Frequency- and Time-limited H2-optimal Model Order Reduction within Projection Framework
aa r X i v : . [ ee ss . S Y ] F e b Frequency- and Time-limited H -optimal Model Order Reductionwithin Projection Framework Umair Zulfiqar a , Victor Sreeram a , and Xin Du b a School of Electrical, Electronics and Computer Engineering, The University of WesternAustralia (UWA), Perth, Australia; b School of Mechatronic Engineering and Automation,and Shanghai Key Laboratory of Power Station Automation Technology, ShanghaiUniversity, Shanghai, China
ARTICLE HISTORY
Compiled February 9, 2021
ABSTRACT
In this short note, the problems of frequency-limited and time-limited H -optimalmodel order reduction are considered within the projection framework. It is shownthat it is inherently not possible to satisfy all the necessary conditions for the localminimizer in the projection framework. The conditions for the exact satisfaction ofthe optimality conditions are also discussed. The equivalence between the tangentialinterpolation conditions and the gramians-based necessary condition is established.The stationary point iteration algorithms that satisfy two out of three necessaryconditions for the local minimizer are also proposed. The efficacy of the proposedalgorithms is validated by considering an illustrative example. KEYWORDS H -optimal; frequency-limited; model order reduction; oblique projection;pseudo-optimal; time-limited
1. Preliminaries
This section covers the preliminary details of the problems under consideration. First,the problem of model order reduction (MOR) via projection is introduced. Second,the frequency- and time-limited H -optimal MOR problems are formulated, and thenecessary conditions for the local minima are detailed. The mathematical notationsused throughout the text are given in Table 1. Let G ( s ) be an n th -order p × m transfer function of a stable linear time-invariantsystem, which is related to its state-space realization ( A, B, C, D ) as G ( s ) = C ( sI − A ) − B + D where A ∈ R n × n , B ∈ R n × m , C ∈ R p × n , and D ∈ R p × m . CONTACT Umair Zulfiqar. Email: umair.zulfi[email protected] able 1.: Mathematical NotationsNotation Meaning (cid:2) · (cid:3) ∗ Hermitian tr ( · ) Trace L [ · ] Fr´echet derivative of the matrix logarithm. Ran ( · ) Range orth ( · ) Orthogonal basis span i =1 , ··· ,r {·} Span of the set of r vectorsThe MOR problem is to construct an r th -order p × m transfer function ˆ G ( s ) thatclosely approximates G ( s ) where r ≪ n . Let ˆ G ( s ) be related to its state-space realiza-tion ( ˆ A, ˆ B, ˆ C, D ) as ˆ G ( s ) = ˆ C ( sI − ˆ A ) − ˆ B + D where ˆ A ∈ R r × r , ˆ B ∈ R r × m , and ˆ C ∈ R p × r .Let ˆ V ∈ R n × r and ˆ W ∈ R n × r be the input and output reduction subspaces, respec-tively, which project G ( s ) onto an r -dimensional subspace where Π = ˆ V ˆ W T is theoblique projection and ˆ W T ˆ V = I . The columns of ˆ V span the reduced subspace alongthe kernel of ˆ W T . The r th -order reduced-order model (ROM) obtained by projectionis given by ˆ A = ˆ W T A ˆ V , ˆ B = ˆ W T B, ˆ C = C ˆ V . (1)The error transfer function E ( s ) = G ( s ) − ˆ G ( s ) has the following equivalence with itsstate-space realization ( A e , B e , C e ) E ( s ) = C e ( sI − A e ) − B e where A e = (cid:20) A
00 ˆ A (cid:21) , B e = (cid:20) B ˆ B (cid:21) , C e = (cid:2) C − ˆ C (cid:3) . Let P e,ω and Q e,ω be the frequency-limited controllability and the frequency-limitedobservability gramians, respectively, of the realization ( A e , B e , C e ) within the fre-quency interval [ − ω, ω ] rad/sec (Gawronski and Juang, 1990), which solve the fol-lowing Lyapunov equations A e P e,ω + P e,ω A Te + B e,ω B Te + B e B Te,ω = 0 ,A Te Q e,ω + Q e,ω A e + C Te,ω C e + C Te C e,ω = 02here B e,ω = F ω [ A e ] B e , C e,ω = C e F ω [ A e ] ,F ω [ A e ] = j π log (cid:0) ( jωI + A e )( − jωI + A e ) − (cid:1) . The energy of the impulse response of E ( s ) within the frequency interval [ − ω, ω ]rad/sec is quantified by the frequency-limited H -norm (Petersson and L¨ofberg, 2014),which is related to P e,ω and Q e,ω as || E ( s ) || H ,ω = q tr ( C e P e,ω C Te ) = q tr ( CP ω C T − C ¯ P ω ˆ C T + ˆ C ˆ P ω ˆ C T )= q tr ( B Te Q e,ω B e ) = q tr ( B T Q ω B − B T ¯ Q ω ˆ B + ˆ B T ˆ Q ω ˆ B ) .P ω , Q ω , ¯ P ω , ¯ Q ω , ˆ P ω , and ˆ Q ω solve the following linear matrix equations AP ω + P ω A T + BB Tω + B ω B T = 0 , (2) A T Q ω + Q ω A + C T C ω + C Tω C = 0 (3) A ¯ P ω + ¯ P ω ˆ A T + B ˆ B Tω + B ω ˆ B T = 0 , (4) A T ¯ Q ω + ¯ Q ω ˆ A + C T ˆ C ω + C Tω ˆ C = 0 , (5)ˆ A ˆ P ω + ˆ P ω ˆ A T + ˆ B ˆ B Tω + ˆ B ω ˆ B T = 0 , (6)ˆ A T ˆ Q ω + ˆ Q ω ˆ A + ˆ C T ˆ C ω + ˆ C Tω ˆ C = 0 (7)where B ω = F ω [ A ] B, ˆ B ω = F ω [ ˆ A ] ˆ B, C ω = CF ω [ A ] , ˆ C ω = ˆ CF ω [ ˆ A ] . In the frequency-limited MOR problem, the frequency response of E ( s ) is soughtto be small within the desired frequency interval Ω = [ − ω, ω ] rad/sec, which is gener-ally quantified by the H ,ω -norm. Thus H ,ω -MOR problem under consideration is toconstruct ˆ G ( s ) such that || E ( s ) || H ,ω is small, i.e.,min ˆ G ( s )order= r || E ( s ) || H ,ω . Let P e,τ and Q e,τ be the time-limited controllability and the time-limited observ-ability gramians, respectively, of the realization ( A e , B e , C e ) within the time interval τ = [0 , t ] sec (Gawronski and Juang, 1990), which solve the following Lyapunov equa-tions A e P e,τ + P e,τ A Te + B e B Te − B e,τ B Te,τ = 0 ,A Te Q e,τ + Q e,τ A e + C Te C e − C Te,τ C e,τ = 0 . where B e,τ = e A e t B e and C e,τ = C e e A e t . E ( s ) within the time interval τ = [0 , t ] rad/sec isquantified by the time-limited H -norm (Goyal and Redmann, 2019), which is relatedto P e,τ and Q e,τ as || E ( s ) || H ,τ = q tr ( C e P e,τ C Te ) = q tr ( CP τ C T − C ¯ P τ ˆ C T + ˆ C ˆ P τ ˆ C T )= q tr ( B Te Q e,τ B e ) = q tr ( B T Q τ B − B T ¯ Q τ ˆ B + ˆ B T ˆ Q τ ˆ B ) .P τ , Q τ , ¯ P τ , ¯ Q τ , ˆ P τ , and ˆ Q τ solve the following linear matrix equations AP τ + P τ A T + BB T − B τ B Tτ = 0 , (8) A T Q τ + Q τ A + C T C − C Tτ C τ = 0 (9) A ¯ P τ + ¯ P τ ˆ A T + B ˆ B T − B τ ˆ B Tτ = 0 , (10) A T ¯ Q τ + ¯ Q τ ˆ A + C T ˆ C − C Tτ ˆ C τ = 0 (11)ˆ A ˆ P τ + ˆ P τ ˆ A T + ˆ B ˆ B T − ˆ B τ ˆ B Tτ = 0 , (12)ˆ A T ˆ Q τ + ˆ Q τ ˆ A + ˆ C T ˆ C − ˆ C Tτ ˆ C τ = 0 (13)where B τ = e At B, ˆ B τ = e ˆ At ˆ B, C τ = Ce At , ˆ C τ = ˆ Ce ˆ At . In the time-limited MOR problem, the time response of E ( s ) is sought to be smallwithin the desired time interval τ = [0 , t ] sec. This is generally quantified by the H ,τ -norm. Thus H ,τ -MOR problem under consideration is to construct ˆ G ( s ) such that || E ( s ) || H ,τ is small, i.e., min ˆ G ( s )order= r || E ( s ) || H ,τ . Let ˆ Q be the observability gramian of the pair ( ˆ A, ˆ C ), and ¯ Q solve the followingSylvester equation A T ¯ Q + ¯ Q ˆ A + C T ˆ C = 0 . Then ( ˆ A, ˆ B, ˆ C ) is a local minimizer for || E ( s ) || H ,ω if it satisfies the following gramian-based conditions (Petersson and L¨ofberg, 2014)¯ Q T ¯ P ω − ˆ Q ˆ P ω + Z ω = 0 , (14)¯ Q Tω B − ˆ Q ω ˆ B = 0 , (15) C ¯ P ω − ˆ C ˆ P ω = 0 (16)4here Z ω = Re (cid:0) jπ L ( − ˆ A − jωI, ˆ C T ˆ C ˆ P ω − ˆ C T C ¯ P ω ) (cid:1) . Let G ( s ) and ˆ G ( s ) have simple poles and the following pole-residue forms G ( s ) = n X i =1 l i r Ti s − λ i + D, ˆ G ( s ) = r X i =1 ˆ l i ˆ r Ti s − ˆ λ i + D. Now define T ω ( s ) and ˆ T ω ( s ) as T ω ( s ) = G ω ( s ) + G ( s ) F ω [ − s ] , ˆ T ω ( s ) = ˆ G ω ( s ) + ˆ G ( s ) F ω [ − s ]where G ω ( s ) = n X i =1 l i r Ti s − λ i F ω [ λ i ] , ˆ G ω ( s ) = r X i =1 ˆ l i ˆ r Ti s − ˆ λ i F ω [ˆ λ i ] . Then ˆ G ( s ) is a local minimizer for || E ( s ) || H ,ω if it satisfies the following bi-tangentialHermite interpolation conditions (Vuillemin, 2014)ˆ l Ti T ′ ω ( − ˆ λ i )ˆ r i = ˆ l Ti ˆ T ′ ω ( − ˆ λ i )ˆ r i , (17)ˆ l Ti T ω ( − ˆ λ i ) = ˆ l Ti ˆ T ω ( − ˆ λ i ) , (18) T ω ( − ˆ λ i )ˆ r i = ˆ T ω ( − ˆ λ i )ˆ r i . (19)On similar lines, ( ˆ A, ˆ B, ˆ C ) is a local minimizer for || E ( s ) || H ,τ if it satisfies thefollowing gramians-based conditions (Goyal and Redmann, 2019)¯ Q T ¯ P τ − ˆ Q ˆ P τ + Z τ = 0 , (20)¯ Q Tτ B − ˆ Q τ ˆ B = 0 , (21) C ¯ P τ − ˆ C ˆ P τ = 0 (22)where Z τ = t (cid:0) ˆ Qe ˆ At ˆ B ˆ B T e ˆ A T t − ¯ Q T e At B ˆ B T e ˆ A T t (cid:1) . Now suppose G ( s ) and ˆ G ( s ) have simple poles and define T τ ( s ) and ˆ T τ ( s ) as T τ ( s ) = G τ ( s ) + G ( s ) , ˆ T τ ( s ) = ˆ G τ ( s ) + ˆ G ( s )where G τ ( s ) = − e − st C ( sI − A ) − e At B, ˆ G τ ( s ) = − e − st ˆ C ( sI − ˆ A ) − e ˆ At ˆ B. G ( s ) is a local minimizer for || E ( s ) || H ,τ if it satisfies the following bi-tangentialHermite interpolation conditions (Sinani and Gugercin, 2019)ˆ l Ti T ′ τ ( − ˆ λ i )ˆ r i = ˆ l Ti ˆ T ′ τ ( − ˆ λ i )ˆ r i , (23)ˆ l Ti T τ ( − ˆ λ i ) = ˆ l Ti ˆ T τ ( − ˆ λ i ) , (24) T τ ( − ˆ λ i )ˆ r i = ˆ T τ ( − ˆ λ i )ˆ r i . (25)
2. Existing Projection-based MOR Techniques
