Table 4 Correlation functions used in “Inuence of the correlation kernel” section and relative truncation errors, number of retained KLE terms and continuity errors, with \(l_c=0.15L\) in all the cases and \(\tau =|s-t|\)
From: Large scale random fields generation using localized Karhunen–Loève expansion
Kernel name | Function | Truncation error \(\varvec{\epsilon _{KL}^2}\) | Retained terms \(\varvec{N}\) | Continuity error \(\varvec{\epsilon _c}\) |
|---|---|---|---|---|
Exponential | \(\exp \left( -\dfrac{\tau }{l_c} \right) \) | 0.001 | 98 | \(1.7\times 10^{-6}\) |
Triangular | \(\max \left( 1-\dfrac{\tau }{l_c},0 \right) \) | 0.001 | 86 | \(1.8\times 10^{-3}\) |
Damped sine | \(\dfrac{l_c}{10\tau }\sin \left( \dfrac{10\tau }{l_c} \right) \) | 0.001 | 24 | \(1.9\times 10^{-2}\) |
Gaussian | \(\exp \left( -\dfrac{\tau }{l_c},\right) ^2\) | 0.001 | 12 | \(9.2\times 10^{-3}\) |