Karhunen-Loève Expansion for the Discrete Transient Heat Transfer Equation

Abstract

In this paper we perform and analyze a Karhunen-Loève expansion on the solution of a discrete heat equation. Unlike the continuous case, several choices can be made from the numerical scheme to the numerical time integration. We analyze some of these choices and compare them. In literature, it is shown that the KL-expansion's error depends on the singular values of the matrix (or the operator) which we attempt to compress, but there is few results on the decay of these singular values. The core of this article is to prove the exponential decay of the singular values. To achieve this result, the analysis is conducted in a classical way by considering the spatial correlation of the temperature. And then, we analyze the problem in a more general view by using the Krylov matrices with Hermitian argument, which are a generalization of the well-known Vandermonde matrices. Some computations are made using MATLAB to ensure the performance of the Karhunen-Loève expansion. This article presents an application of this work to an identification problem where the data are disturbed by a Gaussian white noise.