Model Reduction and Simulation of Nonlinear Circuits via Tensor Decomposition

Transcription

1 Moel Reuction an Simulation of Nonlinear Circuits via Tensor Decomposition The MIT Faculty has mae this article openly available. Please share how this access benefits you. Your story matters. Citation As Publishe Publisher Haotian Liu, Luca Daniel, an Ngai Wong. Moel Reuction an Simulation of Nonlinear Circuits via Tensor Decomposition. IEEE Trans. Comput.-Aie Des. Integr. Circuits Syst. 4, no. 7 July 05): Institute of Electrical an Electronics Engineers IEEE) Version Author's final manuscript Accesse We Aug 5 :49:50 EDT 08 Citable Link Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detaile Terms

2 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. X, NO. X, JANUARY 0XX Moel Reuction an Simulation of Nonlinear Circuits via Tensor Decomposition Haotian Liu, Stuent Member, IEEE, Luca Daniel, Member, IEEE, an Ngai Wong, Member, IEEE Abstract Moel orer reuction of nonlinear circuits especially highly nonlinear circuits), has always been a theoretically an numerically challenging task. In this paper we utilize tensors namely, a higher orer generalization of matrices) to evelop a tensor-base nonlinear moel orer reuction TNMOR) algorithm for the efficient simulation of nonlinear circuits. Unlike existing nonlinear moel orer reuction methos, in TNMOR high-orer nonlinearities are capture using tensors, followe by ecomposition an reuction to a compact tensor-base reuceorer moel. Therefore, TNMOR completely avois the ense reuce-orer system matrices, which in turn allows faster simulation an a smaller memory requirement if relatively lowrank approximations of these tensors exist. Numerical experiments on transient an perioic steay-state analyses confirm the superior accuracy an efficiency of TNMOR, particularly in highly nonlinear scenarios. Keywors Tensor, nonlinear moel orer reuction, reuceorer moel I. INTRODUCTION THE complexity an reliability of moern VLSI chips rely heavily on the effective simulation an verification of circuits uring the esign phase. In particular, mixe-signal an raio-frequency RF) moules are critical an often har to analyze ue to their intrinsic nonlinearities an their large problem sizes. Consequently, nonlinear moel orer reuction becomes necessary in the electronic esign automation flow. The goal of nonlinear moel orer reuction is to fin a reuce-orer moel that simulates fast an yet still captures the input-output behavior of the original system accurately. Compare to the mature moel orer reuction methos in linear time-invariant systems 4, nonlinear moel orer reuction is much more challenging. Several projection-base methos have been evelope in the last ecae. In 5, 6, the nonlinear system is expane into a series of cascae linear subsystems, whereby the outputs from the low-orer subsystems serve as inputs to the higher-orer ones. Then, existing projection-base linear moel orer reuction methos, e.g.,,, can be applie to these linear subsystems recursively. We refer to the metho in 5, 6 as the stanar projection approach. Nonetheless, in this metho, the imension of the resulting reuce-orer moel grows H. Liu an N. Wong are with the Department of Electrical an Electronic Engineering, The University of Hong Kong, Hong Kong SAR {htliu, nwong}@eee.hku.hk). L. Daniel is with the Department of Electrical Engineering an Computer Science, Massachusetts Institute of Technology, Cambrige, MA 09, USA luca@mit.eu). exponentially with respect to the orers of the subsystems. Moreover, to reuce the size of Arnoli starting vectors, lower orer projection subspaces are use to approximate the column spaces of the higher orer system matrices. Consequently, approximation errors in lower orer subspaces can easily propagate an accumulate in higher orer subsystems. To tackle this accuracy issue, a more compact nonlinear moel orer reuction scheme, calle NORM, is propose in 7 where each explicit moment of high-orer Volterra transfer functions is matche. For weakly nonlinear circuits, NORM exhibits an extraorinary improvement in accuracy over the stanar projection approach since lower orer approximations are completely skippe. The resulting reuceorer moel tens to be more compact as the sizes of lower orer reuce subsystems will not carry forwar to higher orer ones. However, this approach still nees to buil the reuce but ense system matrices whose imensions grow exponentially as the orer increases. This limits the practicality of NORM as simulation of small but ense problems is sometimes even slower than simulating the large but sparse original system. To overcome the curse of imensionality, rather than treating the exponentially growing system matrices as -imensional matrices, their nature shoul be recognize. To this en, tensors, as high imensional generalization of matrices, can be utilize. In recent years, there has been a strong tren towar the investigation of tensors an their low-rank approximation 8, ue to their high imensional nature ieal for complex problem characterization an their efficient compact representation ieal for large scale ata analyses. Therefore, it is natural to characterize circuit nonlinearities by tensors whereby the tensor structure can be exploite to reuce the original nonlinear system. In this paper, we propose a tensor-base nonlinear moel orer reuction TNMOR) scheme for typical circuits with a few nonlinear components. The work is a variation of the Volterra series-base projection methos 5 7. The nonlinear system is moele by a truncate Volterra series up to a certain high orer. However, in the propose metho, the higher orer system matrices are moele by high-orer tensors, so that the high imensional ata can be approximate by the sum of only a few outer proucts of vectors via the canonical tensor ecomposition 8,, 4. Next, the projection spaces are generate by matching the moments of each subsystem, in terms of those ecompose vectors. Finally, the reuce-orer moel is represente in the canonical tensor ecomposition form, where the sparsity of the high imensional system Copyright c 05 IEEE. Personal use of this material is permitte. However, permission to use this material for any other purposes must be obtaine from the IEEE by sening an to pubs-permissions@ieee.org.

