循环嵌入方法用于采样平稳随机场的分析
Analysis of circulant embedding methods for sampling stationary random fields
AI总结：
本文分析了循环嵌入方法中循环矩阵正定性的保证条件，针对Matérn协方差函数证明了在特定条件下正定性成立，并通过数值实验验证了相关结论。
摘要：
本文证明，在温和条件下，循环嵌入方法中出现的循环矩阵的正定性始终得到保证，只要包围立方体足够大。我们详细研究了Matérn协方差函数的情况，并证明（对于固定相关长度）当h₀→0时，若随机场在比物理域大order(1 + ν^{1/2} log h₀^{-1})的立方体上采样，则正定性得到保证。（此处h₀是正则网格的网格间距，ν是Matérn平滑参数。）我们证明当h₀和ν固定时，采样立方体可以随着相关长度的减小而变小。我们的结果通过数值实验得到验证。我们还证明了循环矩阵特征值衰减的若干结果。这些结果导致了猜想，通过数值实验验证了该猜想，即特征值衰减速率与协方差算子的Karhunen--Loève特征值衰减速率相同。
英文摘要：
In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix appearing in the circulant embedding method is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail the case of the Matérn covariance, and prove (for fixed correlation length) that, as $h_0\rightarrow 0$, positive definiteness is guaranteed when the random field is sampled on a cube of size order $(1 + ν^{1/2} \log h_0^{-1})$ times larger than the size of the physical domain. (Here $h_0$ is the mesh spacing of the regular grid and $ν$ the Matérn smoothness parameter.) We show that the sampling cube can become smaller as the correlation length decreases when $h_0$ and $ν$ are fixed. Our results are confirmed by numerical experiments. We prove several results about the decay of the eigenvalues of the circulant matrix. These lead to the conjecture, verified by numerical experiment, that they decay with the same rate as the Karhunen--Loève eigenvalues of the covariance operator.
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