The name is absent

66

Algorithm 2 : DEIM (Discrete Empirical Interpolation Method)

Input: {W_i}*Zι ɑ r^jv linearly independent

Output: z = [z₁,..., z_kf]^τ ∈ R^fcy

1: Define zɪ = argmax∣Wι(j)∣

J

2: W = [W₁], P = [e_zι], z = [z₁]

3: for i = 2 to kf do

4: Solve (P^τW)s = P^τW_i for s

5:   r ≡ Wj — Ws

6:   Define z_i = argmax∣r(j)∣

j

7: W ÷- [W W₁], P <— [P e_zJ, z ÷-

8: end for

term can be approximated, thus reducing the complexity of N to kf ≪ N (Barrault
et al., 2004). This method, originally described for use with finite elements, has
been extended to our case of finite differences via the discrete empirical interpolation
method (DEIM) as given by (Chaturantabut and Sorensen, 2009).

We begin with the basis W for the snapshot set F of nonlinear terms and seek a
“maximally independent basis set” for W (Nguyen et al., 2008). The first interpolation
point Zi is the index of the entry of W_i with the largest magnitude, where wɪ is the
first column of W. For i = 2,..., kf each point z_i is chosen as the index of the entry
of the largest magnitude of the residual vector r ≡ W_i — Ws, where P is the matrix of
the г-l coordinate vectors corresponding to the previous interpolation points, W is
the matrix of the previous i — 1 DEIM basis vectors for N (that is, W = W(:, 1 : i — 1)),
and s is the coefficient vector of components of W in W_i relative to the previous i — 1

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