Interdisciplinary Applied Mathematics

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where V contains the eigenvector as columns and S contains the eigenvalues in the diagonal. By taking a* = v* for i = 1, 2,…,n, the minimization of the expression in equation (17.18) is accomplished.

The KL approach (Sirovich, 1987) is a procedure for extracting an empirical basis for a modal decomposition from an ensemble of signals. Assuming that the signals are an ensemble of the functions u*, the objective is to find a single deterministic function that is most similar to members of u on average. In other words, one needs to find a function that maximizes the inner product with the field u*. That is, one needs to maximize

}_((Ф, uQ2)


where {) is the averaging operator, which can be a time, space, or ensemble average, and (ф, u*) = JQ ф(х)и*(x)dil is the inner product defined in the function space П. It turns out that this condition is met when ф(х) is an eigenfunction of the two-point tensor given by

Кф =    K (x,x’ ^(x’)dx’ = Хф(х),


where К(x,x’) is a nonnegative Hermitian operator given by

К (x,x’) = (ui(x)ui(x’))





^2ud(x)uI (x0. *=1

The above equation can be solved by the direct method or by the method of “snapshots” (Sirovich, 1987). In the method of “snapshots” the eigenfunction can be represented as the summation of snapshots u, i.e.,

фИ=£ akuk(x).    (17.19)


Substituting this into the two-point tensor equation yields a matrix eigenvalue problem that determines the eigenvalues and eigenvectors a. Substituting this set of eigenvectors in equation (17.19) yields a set of eigenfunctions ф^), which is the set of global basis functions. It is important to note that the eigenfunction corresponding to the largest eigenvalue corresponds to the most energetic of the snapshots ensemble followed by the eigenfunction corresponding to the next-largest eigenvalue, and so on. After the basis functions are generated by using either the SVD or the KL method, the Galerkin procedure is employed with the basis functions to

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