Efficient Computation of Karhunen–Loéve Decomposition

Abstract

In structural analysis and design, it is important to consider the effects of uncertainties in loading and material properties in a rational way. Such uncertainties can be mathematically modeled as random field. For computational purpose, it is essential to represent and discretize the random field. For a field with known second-order statistics, such a representation can be achieved by the Karhunen-Lo‘eve expansion. Accordingly, the random field is represented in a truncated series expansion using a few eigenvalues and associated eigenfunctions of the covariance function and random coefficients. The eigenvalues and eigenfunctions are found by solving an integral equation with its kernel as the covariance function. Since the analytical solutions for the integral equation may not be present, therefore, it is important to find an approximate solution. It is important to consider both accuracy of the solution and the cost of computing the solution. This work is focused on exploring a few numerical methods for the numerical solution of this integral equation. A rectangular spatial domain with an uncertainty is modeled as a Gaussian random field with a known covariance function. Three different methods (1) using finite element bases, (2) approximating the covariance function by averaging locally, and (3) by the Nyström method are implemented and numerically studied.