Index-aware Model Order Reduction Methods: Applications to by N. Banagaaya, Giuseppe Alì, Wil H.A. Schilders

By N. Banagaaya, Giuseppe Alì, Wil H.A. Schilders

The major target of this booklet is to debate version order aid (MOR) tools for differential-algebraic equations (DAEs) with linear coefficients that utilize splitting options earlier than using version order aid. The splitting produces a approach of normal differential equations (ODE) and a procedure of algebraic equations, that are then lowered individually. For the relief of the ODE approach, traditional MOR equipment can be utilized, while for the aid of the algebraic platforms new equipment are mentioned. The dialogue specializes in the index-aware version order aid strategy (IMOR) and its diversifications, equipment for which the so-called index of the unique version is instantly preserved after reduction.

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Additional resources for Index-aware Model Order Reduction Methods: Applications to Differential-Algebraic Equations

Example text

Zpj−2 zqj−1 , If, 2 ≤ j ≤ μ − 2, ⎪ ⎪ ⎩ ∗T If, j = μ − 1. q0 AQ0,j p0 zp0 . . zpμ−3 , CTq,0 = CT q0 ∈ R ×k0 , CTq,1 = CT p0 zq0 ∈ R ×k1 , CTq,i = CT p0 zp0 . . zpi−2 zqi−1 ∈ R ×ki . We can observe that Eq. 22b) where ξq = (ξq,μ−1 , . . , ξq,0 )T ∈ Rn , Bq = (Bq,μ−1 , . . , Bq,0 )T ∈ Rn×m,Cq = (CTq,μ−1 , . . , CTq,0 )T ∈ Rn× , L ∈ Rn×n is a strictly lower triangular nilpotent matrix of index μ. 23) i=0 where u(i) ∈ Rm is the ith derivative of the input data. We observe that, we have only algebraic equations and their solutions can be computed exactly.

10b) μ−1 ξq,i = Aq,i ξp + Bq,i u + Aqi,j ξq,j , i = μ − 2, . . 10f) i=0 where ∗T ∗T np ×np ∗T ∗T np ×m Ap := z∗T , Bp := z∗T , pμ−2 . . zp0 p0 AP p0 zp0 . . zpμ−2 ∈ R pμ−2 . . zp0 p0 BP ∈ R ∗T ∗T ∗T kμ−1 ×np Aq,μ−1 := z∗T , qμ−2 zpμ−3 . . zp0 p0 AQ,μ−1 p0 zp0 . . zpμ−2 ∈ R ∗T ∗T ∗T kμ−1 ×m Bq,μ−1 := z∗T , qμ−2 zpμ−3 . . zp0 p0 BQ,μ−1 ∈ R ∗T ∗T ∗T ki ×np Aq,i := z∗T , qi−1 zpi−2 . . zp0 p0 AQ,i p0 zp0 . . zpμ−2 ∈ R ∗T ∗T ∗T ki ×m Bq,i := z∗T , qi−1 zpi−2 . . zp0 p0 BQ,i ∈ R ∗T ∗T ∗T ki ×kj Aqi,j := z∗T qi−1 zpi−2 .

We denote by (z∗T p0 , zq0 ) the inverse of (zp0 , zq0 ), such that ∗T ∗T ∗T z∗T p0 zp0 = In1 , zp0 zq0 = 0, zq0 zp0 = 0, zq0 zq0 = Ik1 . 1) are {q0 , p0 zq0 , p0 zp0 }, respectively. 5) μ−1 with Zp1 := zp∗T p∗T 0 0 i−1 Qi p0 zp0 + ∗T z∗T p0 p0 μ−1 p0 zp0 , ∗T Zq1 := z∗T p0p0 1 Q2 p0 zp0 . We i=3 μ>3 can also see that the projectors are mutually orthogonal projectors, acting in Rn1 . If, we let zp1 ∈ Im Zq1 and zp1 ∈ Im Zp1 . Let k2 = dim(Im Zq1 ), and n2 = n1 − k2 , and let us consider a basis matrix (zp1 , zq1 ) ∈ Rn1 made of n2 independent columns of projection matrix Zp0 and k2 independent columns of the complementary projection matrix Zq1 .

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