Scalable Symbolic Model Order Reduction

Transcription

1 Scalable Sybolic Model Order Reduction Yiyu Shi Lei He -J Richard Shi Electrical Engineering Dept, ULA Electrical Engineering Dept, UW Los Angeles, alifornia, 924 Seattle, WA, 985 {yshi, lhe}eeuclaedu {cjshi}uwashingtonedu ASTRAT Sybolic odel order reduction (SMOR) is to reduce the coplexity of a odel with sybolic paraeters It is an iportant proble in analog circuit synthesis and digital circuit odeling with process variations However, existing sybolic odel order reduction (SMOR) ethods do not scale well with the nuber of sybols or with the odel order This paper presents a scalable SMOR algorith, naely S 2 MOR We first separate the original ulti-port ulti-sybol syste into a set of single-port systes by superposition theore, and then integrate the together to for a lower-bordered block diagonal (LD) structured syste Each block is reduced independently, with a stochastic prograing to distribute the given overall odel order between blocks for best accuracy The entire syste is efficiently solved by lowrank update opared with existing SMOR algoriths, given the sae eory space, S 2 MOR iproves accuracy by up to 78% at a siilar reduction tie In addition, the factorization and siulation of the reduced odel by S 2 MOR is up to 7 faster INTRODUTION Sybolic circuit techniques are playing an increasingly iportant role with the advance of design technology, especially when we have entered the nano regie Plenty of algoriths exist in literature discussing how to analyze and siulate those sybolic circuits [ 3] However, all those ethods are practical only if the circuit has a oderate size Unfortunately, to guarantee reasonable odel accuracy, the circuits fro physical extraction usually contain illions of nodes, thus rendering those ethods inefficient Towards this end, nuerous odel order reduction (MOR) techniques have been successfully applied to the reduction of linear large scale circuits over the past decade ( [4, 5],etc) There were also efforts to extend those ethods to nonlinear circuits ( [6, 7], etc) and paraetrized circuits ( [8, 9], etc) However, despite their wide application, unsolved probles do exist when directly extending the to sybolic circuits The idea of sybolic odel order reduction (SMOR) was first introduced in [], which contains three different ethods: sybolic isolation, noinal projection and first order expansion The sybol isolation ethod first reoves all the sybols fro the circuits, and the nodes to which the sybols are connected are odeled as ports As such, the sybolic circuit becoes a sybol-free circuit with assive ports, and can be reduced by any traditional ethods [4,5, 3] However, the size of the reduced circuit is proportional to the port nuber ultiplying the nuber of oents to be atched Accordingly, the tie and space coplexity for the reduced odel increases cubically This work is supported in part by NSF AREER Award R-36682, Defense Advanced Projects Agency under contract W 3P 4θ 5 O R64 and in part by Defense Threat Reduction Agency under contract #F A This work was perfored while Yiyu Shi was an intern at Orora Design Technologies, Inc, Redond, WA with the nuber of ports and thus the nuber of sybols, which renders those ethods of less practical value for circuits with any sybols The noinal projection ethod uses the noinal values of the sybols to copute the projection atrix and is accurate only when the sybol values slightly deviate fro the noinal value The first order expansion ethod uses the first order expansion of the atrix inversion and ultiplication to find the projection atrix, which is first order atrix polynoial wrt all the sybols Again, no large change is allowed for the sybols in order for the ethod to be accurate In addition, all the above three sybolic MOR ethods suffer fro the following three probles: First, they don t scale well with the nuber of sybols or with the reduced order In literature only the experiental results for circuits with less than ten sybols are reported Second, if the original circuit has no special pattern to be explored, the reduced odel is dense and therefore is