Electrical Engineering Department, University of Texas at Dallas, Richardson, TX, USA 3

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1 Effiient Performane odeling via Dual-Prior Baesian odel Fusion for Analog and ixed-signal Ciruits Qiheng Huang Chenlei Fang Fan Yang * Xuan Zeng * Dian Zhou and Xin i 3 State Ke ab of ASIC & Sstem iroeletronis Department Fudan Universit Shanghai P R China Eletrial Engineering Department Universit of exas at Dallas Rihardson X USA 3 Eletrial & Computer Engineering Department Carnegie ellon Universit Pittsburgh PA USA ABSRAC In this paper we propose a novel Dual-Prior Baesian odel Fusion (DP-BF) algorithm for performane modeling Different from the previous BF methods whih use onl one soure of prior knowledge DP-BF takes advantage of multiple soures of prior knowledge to full exploit the available information and hene further redue the modeling ost Based on a graphial model an effiient Baesian inferene is developed to fuse two different prior models and ombine the prior information with a small number of training samples to ahieve high modeling aura Several iruit examples demonstrate that the proposed method an ahieve up to 83 ost redution over the traditional one-prior BF method without surrendering an aura INRODUCION he ontinuous saling of integrated iruits (ICs) leads to severe proess variations hese devie level proess variations (eg Vth ox et) pose large-sale unertainties in iruit performanes and hene impat the parametri ield of analog and mixed-signal (AS) iruits [] o model analze and optimize proess variations at all levels of design hierarh various tehniques have been developed for performane modeling during the past deades []-[4] he objetive is to desribe the performane of interest (eg offset of an operational amplifier) b an analtial (eg linear quadrati et) funtion of devie-level variations and/or environmental onditions One the performane models are reated the an be applied to various appliations suh as parametri ield predition [5] and worstase orner extration [6] Although man performane modeling tehniques were developed the evolution of AS iruits espeiall the inrease of iruit size and omplexit has posed a number of new hallenges in this area On one hand a large number of random variables have to be used to model the proess variations assoiated with large-sale iruits In onsequene a huge amount of simulation samples must be generated for highdimensional modeling On the other hand the omputational ost of iruit simulation inreases signifiantl due to inreasingl large iruit size whih makes iruit simulation extremel timeonsuming hese reent trends have made performane modeling prohibitivel expensive toda [7] o address this hallenging issue of modeling ost several advaned performane modeling tehniques (eg sparse * Corresponding authors: {angfan xzeng}@fudanedun Permission to make digital or hard opies of all or part of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for profit or ommerial advantage and that opies bear this notie and the full itation on the first page Coprights for omponents of this work owned b others than AC must be honored Abstrating with redit is permitted o op otherwise or republish to post on servers or to redistribute to lists requires prior speifi permission and/or a fee Request permissions from Permissions@amorg DAC '6 June Austin X USA 06 AC ISBN /6/06 $500 DOI: regression [8] elasti net regularization [9] et) have been proposed In partiular a framework of Baesian odel Fusion (BF) was developed for effiient high-dimensional performane modeling [0]-[] BF optimall ombines the earl-stage (eg shemati-level) information and a small number of late-stage (eg post-laout) samples via Baesian inferene he late-stage model oeffiients are then determined b maximizing the posterior distribution An extended version of BF referred to as Co-earning BF (C-BF) was reentl proposed to further redue the modeling ost [] C-BF trains an extra lowomplexit model to generate pseudo samples for fitting a highomplexit performane model In this wa it greatl redues the number of required phsial samples and hene the overall modeling ost he aforementioned BF approahes attempt to exploit onl one soure of prior knowledge (ie earl-stage model oeffiients) In pratie we an often obtain useful knowledge from multiple soures to failitate late-stage performane modeling For example to model the performane metris based on post-silion measurements we an take advantages of the models fitted b (i) the pre-silion data olleted from simulation and (ii) the postsilion data measured from a previous tape-out Consider another important appliation of modeling the aging behavior for analog iruits o apture the aged performane metris at the postlaout stage we an borrow the prior knowledge from the models fitted b (i) the shemati-level simulation data for the aged performane metris and (ii) the post-laout simulation data at t = 0 hese various soures of prior knowledge are expeted to provide the useful information that failitates us to effiientl fit the performane model of interest herefore a new BF framework must