The name is absent

108

We take the SVD of this matrix, V = UΣX^τ, and choose a strong reduced system
size k_s < N_s. The POD matrix is then the first k_s columns of U, i.e. U = (7(:, 1 : fc_s).
Now we define the reduced variable by

v_s = Uv_s, v_s ∈ R^fes, U ∈ R,^jv≡×^fe≈_j

and then we substitute into (4.16) to obtain

∂t

Q I z_s 0
(c∕2)C H_s

oʃl ¹

Φ 2πα∆a?

⅞ι_w(t)^
. ш.

N(Uv_s(t), w_s(t))

Multiplying through by the projection matrices U and U^τ, we obtain the IRKA+P0D

reduced system
where the new “Hines” matrix is now completely dense, but of dimension (k_w + k_s) ×
( k_w 4^- k_s ).

∂t

(Z_s 0) U ^
U⁷ H_sU

’ 0_l , ɪ ∣^BI_w(t)

U^τΦ] 27rα∆rr [u^τI_s(t)

0
U^rN(Uv_s,w_s)

(4.17)

The final step is to reduce the nonlinear term, which follows the same procedure
as in §3.2.2. In this way we obtain a basis for the nonlinear term by taking snapshots
of N, and then using the DEIM to obtain the set of k_s spatial interpolation points z.

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