still tie consuing for siulation Third, when the values of the sybols change, it is difficult to directly update the factorization of the reduced odel instead of redo it fro scratch, especially for the noinal projection and first order expansion ethods A ore detailed analysis of those sybolic MOR ethods are provided in Section 2 The ain contribution of our paper is as follows: A scalable SMOR algorith, naely S 2 MOR is presented We first separate the original ulti-port ulti-sybol syste into a set of single-port systes by superposition theore, then integrate the together to for a lowerbordered block diagonal (LD) structure Each block is reduced independently, with a stochastic prograing to distribute the given overall odel order between blocks for best accuracy And the entire syste is efficiently solved by low-rank update Experients using analog designs show that S 2 MOR handles circuits with up to nodes and 66 sybols in 5 inutes opared with existing SMOR algoriths, S 2 MOR iproves accuracy by up to 78% at a siilar reduction tie and a sae eory space In addition, the factorization and siulation of the reduced odel by S 2 MOR is up to 7 faster The reaining of the paper is organized as follows: Section 2 reviews the existing approach for handling sybolic circuits and its disadvantages Section 3 presents our LD transforation and syste projection algorith We also discuss how to fully utilize the special sparsity pattern of the reduced odel for the efficient siulation and update of the reduced odel Experiental results are presented in Section 4 and concluding rearks are given in Section 5 2 PRELIMINARIES In this section, we briefly review the three SMOR ethods proposed in [], and address their disadvantages by coplexity analysis wrt the reduced order q For later coparison with our ethod, the coplexity for the three different ethods are suarized in the first three rows of Table 2 Sybol Isolation The ain idea of sybol isolation is to reove all the sybols fro

2 the circuit, and odel the nodes to which the sybols are connected as ports Then any traditional MOR ethods can be applied Generally speaking, a circuit with a total nuber of a sybols and p ports is converted to an equivalent sybol-free circuit with p + a ports, assuing all the sybols are two-terinal A general reduction ethod such as PRIMA [4] states that in order to atch the first q oents, the reduced odel will have a size of q() with a cost of O(() 2 q 2 ) for the reduction (the orthonoralization of q(p + a) vectors plus a constant atrix factorization cost) With no special structure, the tie coplexity to factorize the reduced odel is O(q 3 (p + a) 3 ) which increases cubically with a, and the space coplexity to store the reduced odel is O(q 2 (p + a) 2 ) Moreover, to update the reduced odel with a new set of sybol values, the cost is a() 2 q 2 by using the ethod of low rank update Therefore, such a ethod is useful when the circuits contain only a few sybols 22 Noinal Projection The ain idea of noinal projection ethod is to use the noinal values of the sybols to copute the projection atrix The odified nodal analysis (MNA) equation sybolic circuit can be written as (G + G) + s( + ) = u, () where G and are the noinal atrices, and G, contain the staping of the sybols Then the noinal projection ethod coputes the projection atrix V such that V κ q(a, R ), (2) where A = G and R + G, and κ q(a, R ) is the q th order Krylov subspace spanned by A and R This ethod is very efficient as the projection atrix does not depend on the nuber of sybols In order to approxiately atch q oents, the reduced circuit will have a size of pq with the cost of p 2 q 2 for the reduction (the orthonoralization of pq vectors) Accordingly, the tie coplexity for analyzing the reduced odel is O(q 3 p 3 ), and the space coplexity for the reduced odel including the coefficient atrices for all the sybols is O(ap 2 q 2 ) Moreover, to update the reduced odel with a new set of sybol values costs O(p 3 q 3 ) because it requires new factorization of the reduced odel As the values of the sybols deviate fro the noinal values, the projection atrix no longer contains