Approximations of the modified Bessel functions of the second kind $K_{\nu }$. Applications in random field generation

We propose an analytical approximation for the modified Bessel function of the second kind $K_{\nu }$. The approximation is derived from an exponential ansatz imposing global constrains. It yields local and global errors of less than one percent and a speed-up in the computing time of $3$ orders in magnitude in comparison with traditional approaches. We demonstrate the validity of our approximation for the task of generating long-range correlated random fields.