In this section, the important projection-based algorithms for H ,ω - and H ,τ -MORproblems are briefly reviewed. FLTSIA (Du et al., 2021; Vuillemin et al., 2013) is a heuristic generalization of(Xu and Zeng, 2011) based on analogy and experimental results. Staring with aninitial guess of the ROM, the reduction subspaces are updated as ˆ V = ¯ P ω andˆ W = ¯ Q ω . To ensure the oblique projection condition ˆ W T ˆ V = I , the correctionequation ˆ W = ˆ W ( ˆ V T ˆ W ) − is used. The ROM is generated by using these reduc-tion subspaces, and the process is repeated until the algorithm converges. The ROMconstructed by FLTSIA may not satisfy the optimality conditions (14)-(16). Let us define B Ω , C Ω , ˆ B Ω , and ˆ C Ω as B Ω = (cid:2) B B ω (cid:3) , C Ω = (cid:2) C T C Tω (cid:3) T , ˆ B Ω = (cid:2) ˆ B ˆ B ω (cid:3) , ˆ C Ω = (cid:2) ˆ C T ˆ C Tω (cid:3) T . Now define G Ω ( s ), ˆ G Ω ( s ), H Ω ( s ), and ˆ H Ω ( s ) as G Ω ( s ) = C ( sI − A ) − B Ω , ˆ G Ω ( s ) = ˆ C ( sI − ˆ A ) − ˆ B Ω ,H Ω ( s ) = C Ω ( sI − A ) − B, ˆ H Ω ( s ) = ˆ C Ω ( sI − ˆ A ) − ˆ B. It is shown in (Zulfiqar et al., 2020a) that ˆ G ( s ) satisfies the optimality conditions (16)and (15) if the following tangential interpolation conditions are respectively satisfied G Ω ( − ˆ λ i )¯ r i = ˆ G Ω ( − ˆ λ i )¯ r i , (26)¯ l Ti H Ω ( − ˆ λ i ) = ¯ l Ti ˆ H Ω ( − ˆ λ i ) (27)where ¯ r i = (cid:20) F ω [ˆ λ i ]ˆ r i ˆ r i (cid:21) and ¯ l i = (cid:20) F ω [ˆ λ i ]ˆ l i ˆ l i (cid:21) . 6he input Krylov subspace ˆ V in FLPORK (Zulfiqar et al., 2020a) is obtained as Ran ( ˆ V ) = span i =1 , ··· ,r { (ˆ σ i I − A ) − B Ω ¯ b i } where ˆ σ i is the interpolation point, ˆ b i is the associated tangential direction, and ¯ b i = (cid:20) F ω [ − ˆ σ i ]ˆ b i ˆ b i (cid:21) . Now define the oblique projection Π = ˆ V ˆ W T wherein ˆ W is arbitrary.Also, define B ⊥ , ˆ S , and ¯ L as B ⊥ = ( I − Π) B Ω , ¯ L = ( B T ⊥ B ⊥ ) − B ⊥ ( I − Π) A ˆ V ˆ S = ˆ W T ( A ˆ V − B Ω ¯ L ) . Further, partition ¯ L as ¯ L = (cid:20) ˆ L ω ˆ L (cid:21) where ˆ L ω = ˆ LF ω [ − ˆ S ]. The frequency-limited ob-servability gramian ˆ Q s,ω of the pair ( − ˆ S, ˆ L ) solves the following Lyapunov equation − ˆ S T ˆ Q s,ω − ˆ Q s,ω ˆ S + ˆ L T ˆ L ω + ˆ L Tω ˆ L = 0 . Then ˆ G ( s ) that satisfies the interpolation condition (26) can be obtained asˆ A = − ˆ Q − s,ω ˆ S T ˆ Q s,ω , ˆ B = − ˆ Q − s,ω ˆ L T , ˆ C = C ˆ V .
A dual result also exists that constructs a ROM, which satisfies the interpolationcondition (27); see (Zulfiqar et al., 2020a) for details.
TLIRKA (Goyal and Redmann, 2019) is a heuristic generalization of the iterativerational Krylov algorithm (IRKA) (Gugercin et al., 2008) for the time-limited case.Let ˆ G ( s ) has simple poles and ˆ A = ˆ R ˆΛ ˆ R − be the spectral factorization of ˆ A where ˆΛ = diag (ˆ λ , · · · , ˆ λ r ). Starting with a random guess of the ROM, the reduction subspacesin TLIRKA are computed as ˆ V = ¯ P τ ˆ R −∗ and ˆ W = ¯ Q τ ˆ R . To ensure the obliqueprojection condition ˆ W ∗ ˆ V = I , ˆ V and ˆ W are updated as ˆ V = orth ( ˆ V ), ˆ W = orth ( ˆ W ),and ˆ W = ˆ W ( ˆ V ∗ ˆ W ) − . The ROM is generated by using these reduction subspaces,and the process is repeated until the algorithm converges. The ROM constructed byTLIRKA may not satisfy the optimality conditions (20)-(22). Let us define B T , C T , ˆ B T , and ˆ C T as B T = (cid:2) B − B τ (cid:3) , C T = (cid:2) C T − C Tτ (cid:3) T , ˆ B T = (cid:2) ˆ B − ˆ B τ (cid:3) , ˆ C T = (cid:2) ˆ C T − ˆ C Tτ (cid:3) T . G T ( s ), ˆ G T ( s ), H T ( s ), and ˆ H T ( s ) as G T ( s ) = C ( sI − A ) − B T , ˆ G T ( s ) = ˆ C ( sI − ˆ A ) − ˆ B T ,H T ( s ) = C T ( sI − A ) − B, ˆ H T ( s ) = ˆ C T ( sI − ˆ A ) − ˆ B. It is shown in (Zulfiqar et al., 2020b) that ˆ G ( s ) satisfies the optimality conditions (22)and (21) if the following tangential interpolation conditions are respectively satisfied G T ( − ˆ λ i )˜ r i = ˆ G T ( − ˆ λ i )˜ r i , (28)˜ l Ti H T ( − ˆ λ i ) = ˜ l Ti ˆ H T ( − ˆ λ i ) (29)where ˜ r i = (cid:20) ˆ r i e ˆ λ i t ˆ r i (cid:21) and ˜ l i = " ˆ l i e ˆ λ i t ˆ l i .The input Krylov subspace ˆ V in TLPORK (Zulfiqar et al., 2020b) is obtained as Ran ( ˆ V ) = span i =1 , ··· ,r { (ˆ σ i I − A ) − B T ˜ b i } where ˜ b i = (cid:20) ˆ b i e − ˆ σ i t ˆ b i (cid:21) . Now define the oblique projection Π = ˆ V ˆ W T wherein ˆ W isarbitrary. Also, define B ⊥ , ˆ S , and ¯ L as B ⊥ = ( I − Π) B T , ¯ L = ( B T ⊥ B ⊥ ) − B ⊥ ( I − Π) A ˆ V ˆ S = ˆ W T ( A ˆ V − B T ¯ L ) . Further, partition ¯ L as ¯ L = (cid:20) ˆ L ˆ L τ (cid:21) where ˆ L τ = ˆ Le − ˆ St . The time-limited observabilitygramian ˆ Q s,τ of the pair ( − ˆ S, ˆ L ) solves the following Lyapunov equation − ˆ S T ˆ Q s,τ − ˆ Q s,τ ˆ S + ˆ L T ˆ L − ˆ L Tτ ˆ L τ = 0 . Then ˆ G ( s ) that satisfies the interpolation condition (28) can be obtained asˆ A = − ˆ Q − s,τ ˆ S T ˆ Q s,τ , ˆ B = − ˆ Q − s,τ ˆ L T , ˆ C = C ˆ V .