the Krylov subspace spanned by the circuit atrix Accordingly, the odel accuracy decreases [] showed that the error of the reduced odel increases draatically even if the value of the sybols deviates only a few percent fro the noinal value 23 First Order Approxiation Siilar proble exists for the first order approxiation ethod The ethod coputes the projection atrix using first order expansion, ie, when coputing the projection atrix, the following approxiation is used [(G + G) ( + )] k (G + G) k A k R {A k G G A i G k i GA + k A j G A k j }R (3) j= After first order expansion, the Krylov subspace is directly used without orthonoralization to avoid the expensive cost of sybolic coputation which ay render the reduced odel singular Such a ethod can only work when the values of the sybols do not deviate uch fro the noinal values Siilar to the noinal projection ethod, the reduced circuit will have a size of pq if we approxiately atch q oents Note that the reduction cost is O(pq) because no orthonoralization is eployed The reduce odel has the sae space coplexity and tie coplexity as the noinal projection ethod 3 S 2 MOR ALGORITHM 3 Port Separation and Model Reduction The key idea of the S 2 MOR algorith is to separate the original ultiport ulti-sybol syste into a set of single-port systes by superposition theore, and then integrate the together to for a lower-bordered block diagonal (LD) structure We further show that by using the projection atrices fro each of the single-port systes, we can reduced such LD syste with the structure preserved The whole procedure is detailed below We start with the following odified nodal analysis (MNA) equation, which represents an RL circuit with a sybols and p ports (G + s)x + a P is i (P T i x) = u (4) y = L T x, (5) where G and ( R N N ) are the inductance and capacitance atrices, and L ( R N p ) are the incidence atrices for input and output, u ( R p ) is the input current vector The vector P i ( R N ) is the incidence vector for sybol i and it takes the for P i = ` T with at the j th eleent (positive node) at at the k th eleent (negative node) s i is the operator corresponding to the i v relation of the i th sybol Typically, for an resistive sybol, s i = /R is a constant operator For capacitive sybol, we have s i = s and for inductance, s i = /sl Accordingly, the current fro sybol i can be coputed as (6) w i = s i (P T i x), i a (7) Denote i as the i th colun of and u i as the i th eleent of u Then according to superposition theore, Eq (4) becoes p a (G + s)x = iu i P iw i, (8) Eq (8) can be further divided into a set of p + a equations, with each equation in the for of j (G + s)x (i) = iu i, i p P i pw i p, p + i p + a, (9) x = x (i), () i= where x (i) is the state variable for the i th equation, which corresponds to a syste with single port, and w i is the current through sybol i On the other hand, fro Eq () and Eq (7), we have w i = s i ( Pi T x (j) ) = j= j= s i (Pi T x (j) ), () For siplicity of presentation, we assue that the sybol is a two-terinal device However, it is understood that the algorith is readily to be applied to ulti-terinal devices as well

3 Table : Space and tie coplexity coparison between four different ethods wrt the reduced order q Method Mo Matched Reduced Size Space oplexity Tie oplexity Reduction Factorization Update Sybol Isolation q (p + a)q (p + a) 2 q 2 (p + a) 2 q 2 (p + a) 3 q 3 a(p + a) 2 q 2 Noinal Projection Approx q pq ap 2 q 2 p 2 q 2 p 3 q 3 p 3 q 3 st Order Expansion Approx, q pq ap 2 q 2 apq p 3 q 3 p 3 q 3 S 2 MOR q (p + a)q (p + a)q 2 (p + a)q 2 (p + a)q 3 const where we have used the fact that s i is a scalar for RL eleents We introduce and Eq (7) becoes z = ` x ()T x (2)T ()T T x w i = ` s i P T i s i P T i s i P T i (2) z, i a (3) Therefore, Eq (9)-Eq () can be cast into a copact atrix for as (Ĝ + sĉ)z = ˆu, (4) where Ĝ is a lower bordered block diagonal (LD) atrix and Ĉ is a block diagonal atrix as shown in Eq (5) at the top of next page In addition, we have ˆ = p (6) A With the augented syste, the output can be coputed as where y = ˆLz, (7) ˆL = ` L L L T (8) One of the advantages of the