دانلود رایگان مقاله خطای برشی تبدیل Karhunen - Loève برای توابع دو متغیره

Abstract

Karhunen–Loeve 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–Loeve’s ` truncation error decreases exponentially fast with respect to the cut-off frequency, for some interesting transient temperature fields despite their lack of smoothness

4. Transient temperature

We investigate two examples of transient heat transfer problems. Writing down the temperature field as a Fourier series is as old as the first closed expressions of the solutions of the heat equation. The infinite Fourier sum involves separation of both time and space variables. Starting from this Fourier series expression, our aim is to come up with a new infinite sum representation that enhances that separation of time and space variables so that a low truncation is possible, preserving the main features of the temperature field with high accuracy. The purpose is therefore to illustrate the impact of the orthogonality defect suffered by the Fourier expansion of the field T and to show how the Karhunen–Loeve decomposition allows a better expression regarding the accuracy of variables separation.