A dual result also exists that constructs a ROM, which satisfies the interpolationcondition (29); see (Zulfiqar et al., 2020b) for details. H ,ω - and H ,τ -optimal MOR In this section, the problems of H ,ω - and H ,τ -optimal MOR are considered within theprojection framework. The inherent difficulty in constructing a local optimum withinthe projection framework is discussed. The conditions for exact satisfaction of theoptimality conditions are also discussed. The equivalence between the gramians-basedand interpolation-based optimality conditions is also established.8 .1. Limitation in the Projection Framework Before beginning the discussion on the inherent limitation in the projection frameworkto satisfy the optimality conditions (14)-(16) and (20)-(22), let us represent the opti-mality conditions (14) and (20) a bit differently so that it resembles Wilson’s conditionof the standard H -optimal MOR (Wilson, 1970). Proposition 3.1. ˆ G ( s ) satisfies the optimality conditions (14) and (20) if ¯ Q Tω ¯ P ω − ˆ Q ω ˆ P ω + X ω = 0 and ¯ Q Tτ ¯ P τ − ˆ Q τ ˆ P τ + X τ = 0 , respectively, where X ω = F ω [ ˆ A T ] Z ω − ¯ Q T F ω [ A ] ¯ P ω + ˆ QF ω [ ˆ A ] ˆ P ω ,X τ = Z τ + e ˆ A T t ¯ Q T e At ¯ P τ − e ˆ A T t ˆ Qe ˆ At ˆ P τ . Proof.
Note that¯ Q Tω ¯ P ω = (cid:0) F ω [ ˆ A T ] ¯ Q T + ¯ Q T F ω [ A ] (cid:1) ¯ P ω and ˆ Q ω ˆ P ω = (cid:0) F ω [ ˆ A T ] ˆ Q + ˆ QF ω [ ˆ A ] (cid:1) ˆ P ω . Thus F ω [ ˆ A T ] ¯ Q T ¯ P ω = ¯ Q Tω ¯ P ω − ¯ Q T F ω [ A ] ¯ P ω , (30) F ω [ ˆ A T ] ˆ Q ˆ P ω = ˆ Q ω ˆ P ω − ˆ QF ω [ ˆ A ] ˆ P ω . (31)By multiplying F ω [ ˆ A T ] from the left, the equation (14) becomes F ω [ ˆ A T ] ¯ Q T ¯ P ω − F ω [ ˆ A T ] ˆ Q ˆ P ω + F ω [ ˆ A T ] Z ω = 0 (32)By putting (30) and (31), the equation (32) becomes¯ Q Tω ¯ P ω − ˆ Q ω ˆ P ω + X ω = 0Similarly, note that¯ Q Tτ ¯ P τ = (cid:0) ¯ Q T − e ˆ A T t ¯ Q T e At (cid:1) ¯ P τ and ˆ Q τ ˆ P τ = (cid:0) ˆ Q − e ˆ A T ˆ Qe ˆ At (cid:1) ˆ P τ . Thus ¯ Q T ¯ P τ = ¯ Q Tτ ¯ P τ + e ˆ A T t ¯ Q T e At ¯ P τ , (33)ˆ Q ˆ P τ = ˆ Q τ ˆ P τ + e ˆ A T ˆ Qe ˆ At ˆ P τ . (34)By putting (33) and (34), the equation (20) becomes¯ Q Tτ ¯ P τ − ˆ Q τ ˆ P τ + X τ = 0 . This completes the proof. 9et us assume that ˆ P ω and ˆ Q ω are invertible, then the optimal choices ˆ B and ˆ C according to the optimality conditions (15) and (16), respectively, are given byˆ B = ˆ Q − ω ¯ Q Tω B and ˆ C = C ¯ P ω ˆ P − ω . Similarly, let us assume that ˆ P τ and ˆ Q τ are invertible, then the optimal choices ˆ B andˆ C according to the optimality conditions (21) and (22), respectively, are given byˆ B = ˆ Q − τ ¯ Q Tτ B and ˆ C = C ¯ P τ ˆ P − τ . If the ROMs are obtained within the projection framework, the oblique projectionsfor the optimal choices ˆ B and ˆ C are given by Π = ¯ P ω ˆ P − ω ˆ Q − ω ¯ Q Tω (with ˆ V = ¯ P ω ˆ P − ω and ˆ W = ¯ Q ω ˆ Q − ω ) and Π = ¯ P τ ˆ P − τ ˆ Q − τ ¯ Q Tτ (with ˆ V = ¯ P τ ˆ P − τ and ˆ W = ¯ Q τ ˆ Q − τ ).Further, since ˆ W T ˆ V = I , ¯ Q Tω ¯ P ω − ˆ Q ω ˆ P ω = 0 and ¯ Q Tτ ¯ P τ − ˆ Q τ ˆ P τ = 0. Then thedeviations in the optimal choices of ˆ A are given by X ω and X τ . It can readily beverified that X ω = 0 when F ω [ A ] = ˆ V F ω [ ˆ A ] ˆ W T and ¯ Q = ˆ W ˆ Q . Similarly, X τ = 0when e At = ˆ V e ˆ At ˆ W T and ¯ Q = ˆ W ˆ Q . Note that ˆ W ˆ Q , ˆ V F ω [ ˆ A ] ˆ W T , and ˆ V e ˆ At ˆ W T arethe projection-based approximations of ¯ Q , F ω [ A ], and e At , respectively (Benner et al.,2016; K¨urschner, 2018). Thus ¯ Q = ˆ W ˆ Q , F ω [ A ] = ˆ V F ω [ ˆ A ] ˆ W T , and e At = ˆ V e ˆ At ˆ W T , ingeneral. Resultantly, X ω = 0 and X τ = 0, in general. Therefore, it is inherently notpossible to obtain an optimal choice of ˆ A within the projection framework when theoptimal choices of ˆ B and ˆ C are made. Note that ˆ P − ω and ˆ Q − ω do not change the subspaces but only change the basis of¯ P ω and ¯ Q ω in ˆ V = ¯ P ω ˆ P − ω and ˆ W = ¯ Q ω ˆ Q − ω , respectively. Similarly, ˆ P − τ and ˆ Q − τ only change the basis of ¯ P τ and ¯ Q τ in ˆ V = ¯ P τ ˆ P − τ and ˆ W = ¯ Q τ ˆ Q − τ , respectively(Gallivan et al., 2004). Thus one may think of constructing the ROM using the obliqueprojections Π = ¯ P ω ¯ Q Tω (as done in FLTSI) and Π = ¯ P τ ¯ Q Tτ (as done in TLIRKA). Thischange of basis is harmless in the standard H -optimal MOR; see (Benner et al., 2011;Xu and Zeng, 2011), for instance. However, in the frequency- and time-limited cases,this incurs deviations in the optimal choices of ˆ B and ˆ C . The next two theorems showthat the deviation caused by this change of basis is zero when F ω [ A ] = ˆ V F ω [ ˆ A ] ˆ W T and e At = ˆ V e ˆ At ˆ W T , which is not possible in general. Theorem 3.2.
