Interdisciplinary Applied Mathematics

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Karhunen-Loeve Decomposition Method


The basic idea in the Karhunen-Loeve (KL) decomposition method is similar to    the    basic idea in    the    linear    modes method,    i.e.,    to develop    a    few


global basis functions to represent the entire system by a reduced-order model. In the case of the linear modes method, the linear modes of the system obtained through the solution of the generalized eigenvalue problem form the set of basis functions. Karhunen-Loeve decomposition is another method to generate global basis functions, and the advantage with the KL decomposition is that it works better than the linear modes technique for nonlinear cases. The procedure for the extraction of the basis functions in the KL decomposition method is summarized below:


1. Simulate the entire system dynamics first by using a time-stepping scheme that is stable and known to give accurate results.


2. The spatial distributions of each state variable u(x,t) are sampled at a series of tn different time instants during the simulations. These sampled distributions are stored as a series of vectors Uj, and each vector represents a “snapshot” in time.


3. The basis functions are determined using either the singular value decomposition (SVD) or the KL approach.


In the SVD approach, which is mathematically equivalent to the KL decomposition technique, n orthogonal basis functions [ai,…, an] are determined by minimizing the following expression:


tn


(17.18)


I Uj — proj(uj, span[ai,…, an]) |2 .


This is accomplished by taking the singular value decomposition (SVD) of the matrix U, whose columns are u*. The SVD of U is given by

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