3 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. X, NO. X, JANUARY 0XX matrices is preserve after TNMOR. The main contribution of this work is that unlike previous approaches, simulation of the TNMOR-prouce reuceorer moel completely avois the overhea of solving highorer ense system matrices. This truly allows the simulation to exploit the acceleration brought about by nonlinear moel orer reuction. We remark that the utilization of TNMOR epens on the existence of low-rank approximations of these high-orer tensors, which are generally available for circuits with a few nonlinear components. Moreover, the size of the reuce-orer moel epens only on the tensor rank an the orer of moments being matche for each system matrix. In other wors, it will not grow exponentially as the orer of subsystems increases, which enables nonlinear moel orer reuction of highly nonlinear circuits not amenable before. The paper is organize as follows. Section II reviews the backgrouns of Volterra series, existing nonlinear moel orer reuction approaches, as well as tensors an tensor ecomposition. After that, Section III presents the tensor-base moeling of nonlinear systems. The propose TNMOR is escribe in Section IV an simulation of the TNMOR-reuce moel is iscusse in Section V. Numerical examples are given in Section VI. Finally, Section VII raws the conclusion. II. A. Volterra subsystems BACKGROUND AND RELATED WORK We consier a nonlinear multi-input multi-output MIMO) time-invariant circuit moele by the ifferential-algebraic equation DAE) t q xt)) + f xt)) = But), yt) = LT xt), ) where x R n an u R l are the state an input vectors, respectively; q ) an f ) are the nonlinear capacitance an conuctance functions extracte from the moifie noal analysis MNA); B an L are the input an output matrices, respectively. The nonlinear system can be expane uner a perturbation aroun its equilibrium point x 0 by the Taylor expansion t C x + C x x) + C x x x) + + G x +G x x) + G x x x) + = Bu, ) where enotes the Kronecker prouct an we will use the shorthan x = x x x etc. for the Kronecker powers throughout the paper. The conuctance an capacitance matrices are given by G i = i f i! x i R n ni, C i = i q x=x0 i! x i R n ni. x=x0 ) By Volterra theory an variational analysis 5, 6, the solution x to ) is approximate with the Volterra series xt) = x t)+x t)+x t)+, where x i t) is the response to each of the following Volterra subsystems t C x + G x = Bu, 4a) t C x + G x = C x G x, 4b) t t C x + G x = C x + C x i x i ) G x t G x i x i ), 4c) t C x 4 + G x 4 = C 4 x 4 + C x i x i x i ) 4 t +C x i x i ) 4 G4 x 4 G x i x i x i ) 4 G x i x i ) 4, 4) an so on, where x i x i ) = x x + x x, x i x i x i ) 4 = x x x +x x x +x x x an more generally x i x in ) k = i + +i n =k x i x in, i,..., i n Z +, where Z + enotes the set of positive integers. B. Existing projection-base nonlinear moel orer reuction methos To reuce the original system ), the stanar projection approach 5, 6 treats 4) as a series of MIMO linear systems, with the right han sie of each equation serving as its actual input. Then, the projection-base linear moel orer reuction approach, e.g.,, is applie. Suppose up to k th-orer viz. from 0th to k th) moments of x in 4a) are matche by a Krylov subspace projector x V k x, the number of columns of V k is k + )l. After that, 4b) is recast into a concatenate, stacke escriptor system an V k is use to approximate its input by assuming x V k x ) = V k x, C I ẋ G 0 x ẋ e 0 I x e G V k C V k G = x C x. 5) Consequently, the multiple Krylov starting vectors of 5) G V become k G instea of, therefore the C V k C imensionality of the input is reuce to k + ) l from n. If up to k th orer moments of x are preserve, 4b) can be reuce by another projection x V k x to a smaller linear system with k +)k +) l inputs. Similarly, higher orer projectors V k, V k4, etc. are obtaine by iteratively reucing the remaining subsystems in 4). Suppose we have N linear subsystems in 4) an k = k = k = = k N orer of moments are matche in each subsystem, the stanar projection approach will result in a reuce-orer moel with size Ok N l N ). Thus, in practical circuit reuction examples, k, k, etc. shoul be relatively small otherwise the size of the reuce system may even excee n quickly), which coul hamper the accuracy of the reuce-orer moel. Instea of regaring 4) as a set of linear equations, NOR- M 7 erives frequency-omain high-orer nonlinear Volterra

4 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. X, NO. X, JANUARY 0XX i iii 6 9 i i a) U U i i b) i U n n n n A ) n n Fig.. -moe matricization of a r-orer tensor. n n n Fig.. a) A tensor A R 4. b) Illustration of A U U U. transfer functions H s, s ), H s, s, s ), etc. associate to the subsystems in 4). These transfer functions are expane into multivariate polynomials of s, s,... such that the coefficients moments) can be explicitly matche. In NORM, the size of an Nth-orer reuce-orer moel is in Ok N+ l N ) if k = k = = k N. In both aforementione methos, the final step consists of replacing the original nonlinear system ) by a smaller system via the transformations x = V T x, B = V T B, L = V T L, G i = V T G i V i ), Ci = V T C i V i ), where i =,..., N an V = orthv k, V k, V k,... is the orthogonal projector. Suppose q is the size of the reuce state, G i an C i will be ense matrices with Oq i+ ) entries, espite the sparsity of G i an C i. To store these ense matrices, the memory space require grows exponentially. C. Tensors an tensor ecomposition Some tensor basics are reviewe here, while more tensor properties an ecompositions can be foun in 9. ensors: A th-orer tensor is a -way array efine by 6) A R n n n. 7) For example, Fig. a) epicts a r-orer 4 tensor. In particular, scalars, vectors an matrices can be regare as 0th-orer, st-orer an n-orer tensors, respectively. Matricization is a process that unfols or flattens a tensor into a n-orer matrix. The k-moe matricization is aligning each kth-irection vector fiber to be the columns of the matrix. For example a r-orer n n n tensor A can be -moe matricize into an n n n matrix A ) as illustrate in Fig.. ensor-matrix prouct: The k-moe