The ROM ( ˆ A, ˆ B, ˆ C ) obtained from the oblique projection Π = ¯ P ω ¯ Q Tω (with ˆ V = ¯ P ω and ˆ W = ¯ Q ω ) satisfies the optimality conditions (15) and (16) provided B ω = ˆ V ˆ B ω and C ω = ˆ C ω ˆ W T . Proof.
By multiplying ˆ W T from the left, the equation (4) becomesˆ W T A ¯ P ω + ˆ W T ¯ P ω ˆ A T + ˆ W T B ˆ B Tω + ˆ W T B ω ˆ B T = 0 . Since ˆ W T ˆ V = I and ˆ W T B ω = ˆ B ω ,ˆ A + ˆ A T + ˆ B ˆ B Tω + ˆ B ω ˆ B T = 0 . P ω = I and thus C ¯ P ω − C ˆ P ω = 0.Similarly, by multiplying ˆ V T from the left, the equation (5) becomesˆ V T A T ¯ Q ω + ˆ V T ¯ Q ω ˆ A + ˆ V T C T ˆ C ω + ˆ V T C Tω ˆ C = 0 . Since ˆ V T ˆ W = I and C ω ˆ V = ˆ C ω ,ˆ A T + ˆ A + ˆ C T ˆ C ω + ˆ C Tω ˆ C = 0 . Due to uniqueness, ˆ Q ω = I and thus ¯ Q Tω B − ˆ Q ω B = 0. This completes the proof.ˆ V ˆ B ω and ˆ C ω ˆ W T are the projection-based approximations of B ω and C ω , respec-tively. Thus B ω = ˆ V ˆ B ω and C ω = ˆ C ω ˆ W T , in general. The Krylov-subspace basedmethods to obtain approximations of B ω and C ω as ˆ V ˆ B ω and ˆ C ω ˆ W T , respectively,can be found in (Benner et al., 2016). Theorem 3.3.
The ROM ( ˆ A, ˆ B, ˆ C ) obtained from the oblique projection Π = ¯ P τ ¯ Q Tτ (with ˆ V = ¯ P τ and ˆ W = ¯ Q τ ) satisfies the optimality conditions (21) and (22) provided B τ = ˆ V ˆ B τ and C τ = ˆ C τ ˆ W T . Proof.
By multiplying ˆ W T from the left, the equation (10) becomesˆ W T A ¯ P τ + ˆ W T ¯ P τ ˆ A T + ˆ W T B ˆ B T − ˆ W T B τ ˆ B Tτ = 0 . Since ˆ W T ˆ V = I and ˆ W T B τ = ˆ B τ ,ˆ A + ˆ A T + ˆ B ˆ B T − ˆ B τ ˆ B Tτ = 0 . Due to uniqueness, ˆ P τ = I and thus C ¯ P τ − C ˆ P τ = 0.Similarly, by multiplying ˆ V T from the left, the equation (11) becomesˆ V T A T ¯ Q τ + ˆ V T ¯ Q τ ˆ A + ˆ V T C T ˆ C − ˆ V T C Tτ ˆ C τ = 0 . Since ˆ V T ˆ W = I and C τ ˆ V = ˆ C τ ,ˆ A T + ˆ A + ˆ C T ˆ C − ˆ C Tτ ˆ C τ = 0 . Due to uniqueness, ˆ Q τ = I and thus ¯ Q Tτ B − ˆ Q τ B = 0. This completes the proof.ˆ V ˆ B τ and ˆ C τ ˆ W T are the projection-based approximations of B τ and C τ , respectively.Thus B τ = ˆ V ˆ B τ and C τ = ˆ C τ ˆ W T , in general. The Krylov-subspace based methods toobtain approximations of B τ and C τ as ˆ V ˆ B τ and ˆ C τ ˆ W T , respectively, can be foundin (K¨urschner, 2018). In this subsection, the gramians-based optimality conditions in (Goyal and Redmann,2019; Petersson and L¨ofberg, 2014) and interpolation-based conditions (Zulfiqar et al.,11020a,b) for the H ω - and H ,τ -optimal MOR problems are shown to be equivalent when G ( s ) and ˆ G ( s ) have simple poles. Theorem 3.4.
