scholarly journals Weak Convergence Rates for an Explicit Full-Discretization of Stochastic Allen–Cahn Equation with Additive Noise

2021 ◽  
Vol 86 (3) ◽  
Author(s):  
Meng Cai ◽  
Siqing Gan ◽  
Xiaojie Wang
Keyword(s):  
Weak Convergence ◽  
Additive Noise ◽  
Convergence Rates ◽  
Full Discretization ◽  
Cahn Equation

scholarly journals Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

2020 ◽  
Vol 54 (6) ◽  
pp. 2199-2227
Author(s):  
Mihály Kovács ◽  
Annika Lang ◽  
Andreas Petersson
Keyword(s):  
Finite Element ◽  
Weak Convergence ◽  
Additive Noise ◽  
Convergence Rates ◽  
Fully Discrete ◽  
Finite Element Approximations ◽  
Negative Order ◽  
Stochastic Wave Equation ◽  
Discrete Finite Element ◽  
Spatial Approximation

The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. 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.

scholarly journals Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

2018 ◽  
Vol 39 (4) ◽  
pp. 2096-2134 ◽  
Author(s):  
Charles-Edouard Bréhier ◽  
Jianbo Cui ◽  
Jialin Hong
Keyword(s):  
Convergence Rate ◽  
Strong Convergence ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Splitting Scheme ◽  
Cahn Equation ◽  
Strong Convergence Rate

Abstract This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space–time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.

scholarly journals The Euler–Maruyama scheme for SDEs with irregular drift: convergence rates via reduction to a quadrature problem.

2020 ◽  
Author(s):  
Andreas Neuenkirch ◽  
Michaela Szölgyenyi
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Error Analysis ◽  
Strong Convergence ◽  
General Framework ◽  
Additive Noise ◽  
Convergence Rates ◽  
Convergence Order ◽  
Quadrature Problem ◽  
Weighted Quadrature

Abstract We study the strong convergence order of the Euler–Maruyama (EM) scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev–Slobodeckij-type regularity of order $\kappa \in (0,1)$ for the nonsmooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min \{3/4,(1+\kappa )/2\}-\epsilon$ for the equidistant EM scheme (for arbitrarily small $\epsilon>0$). The cut-off of the convergence order at $3/4$ can be overcome by using a suitable nonequidistant discretization, which yields the strong convergence order of $(1+\kappa )/2-\epsilon$ for the corresponding EM scheme.

scholarly journals Weak Convergence Rates of Population Versus Single-Chain Stochastic Approximation MCMC Algorithms

2014 ◽  
Vol 46 (04) ◽  
pp. 1059-1083 ◽  
Author(s):  
Qifan Song ◽  
Mingqi Wu ◽  
Faming Liang
Keyword(s):  
Monte Carlo ◽  
Weak Convergence ◽  
Stochastic Approximation ◽  
Convergence Rates ◽  
Gain Factor ◽  
Single Chain ◽  
Practical Applications ◽  
Mcmc Algorithms ◽  
Factor Sequence ◽  
Number Of Iterations

In this paper we establish the theory of weak convergence (toward a normal distribution) for both single-chain and population stochastic approximation Markov chain Monte Carlo (MCMC) algorithms (SAMCMC algorithms). Based on the theory, we give an explicit ratio of convergence rates for the population SAMCMC algorithm and the single-chain SAMCMC algorithm. Our results provide a theoretic guarantee that the population SAMCMC algorithms are asymptotically more efficient than the single-chain SAMCMC algorithms when the gain factor sequence decreases slower than O(1 / t), where t indexes the number of iterations. This is of interest for practical applications.

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