Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise
Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise
复制标题
具有空间白噪声的分数椭圆随机偏微分方程的伽辽金近似的弱收敛性
DOI：
10.1007/s10543-018-0719-8
复制
发表时间：
2017
影响因子：
1.5
通讯作者：
M. Kovács
中科院分区：
文献类型：
--
作者：
D. Bolin;Kristin Kirchner;M. Kovács
考虑有界域上具有加性空间白噪声的随机偏微分方程解的数值近似。假设微分算子是整数阶椭圆微分算子的分数幂。该解通过空间中的有限元离散化和邓福德-泰勒微积分分数逆的积分表示的求积近似来近似。对于所得的近似值，对弱误差进行了简明分析。具体来说，对于具有多项式增长二阶导数的两次连续 Fréchet 可微泛函类，推导了显式的弱收敛率，并且表明，源自随机性的收敛率分量与相应的强收敛率相比增加了一倍。不同泛函的数值实验验证了理论结果。
The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford–Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Fréchet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results.