augented syste is that a sybol-free projection atrix can be found, which can preserve the special sparse structure of the Ĝ and Ĉ atrices as well This is detailed in the following theore Theore If orthonoralized atrices V i satisfies j κ V i q{g,, i} i p κ q{g,, P i p} p + i p + a, (9) where κ q{g,, i} is the q th order Krylov subspace spanned by G, and i and κ q{g,, P i p} is the q th order Krylov subspace spanned by G, and P i p Then with the projection atrix V V 2 V = A, (2) V the first q oents of the reduced syste (Ĝr + sĉr)zr = ˆ ru (2) y r = ˆL rz r (22) and the original syste are atched, where Ĝr = VT ĜV, Ĉ r = V T ĈV, ˆr = V T r and ˆL r = V T ˆL Due to the space liit, the proof is oitted here Although the projection atrix in Eq (9) can guarantee the atching of the first q oents, it does not always provide the best accuracy depending on the sybol values Later we will show how to find an optial projection atrix in ters of accuracy, under the given reduced size, by changing the order of the Krylov subspaces spanning each V i in Eq (2) To atch q oents, the reduced atrices Ĝr and Ĉr would have a diension of (p + a)q, which is the sae as the that fro PRIMA However, different fro the dense reduced atrices fro PRIMA, it is easy to see that the Ĝr of the reduced syste still keeps the LD structure, and the Ĉr still keeps the block diagonal structure as shown in Eq (5) Note that for efficiency, the atrix G r should never be fored explicitly Instead, it can be cast as a block diagonal atrix plus with a rank q updates, ie, Ĝ r = D + LH T (23) where D ( R ()q ()q ) is a block diagonal atrix G r, D = A, (24) and L, H ( R ()q a ) can be written as G r, P p+ r, L = P p+2, (25) A Pr,a s P r, T s 2 P T sa P T r,a s P r, 2T s 2 P 2T sa P 2T r,a H =, (26) s P ()T s r, 2 P ()T sa P r,a ()T A where only D and L, H are stored and explicitly fored This sparsity structure facilitates the efficient siulation as well as update of the reduced odel, as will be discuss shortly The tie coplexity of the proposed algorith can be decoposed into two portions: the tie to copute p + a Krylov subspaces, each with order q; Note that the factorization of of G can be shared between different subspaces, and we need to do p + a ties of vector orthonoralization, each with q vectors Accordingly, the tie coplexity is O((p + a)q 2 ) Assuing no special structure for G and atrices in (4), the space needed to store the reduced atrices Ĉ r is O(pq 2 ) since it is block diagonal To store Ĝr, we need to store D, L and H, which cost O((p + a)q 3 ), O(aq) and O(a(p + a)q) respectively Keeping the doinant ters in q, and we can see that the total space coplexity is O((p + a)q 2 ) 32 Siulation and Update of the Reduced Model To siulate the reduced odel, either in tie doain for transient analysis or in frequency doain for A analysis, we would face the proble of efficiently factorizing a LD atrix We will concentrate on the QR factorization as it is shown to be the ost stable factorization algorith [4] We will show that instead of the general factorization cost O(() 3 q 3 ) for non-structuralized atrices with diension ()q, our special LD atrix can be factorized at a uch lower cost For

4 Ĝ = G G P s P T G P s P T P s P T P as a P T a G P as a P T a A Ĉ = A (5) the siplicity of presentation, we will siply discuss the factorization of Ĝ r + sĉr atrix for frequency doain analysis Siilar algorith can be applied for the tie doain analysis as the atrix structure will reain the sae Suppose we want to solve (Ĝr + sĉr)x = ˆ ru (27) for any input u and frequency s To start with, fro the atrix inversion lea we know that (Ĝr + sĉr) = (D + s Ĉ r + LH T ) = (D + s Ĉ r) (D + s Ĉ r) L I + H T (D + s Ĉ r) L H T (D + s Ĉ r) (28) Accordingly, an efficient factorization ethod can be obtained by first factorizing E = D+s Ĉ r, which is block diagonal, and then factorizing M = I + H T (D + s Ĉ r)l (M R a a ) Then for any input u, we can first solve and then solve Finally, solve and the solution can be obtained as Ex = ru, (29) Mx = x (3) Ex = Lx (3) x = x x (32) The ain cost for such an algorith lies in the factorization of E and M, which is O((p + a)q 3 ) and O(a 3 ) Keeping the doinant ter in q, we get that the overall tie coplexity for factorization is O((p + a)q 3 ) Moreover, each tie the values of the sybols are changed, we only need to re-factorize M, the cost of which is O(a 3 ) and does not depend on q The coparison of coplexity between the sybol isolation ethod, the noinal projection ethod, the first order expansion ethod as well as the S 2 MOR ethod is presented in Table The reduced circuits fro the four ethods all atch or approxiately atch the first q oents of the original circuit The best of the four ethod is shown in bold Fro the table we can see that the S 2 MOR ethod outperfors the other three ethods in alost all the coplexity coparisons For the sae nuber of oents to be atched, the S 2 MOR algorith has the lowest space and tie coplexity 33 Min-ax Prograing based Projection Order Decision Theore provides a ethod of coputing the projection atrix which can atch the first q oents of the original circuit with a reduced size of d = (p + a)q It would also be interesting to note the following proble: If we are given the overall reduced size d, how to distribute it between each V i in (2), such that the reduced odel has the best accuracy We will show that unifor distribution between the blocks as in Theore does not necessarily provide the best accuracy Specifically, the freedo of choosing the projection atrix lies in the order of the Krylov subspace for each V i in (9) If we choose a Krylov subspace with order q i, then the error introduced due to the isatching of the (q i + ) th oent can be coputed fro Eq (4) If i p, then i = = L T (G ) q i+ G i L T A q i + R i, (33) where A = G, R i = G i and L is the th colun of L If p + i p + a, then i = L T ((G + P i ps i p Pi p) T ) q i+ (G + P i ps i p P T i p) P i p (34) We can expand (34) to the first order by letting G = P is i Pi T k = q i + in (35), ie, i = L T ((G + P i ps i p P T i p) ) q i+ (G +P i ps i p Pi p) T P i p L T q i + q `Ai Ri p (A i + i F (s i ) and q i A j q i F (s i )A i + j i ) R i p, (35)! j= where F (s i ) = G P i ps i p Pi p T and R i p = G P i p Accordingly, the total error is f = p L T q i A + R + i i=p+ L T q i + `Ai Ri p q i q (A i + i F (s i ) A j q i F (s i )A i + j i ) R i p, (36)! j= which is a function of q i ( i p + a) and s i ( i a) Since we know the reduced size is d, we have the constraint q i = d (37) and they ust be non-negative integer q i Z + {}, i p + a (38)

5 Moreover, assue we have the statistical description for the sybols, ie, s i ω i, (39) where ω i is soe statistical description for s i Then we need to solve the following statistical optiization proble in q,q f(q, q ; s, s a ) (4) st q i = d (4) q i Z + {}, i p + a, (42) s i ω i, i a (43) Note that the objective function in Eq (43) is statistical, and we try to iniize its worst case value for all possible values of the sybols, which gives the following constrained in-ax ixed integer optiization proble in q,q st ax f(q, q; s, sa ) (44) s,s a q i = d (45) q i Z + {}, i p + a, (46) s i ω i, i a (47) Generally the in-ax proble is very hard to solve, especially the above proble is ixed-integer based and is nonlinear elow we propose an efficient heuristic algorith to approxiate solve it, based on the key observation that the constraints for s i and q i can be separated (ie, there are no cross ters) Accordingly, we separate the in-ax probles into the following two sub-probles Siilar algorith has been used in [5] for decoupling capacitance budgeting proble (P :) in q,q f(q, q ; s, s a ) (48) st q i = d (49) q i Z + {}, i p + a, (5) which is a integer-iniization proble for fixed s,, s a, and (P2 :) ax s,,s a f(q,, q ; s, s a ) (5) st s i ω i, (52) which is a axiization proble for fixed q,, q Efficient algoriths exist to solve the above two sub-probles [6] and is beyond the scope of this paper We iteratively solve (P) and (P2) until the solution converges In certain cases, the solution ay fail to converge, and we force the algorith to exit after certain iterations to guarantee the convergence This heuristic algorith can provide optial solution only in the case that (P) is