be reated to properl fuse multiple soures of prior knowledge for performane modeling owards this goal we propose a novel BF tehnique referred to as Dual-Prior Baesian odel Fusion (DP-BF) that takes into aount multiple soures of prior knowledge DP-BF is derived from the Baesian inferene that an be represented as a graphial model [3]-[4] he performane model of interest is built b ombining multiple prior models and a small number of training samples As will be demonstrated b our experimental results in Setion 5 the proposed method an ahieve up to 83 ost redution over the onventional BF approah he reminder of this paper is organized as follows We briefl review the bakground of BF in Setion and then derive our DP-BF method in Setion 3 In Setion 4 several implementation issues are further disussed he effia of the proposed method is demonstrated b two iruit examples in Setion 5 Finall we onlude in Setion 6 BACKGROUND he performane modeling of an AS iruit aims to desribe ertain performanes of interests with an analtial funtion of devie-level variations and/or environmental onditions For example we an approximate the offset of an operational amplifier as a polnomial funtion of variables like Vth and DC bias urrent

2 Generall the performane model of a given iruit an be desribed as: f( ) = α m gm( ) x x () denotes the performane metri to be estimated x is a vetor representing the variations and operation point f is the performane funtion he performane funtion is a linear ombination of basis funtions (eg linear or quadrati polnomials) {gm(x); m = } and {m; m = } are the model oeffiients he unknown oeffiients in () are traditionall determined b solving the following least-squares regression problem: min G () () () () g( x ) g( x ) g ( x ) () () () g( x ) g( x ) g ( x ) 3 G = (3) ( K) ( K) ( K) g( x ) g( x ) g ( x ) 4 = [ α α α ] (4) () () ( K ) 5 = [ ] (5) In ()-(5) stands for the -norm of a vetor K represents the total number of sampling points and x (k) and (k) are the k-th sampling values of x and respetivel is a vetor ontaining all the model oeffiients and is a vetor onsisting of all the samples of When applied to high-dimensional performane modeling the least-squares fitting method requires a huge number of sampling points and thus leads to extremel expensive modeling ost o address the omplexit issue of least-squares fitting sparse regression method [9] has reentl been developed b exploiting the fat that most high-dimension model oeffiients are lose to zero However traditional sparse regression approah onl fits the performane model based on the simulation data of a single stage BF an be applied to further ombine the knowledge from earl stage and thus redue the performane modeling ost In the onventional BF method the prior knowledge of model oeffiients obtained from earl stage data (eg shematilevel simulation data) is enoded into a nonzero-mean Gaussian prior distribution With onl a few samples from late-stage data (eg post-laout simulation data) the estimated late-stage model oeffiients an then be derived as: 6 = η D+ G G η D E+ G (6) = [ α α α ] (7) 7 E E E E 8 D = diag( α E α E α E ) E is the oeffiient vetor of a model fitted b the earl-stage data G and are defined b (3) and (5) respetivel and diag() represents the operator to onstrut a diagonal matrix is a hper-parameter ontrolling the onfidene in prior knowledge In the ase that is suffiientl large (6) an be redued as: 9 E (9) whih reveals that the prior knowledge is ver aurate; In another extreme ase that is ver small we have: 0 ( ) (8) = G G G (0) whih implies that the prior knowledge is inaurate so the leastsquares regression is applied on the late-stage data alone for model fitting he optimal value of an be determined b the ross-validation tehnique [3] While the aforementioned BF method proves to ahieve signifiant speedup over least-squares regression it restrits the prior knowledge to single soure In pratie we often have multiple soures of prior knowledge It is possible to exploit more orrelated information from different aspets to further failitate the late-stage statistial analsis With this motivation we propose a new BF strateg to utilize two soures of prior knowledge 3 PROPOSED APPROACH In this setion we develop our proposed DP-BF method to borrow prior knowledge from two soures of earl-stage data for more effiient performane modeling 3 Problem Formulation Speifiall we denote the two groups of prior model oeffiients as E and E = [ α α α ] () E E E E E = αe αe αe [ ] () he two groups of oeffiients are obtained b fitting from two different soures of existing data respetivel using the same set of basis funtions for late-stage performane modeling herefore we are supposed to know the two groups of oeffiients before fitting the late-stage model he fundamental problem of the proposed method then beomes: with the input information of (i) two groups of earlstage model oeffiients {Em; m = } and {Em; m = } and (ii) a few late-stage samples of x and how to properl estimate the late-stage model oeffiients b borrowing prior knowledge of the two soures o this end a Baesian inferene