When G ( s ) and ˆ G ( s ) have simple poles, the following statements aretrue:(i) The optimality condition (16) is equivalent to the tangential interpolation condition(26).(ii) The tangential interpolation conditions (26) and (19) are equivalent.(iii) The optimality condition (15) is equivalent to the tangential interpolation condi-tion (27).(iv) The tangential interpolation conditions (27) and (18) are equivalent.(v) The optimality condition (22) is equivalent to the tangential interpolation condi-tion (28).(vi) The tangential interpolation conditions (28) and (25) are equivalent.(vii) The optimality condition (21) is equivalent to the tangential interpolation condi-tion (29).(viii) The tangential interpolation conditions (29) and (24) are equivalent. Proof. (i) Let us define ˜ B and ˜ C as˜ B = ˆ R − ˆ B = (cid:2) ˆ r · · · ˆ r r (cid:3) T , ˜ C = ˆ C ˆ R = (cid:2) ˆ l · · · ˆ l r (cid:3) . Now define ¯ P ω as ¯ P ω = ¯ P ω ˆ R − T . By noting that F ω [ ˆ A ] = ˆ RF ω [ ˆΛ] ˆ R − and multiplyingˆ R − T from the right, the equation (4) becomes A ¯ P ω + ¯ P ω ˆΛ + B ˜ B T F ω [ ˆΛ] + B ω ˜ B T = 0 , Since ˆΛ (and resultantly F ω [ ˆΛ]) is a diagonal matrix (Petersson, 2013), ¯ P ω can becomputed column-wise as¯ P ω,i = ( − ˆ λ i I − A ) − BF ω [ˆ λ i ]ˆ r i + ( − ˆ λ i I − A ) − B ω ˆ r i . Further, let us define ˆ P ω as ˆ P ω = ˆ P ω ˆ R − T . By multiplying ˆ R − T from the right, theequation (6) becomes ˆ A ˆ P ω + ˆ P ω ˆΛ + ˆ B ˜ B T F ω [ ˆΛ] + ˆ B ω ˜ B T = 0 . Then ˆ P Ω can be computed column-wise asˆ P ω,i = ( − ˆ λ i I − ˆ A ) − ˆ BF ω [ˆ λ i ] ˆ r i + ( − ˆ λ i I − ˆ A ) − ˆ B ω ˆ r i . Now, the optimality condition (16) can be written as C (cid:2) ¯ P ω, · · · ¯ P ω,r (cid:3) ˆ R T − ˆ C (cid:2) ˆ P ω, · · · ˆ P ω,r (cid:3) ˆ R T = 0 . (35)After multiplying ˆ R − T from the right, each column of (35) becomes G Ω ( − ˆ λ i )¯ r i − ˆ G Ω ( − ˆ λ i )¯ r i = 0 . A be A = R Λ R − where Λ = diag ( λ , · · · , λ n ).Note that B ω = RF ω [Λ] R − B and ˆ B ω = ˆ RF ω [ ˆΛ] ˆ R − ˆ B . Thus G Ω ( s ) and ˆ G Ω ( s ) canbe represented as G Ω ( s ) = hP ni =1 l i r Ti s − λ i P ni =1 l i r Ti s − λ i F ω [ λ i ] i = (cid:2) G ( s ) G ω ( s ) (cid:3) , ˆ G Ω ( s ) = hP ri =1 ˆ l i ˆ r Ti s − ˆ λ i P ri =1 ˆ l i ˆ r Ti s − ˆ λ i F ω [ˆ λ i ] i = (cid:2) ˆ G ( s ) ˆ G ω ( s ) (cid:3) . Then it can readily be noted that G Ω ( − ˆ λ i )¯ r i = T ω ( − ˆ λ i )ˆ r i and ˆ G Ω ( − ˆ λ i )¯ r i = ˆ T ω ( − ˆ λ i )ˆ r i .(iii) Let us define ¯ Q ω as ¯ Q ω = ¯ Q ω ˆ R . By multiplying ˆ R from the right, the equation(5) becomes A T ¯ Q ω + ¯ Q ω ˆΛ + C T ˜ CF ω [ ˆΛ] + C Tω ˜ C = 0 . Then ¯ Q ω can be computed column-wise as¯ Q ω,i = ( − ˆ λ i I − A T ) − C T F ω [ˆ λ i ]ˆ l i + ( − ˆ λ i I − A T ) − C Tω ˆ l i . Now define ˆ Q ω as ˆ Q ω = ˆ Q ω ˆ R . By multiplying ˆ R from the right, the equation (7)becomes ˆ A T ˆ Q ω + ˆ Q ω ˆΛ + ˆ C T ˜ CF ω [ ˆΛ] + ˆ C Tω ˜ C = 0Then ˆ Q ω can be computed column-wise asˆ Q ω,i = ( − ˆ λ i I − ˆ A T ) − ˆ C T F ω [ˆ λ i ]ˆ l i + ( − ˆ λ i I − ˆ A T ) − ˆ C Tω ˆ l i . Further, the optimality condition (15) can be written as B T (cid:2) ¯ Q Ω , · · · ¯ Q Ω ,r (cid:3) ˆ R − ˆ B T (cid:2) ˆ Q Ω , · · · ˆ Q Ω ,r (cid:3) ˆ R = 0 . (36)After multiplying ˆ R − from the right, each column of (36) becomes H T Ω ( − ˆ λ i )¯ l i − ˆ H T Ω ( − ˆ λ i )¯ l i = 0 . (iv) Note that C ω = CRF ω [Λ] R − and ˆ C ω = ˆ C ˆ RF ω [ ˆΛ] ˆ R − . Thus H Ω ( s ) and ˆ H Ω ( s )can be represented as H Ω ( s ) = " P ni =1 l i r Ti s − λ i P ni =1 l i r Ti s − λ i F ω [ λ i ] = (cid:20) G ( s ) G ω ( s ) (cid:21) , ˆ H Ω ( s ) = P ri =1 ˆ l i ˆ r Ti s − ˆ λ i P ri =1 ˆ l i ˆ r Ti s − ˆ λ i F ω [ˆ λ i ] = (cid:20) ˆ G ( s )ˆ G ω ( s ) (cid:21) . Then it can readily be noted that ¯ l Ti H Ω ( − ˆ λ i ) = ˆ l Ti T ω ( − ˆ λ i ) and ¯ l Ti ˆ H Ω ( − ˆ λ i ) =ˆ l Ti ˆ T ω ( − ˆ λ i ). 