Categories and Subject Descriptors B.7.2 [Integrated Circuits]: Design Aids Verification. General Terms Algorithms

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1 5. oward Eicient arge-scae Perormance odeing o Integrated Circuits via uti-ode/uti-corner Sparse Regression Wangyang Zhang entor Graphics Corporation Ridder Park Drive San Jose, CA 953 wangyan@ece.cmu.edu sung-hao Chen and ing-yuan ing entor Graphics Corporation Ridder Park Drive San Jose, CA 953 {howard_chen, ming_ting}@mentor.com Xin i Carnegie eon University 5 Forbes Avenue Pittsburgh, PA 53 xini@ece.cmu.edu ABSRAC In this paper, we propose a nove muti-mode/muti-corner sparse regression (SR agorithm to buid arge-scae perormance modes o integrated circuits at mutipe working modes and environmenta corners. Our goa is to eicienty extract mutipe perormance modes to cover dierent modes/corners with a sma number o simuation sampes. o this end, an eicient Bayesian inerence with shared prior distribution (i.e., mode tempate is deveoped to expore the strong perormance correation among dierent modes/corners in order to achieve high modeing accuracy with ow computationa cost. Severa industria circuit exampes demonstrate that the proposed SR achieves up to 85 speedup over east-squares regression [4] and 6.7 speedup over east-ange regression [7] without surrendering any accuracy. Categories and Subject Descriptors B.7. [Integrated Circuits]: Design Aids Veriication Genera erms Agorithms eywords Process Variations, Perormance odeing. INRODUCION As IC technoogy scaes to nanoscae region, process variation has become a critica issue that must be careuy addressed within today s IC design ow []. odeing, anayzing and optimizing process variation is now a critica task in order to guarantee high parametric yied or siicon chips. For this reason, various response surace modeing (RS techniques have been deveoped and used as an eicient method to anayze circuit variabiity []-[7]. he objective o RS is to approximate the circuit perormance (e.g., deay, gain, etc. as an anaytica (either inear or noninear unction o device parameters (e.g., V H, OX, etc.. Once these perormance modes are created, they can be used or many circuit anaysis and optimization appications [8]- [9], e.g., parametric yied prediction, robust circuit design, etc. Whie RS was extensivey studied in the past, severa recent trends o advanced IC technoogy pose a number o new chaenges in this area. Strong noninearity: As process variation becomes reativey arge, noninear (e.g., quadratic modes are required to accuratey capture perormance variabiity []-[7]. High dimensionaity: As random device mismatch becomes dominant at sub-65nm technoogy node, a arge number o random variabes must be used to mode device-eve variation, rendering a high-dimensiona variation space []-[7]. uti-mode/muti-corner operation: oday s integrated circuits oten operate at mutipe working modes (e.g., high perormance vs. ow power and mutipe environmenta corners (e.g., high temperature vs. ow temperature [], [5]. o accuratey capture perormance variabiity, dierent modes/corners must be characterized by dierent perormance modes. It, in turn, resuts in an enormousy arge modeing probem, as a huge number o perormance modes must be created to cover a modes/corners. he aorementioned issues make RS increasingy diicut. o accuratey mode and anayze perormance variabiity o anaog and mixed-signa integrated circuits, we have to buid hundreds o perormance modes and each mode contains thousands o unknown mode coeicients. Whie a number o new RS techniques have been recenty deveoped []-[7], they remain i-equipped to address the tremendous chaenges that we ace today. For instance, the east-ange regression (AR agorithm in [7] can it a high-dimensiona perormance mode with about 3 simuation sampes. I more than perormance modes are required to cover a working modes and environmenta corners (as wi be demonstrated by severa industria circuit exampes in Section 4, over 5 samping points must be generated by SPICE simuation to buid a these modes. his requires a huge amount o simuation time and, hence, suggests a need to re-think our undamenta strategy or RS and deveop a competey new modeing agorithm to accommodate such a arge probem size. In this paper, we propose a nove muti-mode/muti-corner sparse regression (SR method to address the aorementioned modeing chaenges. he key idea is to reduce the number o required simuation sampes and, hence, modeing cost by exporing the oowing two unique properties observed or nanoscae integrated circuits. Sparse mode coeicients: Whie a arge number o basis unctions must be used to span the high-dimensiona, strongy-noninear variation space, not a o them pay an important roe or a given perormance o interest. In other words, ony a sma subset o mode coeicients corresponding to the important basis unctions are non-zero, rendering to a unique sparse structure [7]. Correated perormance variabiity: he perormance modes associated with dierent modes/corners are not independent. Instead, since these modes capture the perormance variabiity o the same circuit [], [5], they are strongy correated. However, we do not know the ocation o the non-zero mode 897