strateg will be onstruted and represented b a graphial model 3 Graphial odel and ikelihood Funtion We onsider three different performane models: two singleprior model f(x) f(x) and the late-stage model f(x) we aim to fit: 3 f( ) = α m gm( ) x x (3) 4 f( ) = α m gm( ) x x (4) 5 f( ) = α m gm( ) x x (5) {m; m = } and {m; m = } represent the oeffiients of two single-prior models and {m; m = } denote the late-stage model oeffiients to be estimated We assume f(x) f(x) and f(x) share the same set of basis funtions he meaning of single-prior models will be explained soon in the details of the graphial model As shown in figure a graphial model is onstruted to illustrate the main strateg of our method Eah node represents a random quantit and eah direted/undireted edge represents a unidiretional/non-diretional dependen he two small solid irles stand for the two soures of prior knowledge he filled node indiates that the orresponding data (ie the phsial samples of ) have been observed We all f(x) and f(x) singleprior models beause the aim to predit the late-stage model oeffiients based on single soure of prior knowledge (ie E or E) respetivel Our target model f(x) then works as a onsensus funtion to balane the two single-prior models and ombine the useful information the extrat from their respetive

3 prior information We expet that the three models f(x) f(x) and f(x) are onsistent sine the are supposed to predit the same performane metri although in different was Figure A graphial model is shown to illustrate the Baesian inferene strateg of DP-BF Aording to the graphial model in Figure we an derive the joint probabilit densit funtion (PDF) as: ( f f ) pdf( f f f ) exp 6 (6) ( f f) ( f) exp exp Here we assume the distribution of the differene between < f f> < f f> and <f > are all zero-mean Gaussian with and to be the orresponding standard deviations Given a number of independent late-stage samples {(x (r) (r) ); r = K} the joint PDF for all the olleted samples an be desribed as: 7 f f pdf( f f f ) exp f f f exp exp 8 () () ( K ) f f( x ) f( x ) f( x ) 9 () () ( K ) f f( x ) f( x ) f( x ) 0 () () ( K ) f f( x ) f( x ) f( x ) (7) = (8) = (9) = (0) and is defined as (5) Aording to (3)-(5) f f and f an be re-written as: f = G () f= G () 3 f = (3) G and are defined as (3) and (4) respetivel and = [ α α α ] (4) 4 5 = α α α [ ] (5) hen b substituting ()-(3) into the joint PDF (7) we an get the likelihood funtion: G G ( ) pdf exp 6 (6) G G G exp exp he first two -norm terms represent the differenes between the target model f and two single-prior models f and f respetivel and the third -norm term stands for the modeling error of f as ompared to the observed samples of 33 Prior Distribution Definition Sine f(x) and f(x) aim to estimate the late-stage model oeffiients based on prior knowledge E and E respetivel we expet that the model oeffiients {m; m = } are lose to {Em; m = } and the oeffiients {; m = } are lose to {Em; m = } herefore we onstrut two nonzero-mean Gaussian distributions for eah of the two sets of oeffiients: pdf α Gauss α k α (7) 7 ( m) ( E m E m) ( ) 8 pdf ( α ) Gauss( α k α ) ( m = ) (8) m E m E m k and k are two hper-parameters that an be determined b ross-validation as will be disussed in detail in Setion 4 In the prior distribution (7) we assume pdf(m) is peaked at its mean value m = Em impling the earl-stage oeffiient Em and the late-stage m are likel to be similar Also the standard deviation of pdf(m) is assumed to be proportional to Em whih provides eah late-stage oeffiient m with a relativel equal opportunit to deviate from the orresponding earl-stage oeffiient Em he prior distribution pdf(m) in (8) an be interpreted in a similar wa hen we further assume that all late-stage model oeffiients are statistiall independent In this wa we an enode the prior knowledge of and as the joint PDF funtion pdf( ): 9 pdf ( ) E D 0 E E 0 D E (9) exp 30 ( = k diag α E α E α E ) 3 ( = k diag α E α E α E ) D (30) D (3) Sine we have no prior knowledge for f(x) and its oeffiients the ontribution of to the joint distribution is represented as a at the end of (9) 34 aximum-a-posteriori Estimation One the prior distribution pdf( ) is defined b (9) we an ombine it with K late-stage phsial samples {(x (r) (r) ); r = K} and estimate the optimal values of late-stage model oeffiients b maximum-a-posteriori (AP) estimation Based on Baes theorem the posterior distribution is proportional to the prior distribution pdf( ) multiplied b the likelihood funtion pdf( ): 3 pdf( ) pdf( ) pdf( ) (3) AP attempts to find the optimal values of and to maximize the posterior distribution pdf( ) Namel it aims to find the solutions and that are most likel to our aording to the posterior distribution although we atuall onl are about the value of as it ontains the oeffiients of the target model athematiall the AP solution an be found b solving the following optimization problem: 33 max pdf( ) (33) Substituting (3) b (6) and (9) and taking the logarithm for the posterior distribution we an onvert (33) to the following equivalent optimization problem: min h (34) 34 ( )

