# Interdisciplinary Applied Mathematics

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FIGURE 17.11. Basic steps in the Karhunen-Loeve decomposition method.

convert the original nonlinear governing PDE to a set of coupled ODEs much smaller in size. Figure 17.11 explains the various steps for implementing the Karhunen-Loeve decomposition method.

The advantages of this method are: (i) In general, very few basis functions are required. (ii) Nonlinear behavior is efficiently captured. (iii) The same basis functions can be employed to simulate different input parameters, even though regeneration of the basis functions for the new input parameters can provide more accurate results. The disadvantages of the method are: (i) Using a combined state vector instead of independent basis functions can result in a reduced number of basis functions, but this can distort the physical meaning of the problem. (ii) Problems can still occur, since the basis functions chosen may not capture the entire state space. This is usually the case when there are multiple attractors, rare intermittent fluctuations, or bifurcations in the parameter space. (iii) Complications can result in systems exhibiting intermittent chaotic behavior. (iv) The overhead with the initial full-scale simulation is quite high. Low-dimensional flow dynamical systems may converge to erroneous states after long-time integration and can be observed in reduced-order models constructed from the Karhunen-Loeve decomposition method. A dissipative model based on a spectral viscosity diffusion convolution operator has been used in (Sirisup and Karniadakis, 2004), for improving the asymptotic behavior of KL predictions.

In spite of the existing voids, the KL decomposition technique is promising for nonlinear dynamic simulation, especially as the complexity increases. As coupled domain simulation tools improve, this approach can be a powerful tool for microsystem simulation. The standard KL decomposition has been modified under several circumstances to generate more

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