Numerical solution of the Karhunen–Loeve integral equation with examples based on various kernels derived from the Nataf procedure

• A method for solving Fredholm integral equations using an expansion in orthogonal functions.
• Speeding up the convergence by use of the analytical solution of the case with the exponential kernel.
• Numerical comparisons of convergence for 7 Nataf derived kernels.

An efficient solution of the Karhunen–Loeve (KL) integral equation is developed based on an expansion in Legendre polynomials and also an expansion in the eigenfunctions of the Markov exponential kernel for which analytic results are available. We solve the integral equation with the kernels arising from the Nataf procedure for eight different stochastic processes, viz: Markov, uniform, step, triangular, Rayleigh, exponential, log-normal and log-uniform. It may be shown that use of the Markov eigenfunctions leads to a significant improvement in computing speed over that from the Legendre polynomials. We also discuss some curious behavior associated with the convergence of the Markov eigenfunctions.