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Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations

机译:空间谱Galerkin的弱近似误差的下界   随机波动方程的近似

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摘要

Although for a number of semilinear stochastic wave equations existence anduniqueness results for corresponding solution processes are known from theliterature, these solution processes are typically not explicitly known andnumerical approximation methods are needed in order for mathematical modellingwith stochastic wave equations to become relevant for real world applications.This, in turn, requires the numerical analysis of convergence rates for suchnumerical approximation processes. A recent article by the authors proves upperbounds for weak errors for spatial spectral Galerkin approximations of a classof semilinear stochastic wave equations. The findings there are complemented bythe main result of this work, that provides lower bounds for weak errors whichshow that in the general framework considered the established upper bounds canessentially not be improved.
机译:尽管对于许多半线性随机波动方程,从文献中知道了相应求解过程的存在和唯一性结果,但通常并没有明确地知道这些求解过程,并且需要数值逼近方法,以使随机波动方程的数学建模变得与实际应用相关。反过来,这需要对这种数值逼近过程的收敛速度进行数值分析。作者最近发表的一篇文章证明了一类半线性随机波动方程的空间谱Galerkin逼近的弱误差上限。这项工作的主要结果补充了这些发现,该发现为弱错误提供了下界,这表明在一般框架内,已确定的上限根本无法改善。

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