13v) Let us define ¯ P τ as ¯ P τ = ¯ P τ ˆ R − T . By noting that e ˆ At = ˆ Re ˆΛ t ˆ R − and multi-plying ˆ R − T from the right, the equation (10) becomes A ¯ P τ + ¯ P τ ˆΛ + B ˜ B T − B τ ˜ B T e ˆΛ t = 0 , Since ˆΛ (and resultantly e ˆΛ t ) is a diagonal matrix (Goyal and Redmann, 2019), ¯ P τ can be computed column-wise as¯ P τ,i = ( − ˆ λ i I − A ) − B ˆ r i − ( − ˆ λ i I − A ) − B τ e ˆ λ i t ˆ r i . Further, let us define ˆ P τ as ˆ P τ = ˆ P τ ˆ R − T . By multiplying ˆ R − T from the right, theequation (12) becomes ˆ A ˆ P τ + ˆ P τ ˆΛ + ˆ B ˜ B T − ˆ B τ ˜ B T e ˆΛ t = 0 . Then ˆ P τ can be computed column-wise asˆ P τ,i = ( − ˆ λ i I − ˆ A ) − ˆ B ˆ r i − ( − ˆ λ i I − ˆ A ) − ˆ B τ e ˆ λ i t ˆ r i . Now, the optimality condition (22) can be written as C (cid:2) ¯ P τ, · · · ¯ P τ,r (cid:3) ˆ R T − ˆ C (cid:2) ˆ P τ, · · · ˆ P τ,r (cid:3) ˆ R T = 0 . (37)After multiplying ˆ R − T from the right, each column of (37) becomes G T ( − ˆ λ i )˜ r i − ˆ G T ( − ˆ λ i )˜ r i = 0 . (vi) Note that B τ = Re Λ t R − B and ˆ B τ = ˆ Re ˆΛ t ˆ R − ˆ B . Thus G T ( s ) and ˆ G T ( s ) canbe represented as G T ( s ) = hP ni =1 l i r Ti s − λ i − P ni =1 l i r Ti s − λ i e λ i t i = (cid:2) G ( s ) e st G τ ( s ) (cid:3) , ˆ G T ( s ) = hP ri =1 ˆ l i ˆ r Ti s − ˆ λ i − P ri =1 ˆ l i ˆ r Ti s − ˆ λ i e ˆ λ i t i = (cid:2) ˆ G ( s ) e st ˆ G τ ( s ) (cid:3) . Then it can readily be noted that G T ( − ˆ λ i )˜ r i = T τ ( − ˆ λ i )ˆ r i and ˆ G T ( − ˆ λ i )˜ r i = ˆ T τ ( − ˆ λ i )ˆ r i .(vii) Let us define ¯ Q τ as ¯ Q τ = ¯ Q τ ˆ R . By multiplying ˆ R from the right, the equation(11) becomes A T ¯ Q τ + ¯ Q τ ˆΛ + C T ˜ C − C Tτ ˜ Ce ˆΛ t = 0 . Then ¯ Q τ can be computed column-wise as¯ Q τ,i = ( − ˆ λ i I − A T ) − C T ˆ l i − ( − ˆ λ i I − A T ) − C Tτ e ˆ λ i t ˆ l i . Now define ˆ Q τ as ˆ Q τ = ˆ Q τ ˆ R . By multiplying ˆ R from the right, the equation (13)14ecomes ˆ A T ˆ Q τ + ˆ Q τ ˆΛ + ˆ C T ˜ C − ˆ C Tτ ˜ Ce ˆΛ t = 0Then ˆ Q τ can be computed column-wise asˆ Q τ,i = ( − ˆ λ i I − ˆ A T ) − ˆ C T ˆ l i − ( − ˆ λ i I − ˆ A T ) − ˆ C Tτ e ˆ λ i t ˆ l i . Further, the optimality condition (21) can be written as B T (cid:2) ¯ Q τ, · · · ¯ Q τ,r (cid:3) ˆ R − ˆ B T (cid:2) ˆ Q τ, · · · ˆ Q τ,r (cid:3) ˆ R = 0 . (38)After multiplying ˆ R − from the right, each column of (38) becomes H T T ( − ˆ λ i )˜ l i − ˆ H T T ( − ˆ λ i )˜ l i = 0 . (viii) Note that C τ = CRe Λ t R − and ˆ C τ = ˆ C ˆ Re ˆΛ t ˆ R − . Thus H T ( s ) and ˆ H T ( s ) canbe represented as H T ( s ) = " P ni =1 l i r Ti s − λ i P ni =1 l i r Ti s − λ i e λ i t = (cid:20) G ( s ) e st G τ ( s ) (cid:21) , ˆ H T ( s ) = P ri =1 ˆ l i ˆ r Ti s − ˆ λ i P ri =1 ˆ l i ˆ r Ti s − ˆ λ i e ˆ λ i t = (cid:20) ˆ G ( s ) e st ˆ G τ ( s ) (cid:21) . Then it can readily be noted that ˜ l Ti H T ( − ˆ λ i ) = ˆ l Ti T τ ( − ˆ λ i ) and ˜ l Ti ˆ H T ( − ˆ λ i ) =ˆ l Ti ˆ T τ ( − ˆ λ i ). H ,ω - and H ,τ -suboptimal MOR Owing to the inherent difficulty in making an optimal choice of ˆ A within the projectionframework, we now focus on constructing a ROM ( ˆ A, ˆ B, ˆ C ) wherein ˆ B and C are theoptimal choices.Let the ROM be obtained by using the oblique projection Π = ¯ P ω ˆ P − ω ˆ Q − ω ¯ Q Tω (withˆ V = ¯ P ω ˆ P − ω and ˆ W = ¯ Q ω ˆ Q − ω ). Since ¯ P ω , ˆ P ω , ¯ Q ω , and ˆ Q ω depend on the ROM( ˆ A, ˆ B, ˆ C ), finding such a projection is a nonconvex problem. Note that (1) and theequations (4)-(7) can be seen as the following coupled system of equations( ˆ A, ˆ B, ˆ C ) = f ω ( ¯ P ω , ˆ P ω , ¯ Q ω , ˆ Q ω ) , ( ¯ P ω , ˆ P ω , ¯ Q ω , ˆ Q ω ) = g ω ( ˆ A, ˆ B, ˆ C ) . The stationary points of ( ˆ A, ˆ B, ˆ C ) = f ω (cid:0) g ω ( ˆ A, ˆ B, ˆ C ) (cid:1) satisfy the optimality conditions(15) and (16). The pseudo-code of the stationary point iteration algorithm to computethe stationary points of ( ˆ A, ˆ B, ˆ C ) = f ω (cid:0) g ω ( ˆ A, ˆ B, ˆ C ) (cid:1) is given in Algorithm 1, whichis referred to as “Frequency-limited H -suboptimal MOR (FLHMOR)”. The obliqueprojection condition ˆ W T ˆ V = I is ensured by using the biorthogonal Gram-Schmidtmethod (steps (5)-(9)) (Benner et al., 2011).Similarly, if the ROM is obtained by using the oblique projections Π =¯ P τ ˆ P − τ ˆ Q − τ ¯ Q Tτ (with ˆ V = ¯ P τ ˆ P − τ and ˆ W = ¯ Q τ ˆ Q − τ ), (1) and the equations (10)-(13)15 lgorithm 1 FLHMOR
