Karhunen-Loève Expansion for the Discrete Transient Heat Transfer Equation (Journal of Mathematical Study, 2019)
In this paper we perform and analyze a Karhunen-Loève expansion on thesolution of a discrete heat equation. Unlike the continuous case, several choices can bemade from the numerical scheme to the numerical time integration. We analyze someof these choices and compare them. In literature, it is shown that the KL-expansion’serror depends on the singular values of the matrix (or the operator) which we attemptto compress, but there is few results on the decay of these singular values. The coreof this article is to prove the exponential decay of the singular values. To achieve thisresult, the analysis is conducted in a classical way by considering the spatial correlationof the temperature. And then, we analyze the problem in a more general view byusing the Krylov matrices with Hermitian argument, which are a generalization of thewell-known Vandermonde matrices. Some computations are made using MATLAB toensure the performance of the Karhunen-Loève expansion. This article presents anapplication of this work to an identification problem where the data are disturbed bya Gaussian white noise.
Keywords: Karhunen-Loève expansion; discrete temperature field.