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65

choose k_υ ≤ n and kf < n, set

V = U(:,l:k_v) and W = W(: ,1: k_f).

The columns of these matrices form the bases which are the solutions to (3.9) for the
corresponding snapshot sets (Liang et al., 2002).

To complete the POD reduction, we define the reduced voltage variable by

V = Uv                               (3.11)

and, upon substitution into (3.2), we arrive at the reduced-order system

V =U^tHUv -U^tN(Uv,w) + U^tG.(Uv-E_s).             (3.12)

3.2.2 Reduction of the Nonlinear Term via the Discrete Empirical Inter-
polation Method

While the dimension of (3.12) is now k_v ≪ TV, the nonlinear term still depends
on the full dimension N, which indicates that the system has not been truly reduced.
For, the reduced voltage v must be projected up by U to the full subspace before we
can evaluate the nonlinear term, and the result must be projected back down to the
reduced subspace by U^τ. We apply an empirical interpolation method to find a set of
spatial interpolation points z = {¾}*Z₁ from which the behavior of the full nonlinear

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