Computational Aspects of Lévy-Driven SPDE Approximations
Computational Aspects of Lévy-Driven SPDE Approximations
复制标题
Lévy 驱动的 SPDE 近似的计算方面
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
A. Petersson
中科院分区：
文献类型：
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作者：
A. Petersson
为了将解决方案模拟解决方案（SPDE），必须在空间和时间上近似。在本文中，考虑了这样的完全离散的近似值，重点是有限元方法与合理的半群近似结合。有几个关于此错误的概念。其中之一是弱误差，以应用于解决方案的功能的平均值来衡量。为了近似平均值，通常使用蒙特卡洛和多级蒙特卡洛方法，这些方法基于对SPDE的近似解决方案产生大量实现。论文由两篇论文组成。在论文1中，当一个尝试模拟弱误差的蒙特卡洛和多级蒙特卡洛法引起的附加误差将用于分析上限和下限，以说明了不同的方法和模拟说明了结果。当使用多级蒙特卡洛方法估计弱误差以及SPDE的其他特性时，重要的是，所使用的离散化在刻薄的意义上是足够稳定的。在论文2中，建立了一般随机递归方案的渐近平方稳定性的框架。然后将该框架应用于SPDE的几个离散化，这导致一系列足够的稳定条件。这些结果中的一些在模拟中很敏锐。
In order to simulate solutions to stochastic partial differential equations (SPDE) they must be approximated in space and time. In this thesis such fully discrete approximations are considered, with an emphasis on finite element methods combined with rational semigroup approximations. There are several notions of the error resulting from this. One of them is the weak error, measured in terms of the mean of a functional applied to the solution. To approximate the mean, one typically employs Monte Carlo and multilevel Monte Carlo methods that are based on generating a large number of realizations of the approximate solution to the SPDE. The thesis consists of two papers. In Paper 1 the additional error caused by Monte Carlo and multilevel Monte Carlo methods when one attempts to simulate the weak error is analysed Upper and lower bounds are derived for the different methods and simulations illustrate the results. When using multilevel Monte Carlo methods to estimate the weak error, along with other properties of the SPDE, it is important that the discretizations used are sufficiently stable in a mean square sense. In Paper 2 a framework for the analysis of the asymptotic mean square stability of a general stochastic recursion scheme is set up. This framework is then applied to several discretizations of an SPDE, which results in a series of sufficient conditions for stability. Some of these results are found to be sharp in simulations.