convex and (P2) is concave Otherwise, the optiality is not guaranteed However, experiental results show that significant accuracy iproveent is achieved by such heuristic 4 EPERIMENTAL RESULTS In this section, we present experiental results for runtie and accuracy coparison between the S 2 MOR ethod and the sybol isolation, the noinal projection and the first order expansion ethods All the ethods are ipleented in ++ We use the Sparse as the sparse atrix package [7] We run experients on a UNI workstation with Pentiu IV 266G PU and G RAM Finally, we use MOSEK as the linear/quadratic prograing solver [8] for optial projection order decision 4 Algorith Verification Fig illustrates the sparsity of the reduced atrices fro the S 2 MOR ethod The original circuit is a low-noise aplifier (LNA) design with parasitics, which contains 492 nodes, 8 ports sybols The circuit is reduced to order 76 by the S 2 NMOR ethod Fro the figure we can see that although there is no special structure for the original atrices, the reduced Ĝr and Ĉr fro the S2 MOR ethod has LD structure with sparsity 34% and block diagonal structure with sparsity 77% respecitvely nz = 966 (a) Gr nz = 486 (b) r Figure : Sparsity pattern for the atrices Ĝr and Ĉr after reduction fro S 2 MOR Next we verify the effectiveness of our optial projection order decision We perfor Monte arlo siulation on the sybol values for, runs on the sae LNA circuit, and copare the error between unifor projection order decision, assuing the sae projection order for all sybols, and our stochastic prograing based projection order decision The error is defined as the integral of the absolute difference between the original tie doain wavefor and the one of the reduced circuit The error distributions for both ethods are shown in Fig 2 Fro the figure we can see that by our ethod the ean error is reduced by approxiately 3% and the 3σ error by 5%, which verifies the effectiveness of the proposed stochastic prograing based approach ount Stochastic Prograing ased Unifor Error (V*ns) Figure 2: Accuracy oparison between unifor projection order and our stochastic prograing based approach based on the Monte arlo siulation 42 Accuracy oparison In this section, we copare the accuracy of the three existing ethods and the S 2 MOR ethod on soe real industrial designs For fair coparison, we reduce the circuit to different orders for different ethods such that the reduced odels require the sae eory space This

6 Table 2: Testbench inforation ckt nae node # port # sybol # LNA LNA LNA LNA Table 3: Accuracy coparison between sybol isolation (SI), noinal projection (NP), first order expansion (FE) and the S 2 MOR with different variation aount (var) of sybol values All the errors are in the unit of V*ns ckt nae var SI NP FE S 2 MOR LNA % (-24%) 3% (-5%) LNA2 % (-27%) 3% (-78%) LNA3 % (-76%) 3% (-76%) LNA4 % NA 6 (-69%) 3% NA 6 (-69%) iplies that the three existing ethods have the sae reduced order because they are all dense, while the S 2 MOR ethod can have a uch higher reduced order due to its sparsity for the sae eory space We opare the accuracy between the four different ethods on different bencharks, and the results are shown in Table 3 The inforation of the testbenches we used is shown in Table 2 They are all LNA s fro extraction Fro Table 3 we can see that for different circuits and for different variation aount, the S 2 MOR ethod always have the sallest error copared with the other three ethods Specifically, when the variation aount is %, copared with the best one of the other three ethods shown in bold, the S 2 MOR ethod reduces the error by up to 76%; When the variation aount is 3%, the S 2 MOR ethod reduces the error by up to 78% Also note that the first order expansion ethod cannot finish the testbench LNA4 due to its singularity This is a direct result fro the lack of orthonoralization procedure for the projection atrix 43 Runtie oparison In Table 4, we report the runtie for reduction, factorization, and update for the reduced odel on different testbenches All the circuits are reduced to a different size such that the reduced circuits fro different ethods