THE USE OF THE KARHUNEN-LOEVE PROCEDURE FOR THE CALCULATION OF LINEAR EIGENFUNCTIONS

It is shown that the Karhunen-Loeve decomposition may be used to determine the eigenfunctions of a general class of linear operators from an ensemble of realizations that are derived from that system. Given a moderate size data set (either numerical or experimental) from a linear system, good approximations to the eigenfunctions that characterize the underlying equations can be computed by performing the Karhunen-Loeve procedure. Two numerical examples are presented: the vibration of a thin membrane in a rectangular domain and in a stadium. These are used to determine the convergence and accuracy of the method. It is found that this method yields accurate results for the first few eigenfunctions with relatively few realizations. Eigenfunctions with less energy are accurately resolved as the size of the ensemble increase. The method is shown to be an efficient and practical procedure for determining the eigenfunctions of systems in complex geometries and in cases where the governing equations are not known a priori. The effect of random noise contamination of the data set is also investigated and it is found that the Karhunen-Loeve procedure can still achieve accurate results despite the presence of substantial background noise. © 1991.