Interdisciplinary Applied Mathematics

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U W = L2S2 R2T,    (17.21)


where W is an    Ns    x    Ns    diagonal matrix    whose    diagonal    elements    are    the


weighting coefficients for each snapshot, i.e., [W ]M = Wi.


Remarks:


1. If the weighting function matrix is the identity, i.e., wi = 1 (i = 1, 2,…,Ns), then the weighted KL technique and the standard KL technique produce identical bases.


3. Since the snapshot ensemble matrix is multiplied by a diagonal matrix, the computational cost of a weighted KL decomposition based on equation (17.21) is almost the same as the computational cost of the classical KL decomposition.


Significance of Weighting: The concept of assigning different weights to different snapshots is useful when the transient behavior of certain variables (for example, velocity or pressure) changes significantly with time. For example, in the case of electroosmotic transport, the flow gets to steady state at different times for different locations in the channel (see the discussion in (Qiao and Aluru, 2003c), for more details). If a higher weighting is assigned to those snapshots taken during the fast-changing transient, then the basis obtained with SVD will, according to equation (17.20), be able to produce more accurate results. In other words, the new basis obtained with weighted snapshots will be able to represent the system behavior much better than the basis functions obtained with the classical KL decomposition technique. If the transient behavior of the system is gradual, then the use of the weighting function is limited, and both weighted and classical KL decomposition techniques can be expected to produce comparable accuracy results.


A feasible approach for rapidly varying transient solutions is to obtain more snapshots during the time when the solution is changing rapidly and to compute the basis using the classical KL decomposition technique. However, there are several situations in which obtaining snapshots is not straight-forward. For example, when snapshots are obtained from experiments, repeating the experiment to obtain more snapshots can be very expensive. Similarly, if the snapshots are obtained from numerical simulations and if a rapidly varying transient is represented by a few snapshots, repeating the simulation to get more snapshots with a smaller time step can be very expensive. A good compromise in such cases is to use weighted snapshots to get better basis functions, instead of repeating the experiments or the numerical simulations. Many times it is difficult to foresee the various time scales encountered in the system. The concept of weighting in a KL decomposition technique introduces more flexibility and accuracy to represent multiple time scales in a dynamical system.

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