prouct of a tensor A R n n k n with a matrix U R p k n k results in a new tensor A k U R n n k p k n k+ n given by A k U) j j k m k j k+ j = n k j k = A j j k j U mk j k. 8) The complexity refers to the single-point expansion algorithm of NORM. The multi-point version of NORM woul have a lower complexity. We enote tensors by calligraphic letters, e.g., A an G. A conceptual explanation of k-moe prouct is to multiply each kth-irection vector fiber in A by the matrix U. An illustration of the multiplication to a r-orer tensor is shown in Fig. b). The Khatri-Rao prouct is the matching columnwise Kronecker prouct. The Khatri-Rao prouct of matrices A = a, a,..., a k R n k an B = b, b,..., b k R n k is efine by A B = a b, a b,..., a k b k R n n k. If A an B are column vectors, A B = A B. An if A an B are row vectors, A B becomes the Haamar prouct viz. element-by-element prouct) of the two rows. ) Rank- tensors an canonical ecomposition: A rank- tensor of imension can be written as the outer prouct of vectors A = a ) a ) a ), a k) R n k, 9) where enotes the outer prouct. Its element A i i i = a ) i a ) i a ) i, where a k) i k is the i k th entry of vector a k). The CANDECOMP/PARAFAC CP) ecomposition 8, 9, 4, 7 approximates a tensor A by a finite sum of rank- tensors, which can be written by A R r= a ) r a ) r a ) r, a k) r R n k, 0) where R Z +. Concisely, using the factor matrices A k) a k), ak),..., ak) R Rnk R, the right-han sie of the CP 0) can be expresse by the notation A A ),..., A ). Moreover, it is worth mentioning that the k-moe matricization of A coul be reconstructe by these factor matrices Ak) A k) A ) A k+) A k ) A ). ) The rank of the tensor A, ranka), is the minimum value of R in the exact ecomposition 0). A rank-r approximation of a r-orer tensor is shown in Fig.. Several methos have been evelope to compute the CP ecomposition, for example, the alternating least squares AL- S) 8, 7 as well as many of its erivatives) or the optimization methos such as CPOPT. Most of them solve the optimization problem of minimizing the Frobenius norm CANDECOMP canonical ecomposition) by Carroll an Chang 8 an PARAFAC parallel factors) by Harshman 7. They are foun inepenently in history, but the unerlying algorithms are the same.

5 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. X, NO. X, JANUARY 0XX 4 Fig.. ) a ) a ) a ) a ) a ) a A CP ecomposition of a r-orer tensor. of the ifference between the original tensor an its rank-r approximation min fa ),..., A ) ) A A ),..., A ). F ) The ALS algorithm iteratively optimizes one factor matrix A i) at a time, by holing all other factors fixe an solving the linear least square problem ) a R ) a R ) a R min fa ),..., A ) ) ) A i) for the upate A i). Alternatively, CPOPT calculates the graient of the objective function f in ) an uses the generic nonlinear conjugate graient metho to optimize ). For both ALS an CPOPT, the rank R shoul be prescribe an is fixe uring the computation. It is reporte in that the computational complexities for both ALS an CPOPT to approximate an Nth-orer tensor A R n n N are ONQR) per iteration, where Q = N i= n i. It is also mentione in that ALS is several times faster than CPOPT in general. However CPOPT shows an essentially perfect accuracy compare with ALS. A review of ifferent CP methos also can be foun in 9. III. TENSOR-FORM MODELING OF NONLINEAR SYSTEMS To begin with, we give an equivalent tensor-base moeling of the nonlinear system ). Recall the efinitions of G i an C i in ), it is reaily foun that these coefficient matrices are respectively -moe matricizations of i + )th-orer tensors G i an C i, i+ {}}{ G i, C i R n n, 4) where the elements G i ) j0j j i an C i ) j0j j i are coefficients of the Π i k= x j k term in G i an C i, respectively. For instance, G is an n n matrix while G is a r-orer n n n tensor, i.e., G is the -moe matricization of G. Accoring to Proposition.7 in, the Kronecker matrix proucts in ) can be represente by the tensor moe multiplication via G i x i ) = G i x T x T i x T i+ x T an C i x i ) = C i x T x T i x T i+ x T. Therefore, ) is equivalent to C x T + C x T x T + C x T x T 4 x T + t +G x T + G x T x T + G x T x T 4 x T + = Bu. 5) The key to the tensor-form moeling is to pre-ecompose these high imensional tensors via CP. In practical circuit systems, in spite of the growing imensionality, high-orer nonlinear coefficients G i an C i G i an C i ) are almost always sparse. Therefore it is avantageous to make a rank-r CP approximation of G i or C i for a relatively small R. In other wors, we can use a few rank- tensors to express G i an C i by r g,i G i G ) i,..., G i+) i = g ) i,r gi+) i,r, r= r c,i C i C ) i,..., C i+) i = c ) i,r ci+) i,r, r= 6) where i =,..., N, r g,i an r c,i are the tensor ranks of G i an C i, respectively, g k) i,r, ck) i,r Rn, G k) i = g k) i,,..., gk) i,r g,i R n r g,i an C k) i = c k) i,,..., ck) i,r c,i R n r c,i, for k =,..., i +. It shoul be notice that ifferent permutations of inices can result in the same polynomial term. For example, term x x can be represente by any combination of αx x + α)x x. Therefore, the high-orer tensors are not unique for a specific nonlinear system an the consequent lowrank approximations 6) coul be very ifferent. Nonetheless, we use the tensors with minimum nonzero entries in our implementation, as they ten to be sparser such that lower rank approximations are usually available. Using the CP structure 6), the original nonlinear system 5) can be approximate by absorbing tensor proucts of x into the factor matrices C x T + C ) t, xt C ), xt C ) +C ), xt C ), xt C ), xt C 4) + + G x T + G ), xt G ), xt G ) +G), xt G ), xt G ), xt G 4) + = Bu. 7) Applying ), the -moe matricization of 7) is simply ) C x + C ) x T C ) x T C ) + C x T C 4) x T C ) t x T C ) + G ) + + G x + G ) x T G 4) x T G ) x T G ) x T G ) x T G ) + = Bu, 8) an its corresponing -moe matricize Volterra subsystems are given by t C x + G x = Bu, 9a) t C x + G x = ) C ) x T C ) x T C t ) G ) T x T G) x T G), 9b) t C x + G x = ) C ) x T C 4) x T C ) x T C t +C ) x T i C ) x T i C ) ) G ) x T G 4) x T G ) x T G G ) x T i G ) x T i G ), 9c) an so on.

