Karhunen-Loève's Series Truncation for Bivariate Functions

Mejdi Azaïez (1) , Faker Ben Belgacem (2)

1 I2M - Institut de Mécanique et d'Ingénierie de Bordeaux
2 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne

Karhunen-Loève's decompositions (KLD) or the proper orthogonal decompositions (POD) of bivariate functions are revisited in this work. We investigate the truncation error first for regular functions and try to improve and sharpen bounds found in the literature. However it happens that (KL)-series expansions are in fact more sensitive to the liability of fields to approximate to be well represented by a small sum of products of separated variables functions. We consider this very issue for some interesting fields solutions of partial differential equations such as the transient heat problem and Poisson's equation. The main tool to state approximation bounds is linear algebra. We show how the singular value decomposition underlying the (KL)-expansion is connected to the spectrum of some Gram matrices. Deriving estimates on the truncation error is thus strongly tied to the spectral properties of these Gram matrices which are structured matrices with low displacement ranks.

Domaines: Analyse numérique [math.NA]

Mejdi Azaïez, Faker Ben Belgacem. Karhunen-Loève's Series Truncation for Bivariate Functions. 2014. ⟨hal-01063083⟩