Interdisciplinary Applied Mathematics

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efficient macromodels. Nonlinear responses have been well captured by an arc-length-based KL decomposition presented in (Chen and Kang, 2001b). This method is motivated by the fact that while rapidly varying events can occur in a very short time period, they typically traverse a relatively large interval in the phase space. Hence, considering an ensemble average based not    only    on    time    but    also    on    arc    length    in phase space can    lead


to better macromodels. This has been shown in (Chen and Kang, 2001b), for a capacitive pressure sensor, where the arc-length-based approach is found to capture the rapidly changing dynamics of the device better than the standard KL approach. Further analysis of this approach shows that the arc-length-based ensemble average is a weighted time average, with the weighting function equal to the magnitude of the vector field, thereby stressing the event of rapid change.


Weighted Karhunen—Loeve Decomposition Method


Another modification to the standard KL decomposition is the use of a weighting function (Qiao and Aluru, 2003c; Graham and Kevrekidis, 1996; Zhang et al., 2003). The basic idea is that instead of trying to minimize equation (17.18), we assign different weights to different snapshots and try to minimize the weighted residual, i.e.,


Ns


y~] WiUi — proj (wiUi, span{ai,.. ,,aN}) |
30.    (17.20)


i=1


Observe the difference between equations (17.18) and (17.20): wi is the weighting assigned to snapshot ui. In the weighted KL approach, instead of minimizing a least-squares measure of “error” between the linear subspace spanned by the basis functions and the observation space, we minimize the weighted “error” between these two spaces.


By using the fact that the SVD of a snapshot ensemble gives the basis that minimizes equation (17.18), it is easy to show that the basis that minimizes equation (17.20) is the column vector of matrix L2:

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