Mitigating the Influence of the Boundary on PDE-based Covariance Operators
Mitigating the Influence of the Boundary on PDE-based Covariance Operators
复制标题
减轻边界对基于偏微分方程的协方差算子的影响
DOI：
10.3934/ipi.2018045
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
G. Stadler
中科院分区：
文献类型：
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作者：
Y. Daon;G. Stadler
无限维希尔伯特空间上的高斯随机场需要定义适当的协方差算子。使用椭圆PDE算子来构造协方差算子允许构建快速PDE解算器，用于使用所得协方差和精度算子进行操纵。然而，PDE算子需要选择边界条件，并且这种选择可能对高斯随机场产生强烈且通常不期望的影响。我们提出了两种技术，允许改善这些边界效应的大规模问题。第一种方法结合了椭圆PDE算子与罗宾边界条件，其中变化的罗宾系数计算从优化问题。第二种方法通过重新调整协方差算子来规范化逐点方差。这些方法可以单独使用，也可以组合使用。我们研究了这些方法的性质，并讨论了它们的计算复杂度。我们的方法的性能进行了研究，定义在简单和复杂的二维和三维域的随机场。
Gaussian random fields over infinite-dimensional Hilbert spaces require the definition of appropriate covariance operators. The use of elliptic PDE operators to construct covariance operators allows to build on fast PDE solvers for manipulations with the resulting covariance and precision operators. However, PDE operators require a choice of boundary conditions, and this choice can have a strong and usually undesired influence on the Gaussian random field. We propose two techniques that allow to ameliorate these boundary effects for large-scale problems. The first approach combines the elliptic PDE operator with a Robin boundary condition, where a varying Robin coefficient is computed from an optimization problem. The second approach normalizes the pointwise variance by rescaling the covariance operator. These approaches can be used individually or can be combined. We study properties of these approaches, and discuss their computational complexity. The performance of our approaches is studied for random fields defined over simple and complex two- and three-dimensional domains.