Weak Error Estimates for Trajectories of SPDEs Under Spectral Galerkin Discretization
Weak Error Estimates for Trajectories of SPDEs Under Spectral Galerkin Discretization
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DOI：
10.4208/jcm.1607-m2016-0539
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发表时间：
2018-04
影响因子：
0.9
通讯作者：
Charles-Édouard Bréhier;M. Hairer;Andrew M. Stuart
中科院分区：
文献类型：
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作者：
Charles-Édouard Bréhier;M. Hairer;Andrew M. Stuart
我们考虑随机半线性发展方程驱动的添加剂，空间相关，维纳噪声，特别是考虑热方程（解析半群）和阻尼驱动的波动方程（有界半群）类型的问题。我们用谱Galerkin投影离散这些方程，并研究了轨迹概率分布的近似：检验函数是正则的，但依赖于区间[0，T ]上过程的值.我们介绍了一种新的方法在定量弱误差分析的范围内离散化的SPDE。弱误差是使用随机卷积的确定性函数（伊藤映射）时，发现非线性项下降制定。伊藤映射的正则性被利用，特别是二阶泰勒展开，转移到弱误差的兴趣从谱近似的随机卷积的错误。我们证明了在两种情况下，弱收敛速度是强收敛速度的两倍。首先，我们假设协方差算子与半群的生成元交换：弱误差展开中的一阶项由于独立性而抵消。其次，我们删除了交换假设，并扩展了以前的结果，由于一个新的误差项依赖于一个换向器的分析。
We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection , and we study the approximation of the probability distribution of the tra-jectories: test functions are regular, but depend on the values of the process on the interval [0, T ]. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Ito map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Ito map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.