Karhunen-Loève's truncation error for bivariate functions

Karhunen-Loève decompositions (KLD) or equivalently Proper Orthogonal Decompositions (POD) of bivariate functions are revisited in this work. We investigate the truncation error first for regular functions trying to improve and sharpen bounds found in Griebel and Harbrecht (2014). But, it happens that (KL)-series expansions are in fact more sensitive to the capacity of fields (we are concerned with) to be well represented by a sum of few products of separated variables functions. We consider this issue very important for approximating some interesting field problems defined as solutions of partial differential equations such as the transient heat problem and the Poisson equation. The main tool, to establish approximation bounds in this type of problems, is linear algebra. We show how the singular value decomposition underlying the (KL)-expansion is connected to the spectrum of some Gram matrices and that the derivation of the corresponding truncation error is related to the spectral properties of these Gram matrices which are structured matrices with low displacement ranks. This methodology allows us to show that Karhunen-Loève's truncation error decreases exponentially fast with respect to the cut-off frequency, for some interesting transient temperature fields despite their lack of smoothness.