Input:
Original system: (
A, B, C ); Desired frequency interval: [ − ω, ω ] rad/sec; Initialguess: ( ˆ A, ˆ B, ˆ C ). Output:
ROM ( ˆ A, ˆ B, ˆ C ). while (not converged) do Solve the equations (4) and (5) to compute ¯ P ω and ¯ Q ω , respectively. Solve the equations (6) and (7) to compute ˆ P ω and ˆ Q ω , respectively. Set ˆ V = ¯ P ω ˆ P − ω and ˆ W = ¯ Q ω ˆ Q − ω . for i = 1 , . . . , r do ˆ v = ˆ V (: , i ), ˆ v = Q ik =1 (cid:0) I − ˆ V (: , k ) ˆ W (: , k ) T (cid:1) ˆ v . ˆ w = ˆ W (: , i ), ˆ w = Q ik =1 (cid:0) I − ˆ W (: , k ) ˆ V (: , k ) T (cid:1) ˆ w . ˆ v = ˆ v || ˆ v || , ˆ w = ˆ w || ˆ w || , ˆ v = ˆ v ˆ w T ˆ v , ˆ V (: , i ) = ˆ v , ˆ W (: , i ) = ˆ w . end for ˆ A = ˆ W T A ˆ V , ˆ B = ˆ W T B , ˆ C = C ˆ V . end while can be seen as the following coupled system of equations( ˆ A, ˆ B, ˆ C ) = f τ ( ¯ P τ , ˆ P τ , ¯ Q τ , ˆ Q τ ) , ( ¯ P τ , ˆ P τ , ¯ Q τ , ˆ Q τ ) = g τ ( ˆ A, ˆ B, ˆ C ) . The stationary points of ( ˆ A, ˆ B, ˆ C ) = f τ (cid:0) g τ ( ˆ A, ˆ B, ˆ C ) (cid:1) satisfy the optimality conditions(21) and (22). The pseudo-code of the stationary point iteration algorithm to computethe stationary points of ( ˆ A, ˆ B, ˆ C ) = f τ (cid:0) g τ ( ˆ A, ˆ B, ˆ C ) (cid:1) is given in Algorithm 2, which isreferred to as “Time-limited H -suboptimal MOR (TLHMOR)”. The oblique projec-tion condition ˆ W T ˆ V = I is ensured by using the biorthogonal Gram-Schmidt method. Algorithm 2
TLHMOR
Input:
Original system: (
A, B, C ); Desired time interval: [0 , t ] rad/sec; Initial guess:( ˆ A, ˆ B, ˆ C ). Output:
ROM ( ˆ A, ˆ B, ˆ C ). while (not converged) do Solve the equations (10) and (11) to compute ¯ P τ and ¯ Q τ , respectively. Solve the equations (12) and (13) to compute ˆ P τ and ˆ Q τ , respectively. Set ˆ V = ¯ P τ ˆ P − τ and ˆ W = ¯ Q τ ˆ Q − τ . for i = 1 , . . . , r do ˆ v = ˆ V (: , i ), ˆ v = Q ik =1 (cid:0) I − ˆ V (: , k ) ˆ W (: , k ) T (cid:1) ˆ v . ˆ w = ˆ W (: , i ), ˆ w = Q ik =1 (cid:0) I − ˆ W (: , k ) ˆ V (: , k ) T (cid:1) ˆ w . ˆ v = ˆ v || ˆ v || , ˆ w = ˆ w || ˆ w || , ˆ v = ˆ v ˆ w T ˆ v , ˆ V (: , i ) = ˆ v , ˆ W (: , i ) = ˆ w . end for ˆ A = ˆ W T A ˆ V , ˆ B = ˆ W T B , ˆ C = C ˆ V . end while . Illustrative Example Consider a sixth-order model with the following state-space realization A = − . . . − . − . . − . − . − . . . . . − . − . . . . . . − . − . . − . . . . − . − . . . − . − . − . . − . , B = . − . − . . . − . − . − . ,C = [ − . − . − . − . . − . ] . The initial guess of the ROM used in FLHMOR and TLHMOR is given byˆ A (0) = (cid:20) − . − . − . − . (cid:21) , ˆ B (0) = (cid:20) − . − . . . (cid:21) , ˆ C (0) = (cid:2) − . − . (cid:3) . The desired frequency interval in FLHMOR is set to [ − . , .
5] rad/sec. The reductionsubspaces constructed by FLHMOR are given byˆ V = − . − . . − . − . − . . − . − . − . − . − . , ˆ W = − . − . . − . − . − . . − . . . − . − . . The ROM constructed by these reduction subspaces is given byˆ A = (cid:20) − . − . − . − . (cid:21) , ˆ B = (cid:20) . . − . − . (cid:21) , ˆ C = (cid:2) . . (cid:3) . It can be verified that C ¯ P ω = ˆ C ˆ P ω = (cid:2) − . . (cid:3) , ¯ Q Tω B = ˆ Q ω ˆ B = (cid:20) . . − . − . (cid:21) . The desired time interval in TLHMOR is set to [0 , .
1] sec. The reduction subspacesconstructed by TLHMOR are given byˆ V = − . . − . . − . . . . . − . − . . , ˆ W = . . − . . − . . . . − . − . − . . . The ROM constructed by these reduction subspaces is given byˆ A = (cid:20) − . . . − . (cid:21) , ˆ B = (cid:20) . . . . (cid:21) , ˆ C = (cid:2) . − . (cid:3) .
17t can be verified that C ¯ P τ = ˆ C ˆ P τ = (cid:2) − . − . (cid:3) , ¯ Q Tτ B = ˆ Q τ ˆ B = (cid:20) . − . − . . (cid:21) .
6. Conclusion
The H ,ω - and H ,τ -optimal MOR problems via projection are addressed. It is shownthat two out of three optimality conditions can be satisfied within the projectionframework. Two iterative algorithms are proposed for this purpose. The numericalresults confirm that the proposed algorithms construct near-optimal ROMs. References
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