6 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. X, NO. X, JANUARY 0XX 5 IV. TNMOR Without loss of generality, we start from 8) an 9) to erive TNMOR. It can be observe that 9) is a series of linear systems where x is solve by the first equation with input u, x is the solution to the secon linear system with the input epenent on x, an similarly x is solve in the thir system with its input epenent on x an x. Similar to, 5 7, the frequency-omain transfer function of 9a) is given by H s) = sg C + I) G B sa + I) B. 0) To match up to k th-orer moments of 0), its projection space V k is the Krylov subspace of K k +A, B ) if H s) is expane aroun the origin, where K m A, p) = span{p, Ap,..., A m p}. The Krylov subspace can be efficiently calculate by the block Arnoli iteration. The n-orer subsystem 9b) can be recast into C I ẋ G 0 x ẋ e 0 I x e x G ) T = 0 G ) 0 C ) x T G ) ) T x T C ) x T C ), ) Consequently, ) coul be treate as a linear system with r g, + r c, inputs an its transfer function reas G H s) = s C ) G G + I G) C ) sa + I) B. ) Suppose k th-orer moments of H s) are matche, its Krylov subspace is obtaine by K k +A, B ). Thus, we have V k to be the first n rows of K k+a, B ) noticing H s) is a n vector). For the r-orer subsystem 9c), similarly, it coul be represente by another linear system C I 0 0 ẋ G 0 x + ẋ e 0 I x e x T G 4) x T G) x T G) x T C 4) x T C) x T C) 0 C ) 0 C ) x T i G ) x T i G ) x T i C ) x T i C ) G = ) 0 G ) 0 ) with r g, + r c, + r g, + r c, inputs. Moreover, the r-orer transfer function is given by G H s) = s C ) G + I 0 0 G G) 0 G G) 0 0 C ) 0 C ) sa + I) B B. 4) Consequently, the Krylov subspace of the r-orer subsystem shoul be K k +A, B B ) = K k +A, B ) K k +A, B ), if the number of moments being matche is k. However, it is reaily seen that if k k, we have K k +A, B ) K k +A, B ). Since K k +A, B ) has alreay been obtaine from H s), K k+a, B ) oes not nee to be recompute. Therefore, only K k+a, B ) shoul be counte at this stage an we enote its first n rows to be V k. Higher orer linear transfer functions can be obtaine in a similar way. The ith-orer projector V ki is the first n rows of G K ki+a, B i ), where B i = G) i 0 0 C ). i The reucing projector for the nonlinear system is the orthogonal basis of all V ki s, enote by V = orthv k, V k,...). The size of an Nth-orer reuce-orer moel is in Ok l + k r g, + r c, ) + k r g, + r c, ) + + k N r g,n + r c,n )) = ONkr), where k = max{k,..., k N } an r = max{l, r g, + r c,,..., r g,n + r c,n }. Comparing with Ok N l N ) in the stanar projection approach an Ok N+ l N ) in NORM, a slimmer reuce-orer moel can be achieve if low-rank CP of the tensors are available. Finally, the tensor-base reuce-orer moel is given by the following projection x = V T x, B = V T B, L = V T L, ) i+) G i = G i,..., G i = V T G ) i,..., V T G i+) i, ) i+) C i = C i,..., C i = V T C ) i,..., V T C i+) i, G = V T G V, C = V T C V, i =,..., N, 5) which looks similar to 6) except for the tensor CP structure. k) k) By utilizing this structure, only the G i an C i matrices nee to be store, an simulation of the reuce-orer moel can be significantly speee up as will be seen in the next section, as long as low-rank CP approximations of G i an C i are available. Algorithm summarizes the TNMOR assuming a DC expansion point. The computational complexity of TNMOR is ominate by the CP ecompositions of C i an G i. Any CP metho can be use to extract the rank- factors. We use CPOPT to compute the CP in TNMOR. Although CPOPT is not as remarkably fast as ALS, we fin in practice that CPOPT can always achieve a better accuracy uner the same rank. The computational cost for CPOPT to optimize a rank-r g,n approximation of G N is in ONn N+ r g,n ) per iteration. By contrast, the costs for NORM an the stanar projection metho to calculate the Krylov starting vectors for an N th-orer subsystem are in Ok N n N ) an Ok N l N n N ), respectively. It shoul be notice that the CP ecompositions only nee to be one once an the resulting reuce CP structure can be use in ifferent on-going simulations. Another key issue is how to heuristically prescribe/estimate the rank for each tensor. It is reaily foun that accurate lowrank CP approximations of the high imensional tensors is critical to the propose metho. Although it will be shown in Section V that the size of the reuce-orer moel an the time complexity for its simulation are proportional to the tensor ranks r G s an r C s, the efficiency of the propose

7 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. X, NO. X, JANUARY 0XX 6 Algorithm TNMOR Algorithm Input: N, G i, C i, B, L, k i, k N k N k Output: G, C, G i, C i, B, L, i =,..., N : for i = to N o : Gi = G i ; C i = C i ; : G ) i,..., G i+) i = CP G i ); 4: C ) i,..., G i+) i = CP C i ); 5: en for 6: A = G C ; B = G B; 7: V k = K k+a, B ); 8: for i = to N o G 9: A = C G 0 0 ; B i = G G) i 0 0 C ) i 0: V ki = K ki+a, B i ); : en for : V = orthv k, V k,..., V kn ); : G = V T G V ; C = V T C V ; B = V T B; L = V T L; 4: for i = to N o 5: Gi = V T G ) i 6: Ci = V T C ) 7: en for i,..., V T G i+) i ;,..., V T C i+) i ; moel will be compromise if large ranks are require to approximate the tensors. Unfortunately, unlike matrices, there are no feasible algorithms to etermine the rank of a specific tensor; actually it is an NP-har problem 8. Nonetheless, a loose upper boun on the