Karhunen-Loeve-Expansion

The Karhunen-Loeve-Expansion, also known as Proper Orthogonal Decomposition, represents the random field as a infinite linear combination of orthogonal functions chosen as the eigenfunctions of the covariance function [23], [31]. The Karhunen – Loeve-Expansion of the Gaussian random field у is given as:

V(-T C) = Vo (x) + X vbz, (x) Yj(Z) (7)

i=1

= fjV^iZi(x)Yi(Z) xer, Ze£2 (8)

i=1

where A1 > A2 > … > Xi > … > 0 and zi are the eigenvalues and eigenfunctions of the covariance function Cov which is symmetric and positive definite by definition. The deterministic eigenfunctions zi are obtained from the spectral decomposition of the covariance function via solution of

Г Cov (x, y) Zi (y) dy = XiZi (x). (9)

Having the eigenpairs, the uncorrelated Gaussian random variables Yi in equation (8) can be expressed as

Yi(C) = Ay [ V(x, C)zi(x)dx j = 1,2,… (10)

h Jr

with zero mean and unit variance, i. e. E (Yi) = 0 and E (YiYj) = Sij for j = 1,2,…

[5] . In the special case of a Gaussian random field, uncorrelated random variables are independent as well, which is an important property we will need later on for the sparse grid.

Truncating now the Karhunen-Loeve-Expansion after a finite number of terms, we obtain the approximation of the random field у

d

Wd(x, C) = JJ^zi(x)Yi(Q хеГХеП. (11)

i=1

The corresponding covariance function is given by

d

Covd (x, y) = ^hiZi (x) Zi (y). (12)

i=1

In [15], it is shown that the eigenfunction basis {zi} is optimal in the sense that the mean square error resulting from the truncation after the dth term is minimized.

The following approximation error representation is then obtained by Mercer’s theorem [34]

So, yd may provide a suitable approximation of у, if the eigenvalues decay suffi­ciently fast and d is large enough [5]. If one assumes a Gaussian covariance func­tion (cf. (5)), the eigenvalues will exponentially decay towards zero. The proof of this behaviour of the eigenvalues can be found e. g. [11]. This paper also provides a fast algorithm based on a kernel independent fast multipole method to compute the Karhunen-Loeve approximation. Another approach to solve the large eigenvalue problem arising from the Karhunen-Loeve-Expansion can be found in [25]. They in­troduce a Krylov subspace method with a sparse matrix approximation using sparse hierarchical matrix techniques to solve it.