4 35 h X X X X X = + + ( ) ( ) D ( ) k ( ) D ( ) + k + E E E E (35) In this ost funtion the first two terms penalize the disrepan between the predition results of the target model f(x) and the other two single-prior models f(x) and f(x) he third term represents the error of traditional least-squares fitting he last two terms penalize the differenes between the model oeffiients of f(x) f(x) and their prior knowledge E E respetivel We an see the ost funtion aims to ompromise among three parts: f(x) s predition error its similarit to the other two single-prior models and the similarit of the single-prior models to their perspetive prior knowledge Although our target is to get the solution of in (34) the model oeffiients and are solved together B taking partial derivatives of h( ) with respet to and then setting them to zero we an get the AP estimation of the target model oeffiients : 36 = b (36) G G = + + I 4 + k G G k 4 D G G D G G (37) + GG b= + k D k D E GG k D kd E ( GG) G (38) From (36)-(38) we an see the estimated result is ontrolled b five hper-parameters k and k hese parameters ontrol the balane in two aspets: (i) the balane between trusting prior knowledge (E and E) and trusting late-stage samples in ; (ii) the balane between the onfidene in two soures of prior knowledge o alulate the oeffiient vetor in (36) these hper-parameters must be arefull determined In pratie onl three of them are independent and we an determine their values b a two-dimensional ross-validation proess In Setion 4 we will disuss in detail about the respetive influene of these hperparameters and the method to find their optimal values 4 IPEENAION ISSUES o make the proposed DP-BF method pratiall effiient we also need to onsider several implementation issues In this setion we disuss these issues in detail inluding (i) the influene of hper-parameters and how to determine their optimal values and (ii) the detetion method of two highl biased soures of prior knowledge 4 Hper-Parameters he hper-parameters k and k in (37)-(38) ontrol the trust in prior information E prior information E and the late-stage data samples in he should be arefull determined so that we an properl exploit the information from prior knowledge and data samples to get an aurate estimation result In fat are not independent As shown in Figure sine we suppose the distributions of differenes between < f f> and <f > are all zero-mean Gaussian with variane and respetivel the distribution of differene between f and is then Figure he distribution of the differenes between models and observed data zero-mean Gaussian with variane : 39 γ= + (39) B running the onventional single-prior BF in Setion with prior information E and late-stage samples we an then estimate the value of from the variane of modeling error Similarl the distribution of differene between f and is another zero-mean Gaussian distribution with variane : 40 γ = + (40) he value of an be estimated b running another single-prior BF with prior information E One we get the estimated values of and both and an be uniquel determined b hus overall we onl need to find the optimal values for three independent hper-parameters k and k Now we disuss the influene of the hper-parameters k and k b onsidering several extreme ases: Case : k and k are small enough (lose to zero) In this ase the terms onerning k and k in (37)-(38) an all be eliminated and we an then redue (36) to: 4 ( ) G G G (4) whih is exatl the least-squares estimation It indiates that both soures of prior knowledge are inaurate so the model oeffiients are fitted simpl b the late-stage data samples Case : k >> k and k is lose to zero In this ase (37)-(38) an be redued and transformed to: 4 G G = + I k + γ γ D G G (4) ( γ ) E 43 G G b= k k + D D + ( G G) γ γ G (43) whih then ields: 44 E when γ ; (44) γ G G G when (45) 45 ( ) From these ases we an interpret k and k as the trust in prior information E and E respetivel With larger value of k (or k) more weight is assigned to E (or E) in estimation he ratio of k and k then ontrols the balane between two soures of prior knowledge hat is wh we have (4) when k and k are both small and have (44) in the ase that k is obviousl larger than k On the other hand an be interpreted as the distrust in late-stage samples Intuitivel large implies small (or ) whih then indiates that f is muh loser to f (or f) than to the observed samples of herefore the estimation result is similar to the prior information as shown in (44) Small then implies that the estimation based on observed samples of is