maximum rank of a sparse tensor is the number of its nonzero entries. Furthermore, in practical circuit examples, we fin that m to m woul be a possible rank to approximate a sparse tensor with m nonzero elements. As shown by examples in Section VI, this empirical rank works well for systems with fewer than On) nonlinear components. Meanwhile, there is a significant amount of analog an RF circuits with a few usually in O)) transistors, for instance, amplifiers an mixers, which woul potentially amit lowrank tensor approximations. For systems with more than On) nonlinearities, such as ring oscillators or most iscretize nonlinear partial ifferential equations, however, no reuction can be achieve by TNMOR if any of the tensor ranks excees n. It shoul be remarke that such systems may still be reuce by NORM, if relatively low orer of moments is matche. V. SIMULATION OF THE REDUCED-ORDER MODEL Here we show that whenever low-rank tensor approximations of G i an C i exist, TNMOR can help to accelerate simulation an avois the exponential growth of the memory requirement, versus the reuce but ense moels generate by the stanar projection metho or NORM. We escribe the time complexity by means of the two key processes in circuit simulation, namely, function evaluation an calculation of the Jacobian matrices. For the ease of illustration, we assume the conuctance matrix C = I an all higher orer C i = 0, i =,..., N. Extension to general cases is straightforwar. ; A. Function evaluation Rewrite ) for the reuce-orer moel f x) x = G x G x G x + Bu, 6) where f x) is the function to be evaluate in the simulation. Using the reuce-orer moel in 5), the equivalent -moe matricization of 6) is f x) = G x G ) G ) x T G) x T G) x T G4) x T G) x T G) + Bu. 7) It shoul be notice that all x T Gk) i an x T Ck) i are row vectors, therefore correspons to the element-by-element multiplication between matrices. If up to the N th-orer nonlinearity is inclue an the size of the reuce-orer moel is q, the time complexity for evaluating 7) is in ON qr g ) where r g = max{r g,,..., r g,n }, while it is in Oq N+ ) for the moel 6) use in the stanar projection approach an NORM. B. Jacobian matrix Consier the Jacobian matrix of f x) which is often use in time-omain simulation J f x) G G x I + I x) G x x I + x I x + I x x). 8) Its equivalent -moe matricization is given by J f x) = G ) ) G x T G) ) ) T G + G x T G) ) G x T G4) x T G) ) G + x T G4) ) G x T G) ) 4) T + G x T G) x T G). 9) The complexity for the stanar projection metho or NORM to calculate 8) is in ONq N+ ). For the tensor formulation, the complexity for evaluating 9) is further reuce to ONqN + q)r g ). C. Space complexity In our propose metho, the amount of memory space for the reuce-orer moel is etermine by the factor matrices in 5), which shoul be in ONqr), where r = max{r g, + r c,,..., r g,n + r c,n }. For the existing stanar projection approach an NORM, the storage consumption is ominate by the matrices G N an C N whose numbers of elements are in Oq N+ ). VI. NUMERICAL EXAMPLES In this section, TNMOR is emonstrate an compare with the stanar projection approach an NORM. All experiments are implemente in MATLAB on a esktop with an Intel i5 500@.GHz CPU an 6GB RAM. To fairly present the results, all time-omain transient analyses are solve by the trapezoi iscretization with fixe step sizes. In the simulations

8 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. X, NO. X, JANUARY 0XX 7 Fig. 4. Vlo+ Vrf+ Vif+ A ouble-balance mixer. Vrf- Vif- Vlo- TABLE I. SIZES OF THE REDUCED-ORDER MODELS ROM) AND CPU TIMES OF REDUCTION FOR THE MIXER SMALL SIGNAL) Metho k k k CPU time size of ROM stanar projection 5, s 5 NORM s 0 TNMOR 0 4.5s 9 TABLE II. CPU TIMES AND ERRORS OF TRANSIENT SIMULATIONS FOR THE MIXER SMALL SIGNAL) Transient full moel stanar projection 5, 6 NORM 7 TNMOR size CPU time 9s 00s 70s 7s speeup.x error.98%.4%.4% TABLE III. CPU TIMES AND ERRORS OF PERIODIC STEADY-STATE SIMULATIONS FOR THE MIXER SMALL SIGNAL) Perioic steay stanar full moel state projection 5, 6 NORM 7 TNMOR size CPU time 67s 4900s 40s s speeup.x error 0.9% 0.7% 0.8% of the original system, the reuce-orer moels of the stanar projection an NORM approaches, 6) an 8) are use to evaluate the functions an Jacobian matrices, while 7) an 9) are use for the tensor-base moel. The CP in TNMOR is compute by the CPOPT algorithm provie in the MATLAB Tensor Toolbox,, 9. A. A ouble-balance mixer First, we stuy a ouble-balance mixer circuit in Fig. 4 0, where V rf = V rf+ V rf- ) an V lo = V lo+ V lo- ) are the RF an local oscillator LO) inputs, respectively. We assume V rf an V lo are both sinusoial an their frequencies are GHz an 00MHz, respectively. V if+ an V if- are the intermeiate-frequency IF) outputs. The size n of the original system is 9. Firstly, we assume a relatively small V lo swing so that the nonlinear system can be approximate by its r-orer Taylor expansion t C x + G x + G x + G x = Bu, 0) where the numbers of nonzero elements in G an G are 6 an 0, respectively. Then, the stanar projection approach, NORM an TN- MOR are applie to 0). It shoul be notice that all methos are expane at the frequencies GHz an 00MHz. The size of the reuce-orer moel, the orer of the moments k i matche in each subsystem, an the CPU times for each moel orer reuction metho are liste in Table I. It can be notice that ue to the curse of