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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 \(9.2\times 10^{-3}\)