5 aurate so the estimation result tends to largel rel on late-stage samples as (45) shows It is important to find the optimal values of the hperparameters k and k to minimize the modeling error owards this goal we first set the value of as: 46 = λ min ( γ γ) (46) is a sale fator between 0 and sine we an see from (39)-(40) that should be no more than or In pratie we set lose to It is beause the number of late-stage samples is far less than the number of model oeffiients so that simpl estimating from late-stage samples would be ver inaurate whih then leads to large value of hen we use two-dimensional ross-validation [5] to find the optimal values for k and k based on few late-stage samples All ombinations of k and k within a pre-defined range are hosen as andidates For eah ombination we appl a Q-fold rossvalidation strateg We divide the entire set of data samples into Q groups and modeling error is estimated from Q independent runs At eah run Q- groups are used to alulate the model oeffiients and the remaining group is used to estimate the modeling error Different groups are seleted for error estimation in different runs After a omplete ross-validation proess of Q runs we then alulate the average error of the Q modeling errors and use it to indiate the estimation aura of the given hperparameters k and k he ombination with the least modeling error is then seleted as the optimal values of k and k 4 Highl Biased Prior Knowledge Given two soures of earl-stage information eah of whih an failitate the late-stage performane modeling we still need to onsider the senario when there is great ompetene disparit between the two soures of prior knowledge Namel when one soure of prior knowledge provides far more useful information for oeffiient estimation than the other one does Ideall the two soures of prior knowledge are equall ompetent and omplementar to eah other hus we an extrat more useful information for modeling and ahieve higher estimation aura than using single soure With two highl biased soures however we an expet that ross-validation will automatiall assign an extremel small weight value to the less useful prior information When k >> k for example the estimation result is then similar to (44) In this ase modeling b borrowing two soures of prior knowledge annot ahieve better result than using single prior knowledge E sine our result is alwas a ompromise between two soures and one of them now beomes a pure hindrane Fortunatel suh situation an be easil deteted from two signs he first sign is the values of or after appling two times of single-prior BF For example if is muh larger than then obviousl its orresponding prior knowledge E is far less useful than E he seond sign is the ratio of k to k after the two-dimensional ross-validation proess If the ratio is extremel high for instane then it implies we should trust muh more in E than in E beause E is useless If both signs show that the two soures of prior knowledge are highl biased then DP-BF annot do an better than traditional single-prior BF with the more ompetent soure as prior knowledge 43 Summar Algorithm summarizes the major steps of our proposed DP- BF method for performane modeling from two soures of prior knowledge Starting from two groups of given earl-stage model oeffiients as prior knowledge and a set of late-stage samples we first determine the values of hper-parameters b the method in Setion 4 he hper-parameters ontrol the weights of eah soure of prior knowledge and information from late-stage samples One the optimal values of hper-parameters are obtained we estimate the late-stage model oeffiients based on AP Algorithm :Dual-Prior BF for Performane odeling Start from two groups of existing oeffiients {Em; m = } and {Em; m = } and a set of late-stage samples {(x (r) (r) ); r = K} Run single-prior BF method as defined in Setion twie with {Em; m = } and {Em; m = } as prior knowledge respetivel hen estimate and from the fitting error of eah time 3 Determine the value of hper-parameter b (45) and the values of k and k b two-dimensional ross-validation hen alulate the values of and aording to (39) and (40) respetivel 4 Estimate the late-stage model oeffiients b (36)-(38) 5 NUERICA EXAPES In this setion we use two iruit examples to demonstrate the effiien of our proposed DP-BF method Our objetive is to build performane models for pre-silion verifiation of these iruits o illustrate the improvement b using multiple soures of prior knowledge we ompare three different performane modeling methods: (i) single-prior BF using one soure of prior information (ii) single-prior BF using another soure of prior information and (iii) the proposed DP-BF using both soures of prior information All experiments are performed on a server with 5GHz dual-ore CPU and 6GB memor 5 Operational Amplifier In this example we use a two-stage operational amplifier (Opamp) designed in a 45nm COS proess We onsider the postlaout verifiation as late-stage and generate the data samples b post-laout simulation he first soure of prior knowledge is from least-squares fitting from man existing data samples of shemati-level simulations On the other hand b exploiting the underling sparsit of model oeffiients we appl the sparse regression method [8] on a small set of obtained post-laout samples to get a group of oeffiients as the seond soure of prior knowledge We aim to auratel estimate the model oeffiients of ertain performane metris in post-laout simulation b borrowing the prior knowledge of two groups of the given oeffiients It is noted that the soures of prior knowledge are not restrited to these we use Other orrelated information from simulation/measurement data