imensionality, the stanar projection approach generates a larger an enser) reuce-orer moel but fewer orers of moments are persevere. TNMOR takes.s to optimize a best rank-6 r g, = 6) approximation of G with the error G G F G F = , an.s for a best rank-9 r g, = 9) approximation of G with an error A transient simulation from 0 to T = 5ns with a t = 5ps step size is performe on each reuce-orer moel. The runtimes an overall errors of ifferent methos are summarize in Table II. The overall error is efine as the relative error between the vector of V if+ at subsequent timesteps V = V if+ 0), V if+ t),..., V if+ T ) T of the reuce-orer moels an the corresponing vector compute with the full moel, i.e., V V full V full. The first 7.5ns of the transient waveforms of V if+ an their relative errors are plotte in Figs. 5a) an 5b), respectively. Next, the perioic steay-state analyses of ifferent moels are achieve by a shooting Newton metho-base perioic steay-state simulator. The CPU times an the overall errors of the frequency responses between 0 to 4GHz are liste in Table III. The relative errors of the frequency responses are plotte in Fig. 5c). It is shown in Tables II an III that although TNMOR generates a larger reuce-orer moel than NORM 9 versus 0), its transient an perioic steay-state analyses are faster ue to the efficient algorithm of function an Jacobian evaluations. In contrast, simulations of the ense reuce-orer moels generate by NORM an the stanar projection approach are much slower than the original large but sparse system, inicating that these methos are impractical for strongly nonlinear systems r-orer nonlinearity in this example). This is mainly because though both 6) an 8) have been use in the simulations, the Kronecker powers in 6) an 8) of the original system never have to be explicitly compute ue to the sparsity of the original G i an C i matrices. Moreover, comparing to NORM, the propose metho provies a competitive accuracy as can be seen in Fig. 5. Practically, mixers are often moele by perioic timevarying systems ue to the existence of large V lo signals 5

9 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. X, NO. X, JANUARY 0XX 8 V if V) V V full / V full f f full / f full Full TNMOR NORM SP time s) x a) TNMOR NORM SP time s) x 0 9 b) SP TNMOR NORM frequency Hz) x 0 9 c) Fig. 5. aransient waveforms of the full moel an reuce-orer moels. b) Relative errors of the transient simulations. c) Relative errors of the perioic steay-state simulations. SP is the acronym for stanar projection 5, 6.) 7,. Fortunately, TNMOR can be easily extene to perioic time-varying systems as well. Following 5 7, suppose all higher orer C i = 0, all the system matrices in 9) become T lo -perioic where T lo is the perio of V lo. Next, a uniform backwar-euler iscretization over sampling points t, t,..., t M where t i = i M T lo is applie on each linear time-varying system in 9), resulting a set of LTV systems with transfer functions Ĥis) where Ĵ + sĉĥis) = ˆB i, ) Ĥ i s) = H T i t, s) H T i t, s)... H T i t M, s) T, ˆB = B T t ) B T t )... B T t M ) T, G t ) G t ) Ĝ =, )... G t M ) IM Bm) Full TNMOR NORM SP RF amplitue mv) Fig. 6. IM of the full moel an reuce-orer moels. SP is the acronym for stanar projection 5, 6.) TABLE IV. an SIZES OF THE REDUCED-ORDER MODELS AND CPU TIMES OF REDUCTION FOR THE MIXER LARGE SIGNAL) Metho k k k CPU time size of ROM stanar projection 5, 6 0.5s 46 NORM-mp 7 0.s 8 TNMOR 0 8s 78 ˆB i = Ĝ) i C t ) Ĉ = I = M I T lo C t ) I I..., C t M ) I, Ĵ = Ĝ + Ĉ ) = G ) i t ) I G ) i t )... G ) i t M ) i. 4), We omit the etaile erivation as it is straightforwar. Similarly, the projector is obtaine by collocating the moments of each orer subsystem in ). It shoul be notice that the factor matrices at each sampling point can be compute iniviually via CP ecompositions. Therefore the computational complexity is proportional to M. Next, we reformulate the mixer by a r-orer nonlinear time-varying system uner a large signal V lo. The 90-variable system is expane aroun M = 0 operating points. This time, TNMOR, multi-point NORM NORM-mp) an the s- tanar projection metho are expane at GHz. The sizes of reuce-orer moels an CPU times are liste in Table IV. The CPU time of TNMOR is mainly spent on sequential CP ecompositions, which coul be further reuce if multi-core parallelization is enable. These reuce-orer moels are simulate for r-orer intermoulation tests. The LO an RF frequencies are fixe

10 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. X, NO. X, JANUARY 0XX 9 TABLE V. CPU TIMES OF IM TESTS FOR THE MIXER LARGE SIGNAL) TABLE VII. CPU TIMES AND ERRORS OF TRANSIENT SIMULATIONS FOR THE BIOCHEMICAL SYSTEM Transient full moel stanar projection 5, 6 NORM 7 TNMOR size CPU time 46 ± 8s 0 ± s ± s 5 ± s speeup.x 6.6x 9.7x TABLE VI. SIZES OF THE REDUCED-ORDER MODELS AND CPU TIMES OF REDUCTION FOR THE BIOCHEMICAL SYSTEM Transient full moel stanar projection 5, 6 NORM 7 TNMOR size CPU time 80 ± 5s 44 ± s 47 ± 69s. ± 0.9s speeup.9x.x 90x error %) 8 ±. ±.0. ±. Metho k k k CPU time size of ROM stanar projection 5, s 9 NORM 7 0.s 45 TNMOR 0 s 9 u u while we sweep the amplitue of the sinusoial RF input from mv to 0mV. Fig. 6 shows the r-orer intermoulation prouct IM) results of the original system an reuceorer moels. The CPU times are summarize in Table V, where they are written as a ± b where a is the average value