We consider the numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise. For the spatial approximation we consider a standard finite element method and for the temporal approximation, a rational approximation of the exponential function. We first show strong convergence of this approximation in both positive and negative order norms. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.
We study an asymptotic preserving scheme for the temporal discretization of a system of parabolic semilinear SPDEs with two time scales. Owing to the averaging principle, when the time scale separation $ε$ vanishes, the slow component converges to the solution of a limiting evolution equation, which is captured when the time-step size $Δt$ vanishes by a limiting scheme. The objective of this work is to prove weak error estimates which are uniform with respect to $ε$, in terms of $Δt$: the scheme satisfies a uniform accuracy property. This is a non trivial generalization of a recent article in …
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…
This paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation (SPDE) driven simultaneously by Gaussian noise and Poisson random measure, more realistic in modeling real world phenomena. The SPDE is discretized in space with the standard finite element method and in time with the linear implicit Euler method or an exponential integrator, more efficient and stable for stiff problems. We prove the strong convergence of the fully discrete schemes toward the mild solution. The results reveal how convergence orders depend on the regularity of the …
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…
This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than autonomous SPDEs while modeling real world phenomena. Numerical approximations for autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet well understood. The non-autonomous SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. We break the complexity in the analysis of the t…
This paper deals with the backward Euler method applied to semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. The SPDE is discretized in space by the finite element method and in time by the backward Euler. We consider a larger class of nonlinear drift functions, which are of Nemytskii type and polynomial of any odd degree with negative leading term, instead of only dealing with the special case of stochastic Allen-Chan equation as in the up to date literature.Moreover our linear operator is of second order and not necessarily self-adjoint, therefo…
Magnus-type Integrator for the Finite Element Discretization of Semilinear Parabolic non-Autonomous SPDEs Driven by multiplicative noise
Western Norway University of Applied Sciences（西挪威应用科学大学）；Center for Research in Computational and Applied Mechanics (CERECAM)（计算与应用力学研究中心）；Department of Mathematics and Applied Mathematics, University of Cape Town（开普敦大学数学与应用数学系）；The African Institute for Mathematical Sciences(AIMS) of South Africa（南非非洲数学科学研究院）；Fakultät für Mathematik, Technische Universität Chemnitz（开姆尼茨工业大学数学学院）
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 under more relaxed conditions. The SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. This extends the current results in the literature to not necessary self-adjoint operator with more general boundary conditions. As a consequence key part of the proof does not rely on the spectral decomposition of the linear operator. We achieve optimal convergence order…
High-order integrator for sampling the invariant distribution of a class of parabolic SPDEs with additive space-time noise
Univ Lyon, Université Claude Bernard Lyon 1（里昂大学，克洛德·贝尔纳里昂大学）；CNRS UMR 5208, Institut Camille Jordan（法国国家科学研究中心联合研究实验室5208，卡米耶·若当研究所）；Université de Genève（日内瓦大学）
Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces
In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter $H>\frac 12$ and Poisson random measure, more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time, more precisely linear implicit method, exponential integrator and …
Weak convergence for a stochastic exponential integrator and finite element discretization of SPDE for multiplicative \& additive noise
Stellenbosh University（斯泰伦博斯大学）；Center for Research in Computational and Applied Mechanics (CERECAM)（计算与应用力学研究中心）；Department of Mathematics and Applied Mathematics, University of Cape Town（开普敦大学数学与应用数学系）；School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal（夸祖鲁-纳塔尔大学数学、统计与计算机科学学院）
Strong convergence of some Magnus-type schemes for the finite element discretization of non-autonomous parabolic SPDEs driven by additive fractional Brownian motion and Poisson random measure
针对 fBm 与 Poisson 随机测度共同驱动的非自治半线性抛物型 SPDE，提出有限元结合随机 Magnus 型或半隐式 Euler 的全离散格式，并证明其强收敛阶。
A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise
Heriot Watt University（赫瑞-瓦特大学）；The African Institute for Mathematical Sciences(AIMS) of South Africa and Stellenbosh University, Center for Research in Computational and Applied Mechanics (CERECAM), and Department of Mathematics and Applied Mathematics, University of Cape Town（南非非洲数学科学研究所（AIMS）和斯泰伦博斯大学，计算与应用力学研究中心（CERECAM），以及开普敦大学数学与应用数学系）
We consider the long-time behavior of an explicit tamed exponential Euler scheme applied to a class of parabolic semilinear stochastic partial differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to appr…
This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. …
Strong convergence of a fully discrete finite element method for a class of semilinear stochastic partial differential equations with multiplicative noise
The University of Tennessee（田纳西大学）；The Ohio State University（俄亥俄州立大学）；The University of North Carolina at Greensboro（北卡罗来纳大学格林斯伯勒分校）
In this work, we provide the first strong convergence result of numerical approximation of a general second order semilinear stochastic fractional order evolution equation involving a Caputo derivative in time of order $α\in(\frac 34, 1)$ and driven by Gaussian and non-Gaussian noises simultaneously more useful in concrete applications. The Gaussian noise considered here is a Hilbert space valued Q-Wiener process and the non-Gaussian noise is defined through compensated Poisson random measure associated to a Lévy process. The linear operator is not necessary self-adjoint. The fractional stocha…
The efficient simulation of the mean value of a non-linear functional of the solution to a linear stochastic partial differential equation (SPDE) with additive Gaussian noise is considered. A Galerkin finite element method is employed along with an implicit Euler scheme to arrive at a fully discrete approximation of the mild solution to the equation. A scheme is presented to compute the covariance of this approximation, which allows for rapid sampling in a Monte Carlo method. This is then extended to a multilevel Monte Carlo method, for which a scheme to compute the cross-covariance between th…
Scheme-Induced Minimum Action Methods for SPDEs: A Variational Framework for Convergence Analysis
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Huazhong University of Science and Technology（华中科技大学）；The Hong Kong Polytechnic University（香港理工大学）
We propose and study a temporal, and spatio-temporal discretisation of the 2D stochastic Navier--Stokes equations in bounded domains supplemented with no-slip boundary conditions. Considering additive noise, we base its construction on the related nonlinear random PDE, which is solved by a transform of the solution of the stochastic Navier--Stokes equations. We show strong rate (up to) $1$ in probability for a corresponding discretisation in space and time (and space-time). Convergence of order (up to) 1 in time was previously only known for linear SPDEs.
A fully implicit numerical approach based on the space-time finite element method is implemented for the semilinear wave equation in 1(space) + 1(time) and 2 + 1 dimensions to explore critical collapse and search for self-similar solutions. Previous work studied this behavior by exploring the threshold of singularity formation using time marching finite difference techniques while this work introduces an adaptive time parallel numerical method to the problem. The semilinear wave equation with a $p = 7$ term is examined in spherical symmetry. The impact of mesh refinement and the time additive …
