种新的分析方法，并通过数值实验验证了理论结果。
Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE
本文研究了非全局利普希茨系数的抛物型随机偏微分方程（SPDE）的不变测度和函数泛函期望的近似问题，采用谱伽辽金方法和时间隐式欧拉方案进行全离散化，通过变分方法和马尔可夫算子理论推导了数值解的先验估计和正则性估计，建立了全离散化的尖锐弱收敛率，并证明在SPDE具有唯一V-一致ergodic不变测度时，全离散化可以近似该不变测度。
Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability an…
Pathwise space approximations of semi-linear parabolic SPDEs with multiplicative noise
本文研究了半线性随机微分方程在乘性噪声下的空间近似收敛率，采用谱伽辽金和有限元方法，证明了强和几乎必然的均匀收敛性。
For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains $D$, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete $hp$-time-stepping approach. We combine this se…
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretise the SPDE in space by the finite element method and propose a new scheme in time appropriate for such equatio…
Stochastic exponential integrators for finite element discretization of SPDEs for multiplicative and additive noise
针对乘性和加性噪声驱动的半线性抛物型随机偏微分方程，采用有限元空间离散和指数积分器时间离散，在迹类噪声假设下证明了均方L2范数收敛性，并通过二维线性反应扩散方程和非线性随机对流扩散反应方程验证了方法。
Quasi-optimality of Petrov-Galerkin discretizations of parabolic problems with random coefficients
研究随机系数抛物型问题的Petrov-Galerkin离散化方法，证明解的矩存在性及近似最优误差估计。
Higher-order time discretizations with ALE finite elements for parabolic problems on evolving surfaces
本文提出利用ALE有限元方法对演变曲面上的抛物问题进行高阶时间离散化，采用Runge-Kutta或后向差分公式，并证明了其无条件稳定性与最优收敛性。
Control-constrained parabolic optimal control problems on evolving surfaces - theory and variational discretization
针对演化曲面上的控制约束线性二次最优控制问题，证明了状态方程弱解的存在唯一性，并提出了变分离散化方法，证明了其收敛性，在弱正则性假设下获得了最优收敛阶。
Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations
本文研究了基于分数阶微积分的空间-时间变分形式的线性抛物型初值问题，证明了压缩空间-时间Galerkin离散化的最优性，并展示了其在稳定性及收敛性方面的优势。
In order to characterize the fluctuation between the ergodic limit and the time-averaging estimator of a full discretization in a quantitative way, we establish a central limit theorem for the full discretization of the parabolic stochastic partial differential equation. The theorem shows that the normalized time-averaging estimator converges to a normal distribution with the variance being the same as that of the continuous case, where the scale used for the normalization corresponds to the temporal strong convergence order of the considered full discretization. A key ingredient in the proof …
We study the discretization of a linear evolution partial differential equation when its Green function is known. We provide error estimates both for the spatial approximation and for the time stepping approximation. We show that, in fact, an approximation of the Green function is almost as good as the Green function itself. For suitable time-dependent parabolic equations, we explain how to obtain good, explicit approximations of the Green function using the Dyson-Taylor commutator method (DTCM) that we developed in J. Math. Phys. (2010). This approximation for short time, when combined with a…
Higher order time discretization method for a class of semilinear stochastic partial differential equations with multiplicative noise
本文针对带乘性噪声且漂移项仅满足单边Lipschitz条件的半线性随机偏微分方程，提出基于Milstein方法的时间离散与插值有限元空间离散方案，证明近1阶强收敛并给出稳定性与几乎最优收敛阶，数值实验验证了理论结果。
Galerkin-type time discretizations for parabolic and hyperbolic problems: stability and a priori error analysis
本文提出了一种基于Galerkin型时间离散化的空间-时间方法分析框架，通过合适测试函数建立解的稳定性，并推导先验误差估计，适用于任意近似次数且对模型参数具有鲁棒性。
We present an algorithm for the solution of a simultaneous space-time discretization of linear parabolic evolution equations with a symmetric differential operator in space. Building on earlier work, we recast this discretization into a Schur-complement equation whose solution is a quasi-optimal approximation to the weak solution of the equation at hand. Choosing a tensor-product discretization, we arrive at a remarkably simple linear system. Using wavelets in time and standard finite elements in space, we solve the resulting system in linear complexity on a single processor, and in polylogari…
A-stable time discretizations preserve maximal parabolic regularity
研究了A稳定的时间离散化方法在抛物问题中保持最大ℓ^p正则性的条件，证明了隐欧拉法、Crank-Nicolson法等方法在满足额外条件时具有均匀正则性。
Weak convergence for a stochastic exponential integrator and finite element discretization of SPDE for multiplicative \& additive noise
本文研究了随机指数欧拉格式在乘性与加性噪声下的SPDE有限元近似中的弱收敛性，探讨了自共轭线性算子下的完整弱收敛率，并为非自共轭算子提供了初步结果。
Magnus-type Integrator for the Finite Element Discretization of Semilinear Parabolic non-Autonomous SPDEs Driven by multiplicative noise
本文研究了由乘性噪声驱动的非自治半线性抛物型随机偏微分方程的数值近似，采用有限元方法和Magnus型积分器进行空间和时间离散，证明了在根均方L²范数下强收敛性，并分析了噪声和初始数据正则性对收敛阶的影响。
Energy-norm error estimates for finite element discretization of parabolic problems
本文基于有限元方法和Runge-Kutta时间步进方案，推导出抛物型初始边界值问题的最优a-先验误差估计，通过精心处理H(-1)-范数中的时间导数和L2投影，避免了人工正则性条件，提高了估计的适用性。
We consider a space-time finite element method on fully unstructured simplicial meshes for optimal sparse control of semilinear parabolic equations. The objective is a combination of a standard quadratic tracking-type functional including a Tikhonov regularization term and of the $L^1$-norm of the control that accounts for its spatio-temporal sparsity. We use a space-time Petrov-Galerkin finite element discretization for the first-order necessary optimality system of the associated discrete optimal sparse control problem. The discretization is based on a variational formulation that employs pi…
This paper analyzes a discretization of a stochastic parabolic optimal control problem, where the diffusion term contains the control variable. With rough data, the convergence of the discretization is derived. In addition, a Monte-Carlo method is presented.
In this paper we deal with an abstract problem which includes the linearized equations of coupled sound and heat flow as an example. Recently, a time discretization of a simultaneous abstract evolution equation applying to some parabolic-hyperbolic phase-field systems has been studied. This paper focuses on a time discretization of an abstract problem applying to the linearized equations of coupled sound and heat flow. Also, this paper gives some parabolic-hyperbolic phase-field systems as examples.
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