an b is the sample stanar eviation. From Fig. 6, goo agreement can be observe for the reuce-orer moels generate by NORM-mp an TNMOR. NORM-mp achieves a smaller size because it only preserves the values of nonlinear transfer functions at specific points, which is particularly useful for matching high-orer istortions. Meanwhile, TNMOR still emonstrates comparable accuracy an better efficiency ue to the benefit of the tensor framework. B. A biochemical reaction system The secon example is a sparse biochemical reaction system moel aapte from. The system is generate by a ranom n-orer polynomial system with a +x function t x + G x + G x 0 + e = Bu, 5) + x where G R 00 00, G R 00 00, B R 00 an e R 00 =, 0,..., 0 T. It shoul be notice that both G an B are ense ranom matrices an the eigenvalues of G are ranomly istribute on 0, so that the nonlinear system is stable at the origin. G is a sparse ranom matrix with 48 nonzero entries. +x is expene by the Taylor series +x x +x x an we control the inputs to guarantee x < uring the simulations. The three nonlinear moel orer reuction approaches are applie on the r-orer polynomial system to generate the reuce-orer moels. We match the moments at the origin in all approaches. The sizes of the reuce-orer moels, the orers of the moments in each subsystem an the CPU times for the reuction have been liste in Table VI. TNMOR optimizes a rank-0 r g, = 0) approximation of G in s with a relative error 0.0. The unique nonzero element in G is 0x, therefore its CP can be obtaine immeiately with the rank r g, =. We fee these reuce-orer moels by 0 sets of sinusoial inputs for transient simulations with the same time perio an the same step size. These sinusoial inputs are uner ifferent CVi,j) a) L L y) y) R i Vi,j i,j+ R i Vi+,j i+,j i i,j x)... i i,j y) rg... i i+,j x) b)... i i+,j y) CVi+,j) rg... i i+,j x) Fig. 7. a) A nonlinear transmission line. b) Moel of the nonlinear transmission line. frequencies. The CPU times an errors have been summarize in Table VII. The CPU times further confirm that for sparse systems, TNMOR can utilize the sparsity while the structure cannot be kept in NORM or the stanar projection approach. C. A -D pulse-narrowing transmission line It is reporte in that linear lossy transmission line woul cause the wave ispersion effect of the input pulse which coul be avoie if certain nonlinear capacitors are introuce. We consier a nonlinear pulse-narrowing transmission line shown in Fig. 7a), 4. There are two pulse inputs injecte at the two corners of the transmission line. We are intereste in the voltage at the center of the shae ege. The length an with of the transmission line are 0cm an 0cm, respectively. It is uniformly partitione into a 0 0 gri, with each noe shown by the lumpe circuit in Fig. 7b). The noes at the shae ege of the transmission line are characterize by the nonlinear state equation C 0 Vi,j = i x) i,j +iy) i,j ix) i+,j iy) i,j+ V i,j /r g ) + N= bv i,j) N ) where C 0 = µf, r g = 0KΩ an b = 0.9, while the nonlinear coefficient b = 0 elsewhere. In Fig. 7b), L = µh an R = 0.Ω. Using MNA, we en up with a system with 570 state variables. First, we approximate the original system by the rorer Taylor expansion, i.e., up to G. There are 78 nonzero elements in each orer G i, i =,,... In this case, all the moments are matche at the origin as well. Again, the sizes of the reuce-orer moels, the orers of the moments in each subsystem an the CPU times for the reuction are liste in Table VIII. Due to the exponential growth of the size of

11 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. X, NO. X, JANUARY 0XX 0 TABLE VIII. SIZES OF THE REDUCED-ORDER MODELS AND CPU TIMES OF REDUCTION FOR THE TRANSMISSION LINE Metho k k k CPU time size of ROM stanar projection 5, 6 0.6s 40 NORM 7 4 9s 4 TNMOR 4 7.6s 46 TABLE IX. CPU TIMES AND ERRORS OF TRANSIENT SIMULATIONS FOR THE TRANSMISSION LINE Transient full moel stanar projection 5, 6 NORM 7 TNMOR size CPU time 58s 440s 49s 8.5s speeup 6.8x error 45.% 0.5% 0.0% the projector, to get a reuce-orer moel fewer than 60 states, only the 0th-orer moments DC term) of the n-orer subsystem an none of the r-orer coul be matche when using the stanar projection approach. TNMOR takes.8s an.5s to optimize the best rank-6 r g, = r g, = 6) matches of G an G, respectively. In this case, the CP ecompositions are exact an the errors of CP are 0. Then, a transient analysis with the pulse inputs shown in Fig. 8a) is teste on each reuce-orer moel. The step size t is chosen to be ms. The CPU times for the reuceorer moels an the overall errors are summarize in Table IX with the resulting waveforms an the relative errors plotte in Figs. 8b) an 8c), respectively. It can be seen that to achieve a goo accuracy error < 0.