requires the sae eory space Fro the table we can see that the reduction tie is alost the sae for all the ethods The factorization tie is reduced by up to 94 for the S 2 MOR ethod on the largest testbench, although the reduced circuit size of the S 2 MOR ethod is alost 3 larger This speedup coes fro the efficient factorization algorith based on the LD structure of the reduced circuit Finally, the factorization update tie for sybol value changes is reduced by 7 for our ethod copared with the sybol isolation ethod, which uses low rank update This is because we only need to refactorize a atrix with size equal to the nuber of sybols All the above results further verify the correctness of our conclusion in Table 5 ONLUSIONS Sybolic odel order reduction (SMOR) is to reduce the coplexity of a odel with sybolic paraeters It is an iportant proble in analog circuit synthesis and digital circuit odeling with process variations However, existing sybolic odel order reduction (SMOR) ethods do not scale well with the nuber of sybols or with the odel order The literature only reports experiental results for circuits with fewer than ten sybols This paper presents a scalable SMOR algorith, naely S 2 MOR, which handles circuits with up to nodes and 66 sybols in 5 inutes fro experiental results opared with Table 4: Runtie coparison between sybol isolation (SI), noinal projection (NP), first order expansion (FE) and the S 2 MOR ethod The reduced sizes are also reported (size) All units are in seconds ckt nae ethod size reduce factor update LNA SI NP FE S 2 MOR LNA2 SI NP FE S 2 MOR LNA3 SI NP FE S 2 MOR LNA4 SI NP FE S 2 MOR existing SMOR algoriths, given the sae eory space, S 2 MOR iproves accuracy by up to 78% at a siilar reduction tie In addition, the factorization and siulation of the reduced odel by S 2 MOR is up to 7 faster 6 REFERENES [] T Halfann, R Soer, and T Sichann, Application of sybolic circuit analysis: An overview and recent resutls in nonlinear odel generation, in International onference on Electronics, ircuits and Systes, pp , 999 [2] -D Tan and -J Shi, Sybolic circuit-noise analysis and odeling with deterinant decision diagras, in ASPDA, pp , 2 [3] G G Gielen and W M Sansen, Sybolic analysis for autoated design of analog integrated circuits: a tutorial overview Kluwer Acadeic Publisher, 99 [4] A Odabasioglu, M elik, and L Pileggi, PRIMA: Passive reduced-order interconnect acro-odeling algorith, IEEE Trans on AD, pp , 998 [5] Y Su and et al, SAPOR: Second-order arnoldi ethod for passive order reduction of RS circuits, in IEEE/AM IAD, pp 74 79, 24 [6] M Rewienski and J White, A trajectory piecewise-linear approach to odel order reduction and fast siulation of nonlinear circuits and icroachined devices, IEEE Trans on AD, pp 55 7, 23 [7] N ond and L Daniel, A piecewise-linear oent-atching approach to paraeterized odel-order reduction for highly nonlinear systes, IEEE Trans on AD, pp , 27 [8] Y Li, Z ai, Y Su, and Zeng, Paraeterized odel order reduction via a two-directional arnoldi process, in IEEE/AM IAD, pp , 27 [9] Y Shi and L He, Epire: An efficient and copact ultiple-paraeterized odel order reduction ethod for physical optiizaion, in ISPD, pp 5 58, 27 [] G Shi, Hu, and J R Shi, On sybolic odel order reduction, IEEE Trans on AD, pp , 26 [] Y Shi, H Yu, and L He, Sason: A generalized second-order arnoldi ethod for reducing ultiple source linear network with susceptance, in ISPD, pp 25 32, 26 [2] P Li and W-P Shi, Model order reduction of linear networks with assive ports via frequency-dependent port packing, in IEEEAM/ DA, pp , 26 [3] Yan, -D Tan, P Liu, and McGaughy, Sbpor:second-order balanced truncation for passive order reduction of rlc circuits, in IEEEAM/ DA, pp 58 6, 27 [4] G H Golub and F V Loan, Matrix oputations altiore, MD: The Johns Hopkins University Press, 3 ed, 989 [5] Y Shi, J iong, Liu, and L He, Efficient decoupling capacitance budgeting considering operation and process variations, IEEE Trans on AD, pp , 28 [6] S oyd and L Vandenberghe, onvex Optiization abridge University Press, 24 [7] T Davis, Direct Methods for Sparse Linear Systes SIAM, 26 [8] in