of different working modes different environment orners or previous time an also be reused as prior knowledge In this example we use 58 independent random variables to model the devie-level proess variations inluding both inter-die variations and random mismathes Our objetive is to approximate the offset of the Op-amp as a linear funtion of these 58 random variables For testing and omparison purpose we generate a set of onte-carlo samples b both shemati-level and post-laout simulations in whih the devie-level variations of all transistors are onsidered We use onte-carlo samples of shemati-level simulation to alulate the model oeffiients as prior information and appl the sparse regression method on 80 samples of postlaout simulation to get another group of oeffiients as prior information A group of another 000 post-laout simulation samples is used as test group to measure the modeling error Figure 4 shows the modeling error as a funtion of the number of late-stage samples he errors are alulated from 50 repeated runs based on independent samples to average out random

6 flutuations Note that the DP-BF method ahieves higher aura than traditional BF using single soure of prior information In partiular DP-BF onl needs 0 samples to ahieve high modeling aura However even the single-prior BF with better performane (denoted as Single-prior ) takes about 0 samples to ahieve the same aura Studing the plot reveals that DP-BF ahieves more than 83 ost redution over single-prior BF We also examine whether DP-BF an effetivel adjust the weights assigned to eah soure of prior knowledge In Figure 4 traditional BF ahieves higher modeling aura with the first soure of prior knowledge whih means that the first soure provides more useful information than the seond one his is refleted in the values of hper-parameters k and k determined b ross-validation he optimized ratio of k to k is relativel small for all the different sample numbers (eg k/k = 0 when post-stage sample number is 40) his implies that we trust the first soure of prior knowledge more beause it provides more useful information odeling error (mv) Single-prior Single-prior Dual-prior #Samples Figure 4 he modeling error is plotted as a funtion of the number of late-stage samples 5 Analog to Digital Converter In this example we onsider a flash analog to digital onverter (ADC) in a 08m COS proess he soures of prior information are the same as the previous example We use 3 independent random variables to model the devie-level proess variations he objetive is to approximate the power of the ADC as a linear funtion of these 3 random variables odeling error (mw) Single-prior Single-prior Dual-prior #Samples Figure 5 he modeling error is plotted as a funtion of the number of late-stage samples We use onte-carlo samples of shemati-level simulation to alulate the model oeffiients as prior information and appl the sparse regression method on 50 samples of post-laout simulation to get another group of oeffiients as prior information A group of another 000 post-laout simulation samples is used as test group to measure the modeling error Figure 5 shows the modeling error as a funtion of the number of late-stage samples Studing the plot reveals that DP-BF ahieves more than 83 ost redution over single-prior BF In addition in Figure 5 we an see that the seond soure of prior knowledge provides more useful information than the first one hat is wh the optimized ratio of k to k is obviousl larger than for all the different sample numbers (eg k/k = 44 when post-stage sample number is 58) his implies that we trust the seond soure of prior knowledge more beause it is more useful 6 CONCUSIONS In this paper we propose a novel performane modeling algorithm DP-BF o ahieve high modeling aura with low modeling ost an effiient Baesian inferene is developed to fuse two different prior models and ombine the prior information with a small number of training samples Several iruit examples demonstrate that the proposed method an ahieve up to 83 ost redution over the traditional method without surrendering an aura 7 ACKNOWEDGEENS his work is supported partl b National Natural Siene Foundation of China researh projet and partl b the Reruitment Program of Global Experts (the housand alents Plan) partl b Chen Guang projet supported b Shanghai uniipal Eduation Commission and Shanghai Eduation Development Foundation and National Siene Foundation under ontrat CCF and grant REFERENCES [] Semiondutor Industr Assoiate (0) International ehnolog Roadmap for Semiondutors [Online] Available: wwwiersnet/inks/0irs/home0htm [] A Singhee et al Beond low-order statistial response surfaes: latent variable regression for effiient highl nonlinear fitting IEEE DAC pp [3] A itev et al Priniple Hessian diretion based parameter redution for interonnet networks with proess variation IEEE ICCAD pp [4] Conagh et al emplate-free smboli performane modeling of analog iruits via anonial-form funtions and geneti programming IEEE rans on CAD vol 8 no 8 pp [5] X i et al Asmptoti probabilit extration for 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