%), the stanar projection approach an NORM may result in a reuce-orer moel which is even slower than the original one. On the other han, TNMOR can reuce the orer while still preserving the sparsity. Finally, we raise the orer of the Taylor expansion to C 5 an G 5 to match the highly nonlinear behavior. The stanar projection approach an NORM woul result in a ense matrix G 5 with q 6 elements where q is the size of the reuce system. The reuce-orer moel is usually impractical to use if an acceptable accuracy is esire. MATLAB faile to construct the matrices G 4 an G 5 even if the projector up to the r-orer subsystem is use while the moments of the 4th- an 5thorer subsystems are ignore. Meanwhile, it takes 6s r g,4 = r g,5 = 6) in total for TNMOR to obtain a 5-state reuceorer moel an another s for the transient simulation of the reuce-orer moel. The transient waveforms of the 5thorer full moel an the reuce-orer moel are shown in Fig. 8), contrasting with the one of the r-orer full moel. VII. CONCLUSION In this paper, a tensor-base nonlinear moel orer reuction scheme calle TNMOR is propose. The high-orer nonlinearities in the system are characterize by high imensional tensors such that these nonlinearities can be represente by just a few vectors obtaine from the canonical tensor ecomposition. Projection-base moel orer reuction is employe on these vectors to generate a compact reuce-orer moel uner the tensor framework. The key feature of TNMOR U V) time s) V V full / V full a) time s) c) u u TNMOR NORM SP V V) Full TNMOR NORM SP time s) V V) b) Full 5th orer) TNMOR 5th orer) Full thir orer) time s) Fig. 8. a) Pulse inputs to the transmission line. bransient waveforms of the full moel an reuce-orer moels. c) Relative errors of the transient simulations. ransient waveforms of the 5th-orer moel an the r-orer moel. SP is the acronym for stanar projection 5, 6.) is that it preserves the inherent sparsity through low-rank tensors, thereby easing memory requirement an speeing up computation via exploitation of ata structures. Examples have been shown to emonstrate the efficiency of TNMOR over existing nonlinear moel orer reuction algorithms. ACKNOWLEDGEMENT This work was supporte in part by the Hong Kong Research Grants Council uner General Research Fun GRF) Projects 78E an 70854, the University Research Committee of The University of Hong Kong, an the MIT-China MISTI program. The authors are also thankful to Prof. Ivan Oseleets of Skoltech an Mr. Zheng Zhang of MIT for their valuable comments an avice. REFERENCES A. Oabasioglu, M. Celik, an L. Pileggi, Prima: passive reuceorer interconnect macromoeling algorithm, Computer-Aie Design of Integrate Circuits an Systems, IEEE Transactions on, vol. 7, no. 8, pp , Aug 998. J. Phillips, L. Daniel, an L. Silveira, Guarantee passive balancing transformations for moel orer reuction, Computer-Aie Design of Integrate Circuits an Systems, IEEE Transactions on, vol., no. 8, pp , Aug 00. P. Felmann an R. Freun, Efficient linear circuit analysis by pae approximation via the lanczos process, Computer-Aie Design of Integrate Circuits an Systems, IEEE Transactions on, vol. 4, no. 5, pp , May B. Bon an L. Daniel, Guarantee stable projection-base moel reuction for inefinite an unstable linear systems, in Computer-Aie Design, 008. ICCAD 008. IEEE/ACM International Conference on, Nov 008, pp )

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E. Afshari an A. Hajimiri, Nonlinear transmission lines for pulse shaping in silicon, Soli-State Circuits, IEEE Journal of, vol. 40, no., pp , 005. B. N. Bon, Stability-preserving moel reuction for linear an nonlinear systems arising in analog circuit applications, Ph.D. issertation, Massachusetts Institute of Technology, Feb. 00. Haotian Liu S ) receive the B.S. egree in microelectronic engineering from Tsinghua University, Beijing, China, in 00, an the Ph.D. egree in electronic engineering from the University of Hong Kong, Hong Kong, in 04. He is currently a Postoctoral Fellow with the Department of Electrical an Electronic Engineering, the University of Hong Kong. In 04, he was a visiting scholar with the Massachusetts Institute of Technology MIT), Cambrige, MA. His research interests inclue numerical simulation methos for very-large-scale integration VLSI) circuits, moel orer reuction, parallel computation an control theory. Luca Daniel S 98 M 0) receive the Ph.D. egree in electrical engineering from the University of California, Berkeley, in 00. He is currently a Full Professor in the Electrical Engineering an Computer Science Department of the Massachusetts Institute of Technology MIT). Inustry experiences inclue HP Research Labs, Palo Alto 998) an Caence Berkeley Labs 00). His current research interests inclue integral equation solvers, uncertainty quantification an parameterize moel orer reuction, applie to RF circuits, silicon photonics, MEMs, Magnetic Resonance Imaging scanners, an the human cariovascular system. Prof. Daniel was the recipient of the 999 IEEE Trans. on Power Electronics best paper awar; the 00 best PhD thesis awars from the Electrical Engineering an the Applie Math epartments at UC Berkeley; the 00 ACM Outstaning Ph.D. Dissertation Awar in Electronic Design Automation; the 009 IBM Corporation Faculty Awar; the 00 IEEE Early Career Awar in Electronic Design Automation; the 04 IEEE Trans. On Computer Aie Design best paper awar; an seven best paper awars in conferences. Ngai Wong S 98 M 0) receive his B.Eng. an Ph.D. egrees in electrical an electronic engineering from the University of Hong Kong, Hong Kong, in 999 an 00, respectively. He was a visiting scholar with Purue University, West Lafayette, IN, in 00. He is currently an Associate Professor with the Department of Electrical an Electronic Engineering, the University of Hong Kong. His current research interests inclue linear an nonlinear circuit moeling an simulation, moel orer reuction, passivity test an enforcement, an tensor-base numerical algorithms in electronic esign automation.