The name is absent

64

(POD) provides a solution to this problem via the singular value decomposition (SVD)
(Liang et al., 2002).

To obtain the POD basis we first take “snapshots” of the voltage and nonlinear
terms at specific (usually equally-spaced) time points during the simulation. For
convenience we denote the nonlinear term by

N(v(t),w(t)) ≡ (Φ(w(t))e).v(t) -Φ(w(t))E_i.

We save the values of v and N(v, w) at times tɪ, t₂, ∙ ∙ ∙ > t_n, where t_j = j∆t. The
snapshots are stored column-wise in matrices

V = [v(tι) v(t₂) ∙ ∙ ∙ v(t_τι)]

F = [N(v(t₁),w(tι)) N(v(t₂),w(t₂)) ∙∙∙ N(v(t_n),w(t_n))].

The matrix V ∈ R^λz×" will be used to build the POD basis, while F ∈ R^7v×" will be
used in §3.2.2.

We begin with the SVDs of the snapshot matrices

V = UΣX^τ, F = WXY^τ,

where U^τU = /, X^τX = I, W^τW⁷ = I, Y^τY = I, and Σ and Λ are diagonal
and non-negative. These diagonal elements are ordered in a descending fashion. We

<<   <   71   72   73   74   75   76   77   >   >>