2 5. coeicients or the correation vaue o the perormance variabiity in advance. he undamenta question is how to deveop an eicient agorithm that can accuratey extract this inormation rom a imited number o simuation sampes. he main contribution o this paper is to borrow a new Bayesian inerence recenty deveoped by statistics community [] to derive an eicient numerica agorithm or SR. his Bayesian ramework aims to buid a shared sparse mode tempate or a modes/corners so that their strong correation can be eectivey taken into account to improve modeing accuracy and/or reduce modeing cost. As wi be demonstrated by severa industria circuit exampes in Section 4, the proposed SR achieves up to 85 speedup over east-squares regression (SR [4] and 6.7 speedup over east-ange regression (AR [7] without surrendering any accuracy. he rest o this paper is organized as oows. In Section, the background on principa component anaysis, response surace modeing, and sparse regression is reviewed. he proposed mutimode/muti-corner sparse regression (SR wi then be described in Section 3. he eicacy o SR wi be demonstrated by severa industria circuit exampes in Section 4, oowed by our concusion in Section 5.. BACGROUND.. Principa Component Anaysis Given N process parameters X = [x x... x N ], the process variation X = X X, where X denotes the mean vaue o X, is oten modeed by mutipe zero-mean, correated Norma distributions []-[7]. Principa component anaysis (PCA [] is a statistica method that inds a set o independent actors to represent the correated Norma distributions. Assume that the correation o X is represented by a symmetric, positive semideinite covariance matrix R. PCA decomposes R as []: R = U Σ U ( where = diag(,,..., N contains the eigenvaues o R, and U = [U U... U N ] contains the corresponding eigenvectors that are orthonorma, i.e., U U = I. (I is an identity matrix. PCA deines a set o new random variabes Y = [y y... y N ] :. 5 ΔY = Σ U ΔX. ( he new random variabes in Y are caed the principa components. It is easy to veriy that a principa components in Y are independent and standard Norma (i.e., zero mean and unit variance. ore detais on PCA can be ound in []... Response Surace odeing Given a circuit design, the circuit perormance (e.g., deay, gain, etc. is a unction o the process variation Y deined in (. RS approximates the perormance unction (Y as the inear combination o basis unctions []-[7]: ΔY α g ΔY (3 3 ( m m ( where { m ; m =,,...,} are the mode coeicients, and {g m (Y; m =,,...,} are the basis unctions (e.g., inear, quadratic, etc.. he unknown mode coeicients in (3 can be determined by soving the oowing inear equation at samping points: 4 G α = F (4 where 5 α = [ α α α ] (5 6 F = ( ( ( (6 [ ] ( ( ( ΔY g ΔY ( ( ( ΔY g ΔY ( ( g ( ΔY ( ( g ( ΔY g 7 g. (7 G = ( ( ( ( Δ ( Δ ( Δ g Y g Y g Y In (5-(7, Y (k and (k are the vaues o Y and (Y at the k-th samping point respectivey. he traditiona east-squares regression (SR [4] attempts to sove the east-squares soution or (4. Hence, the number o sampes ( must be equa to or greater than the number o coeicients (. Such an SR becomes intractabe, i is arge (e.g., 4 ~ 6. For this reason, a number o sparse regression techniques [6]-[7] were recenty proposed to address this computationa cost issue..3. Sparse Regression Whie a arge number o basis unctions must be used to span the high-dimensiona and strongy noninear variation space, ony a ew o them are required to approximate a speciic perormance unction o interest. Sparse regression is motivated by this observation. It assumes that the unknown vector in (4 contains a arge number o zeros and, hence, is sparse. o ind such a sparse soution, the oowing -norm reguarization probem: 8 minimize G α F α (8 subject to α λ is soved by east-ange regression (AR [7]. In (8, and denote the -norm and -norm o a vector, respectivey. he vaue o λ shoud be optimay determined by the cross-vaidation scheme described in [7]. It has been demonstrated that AR can it a high-dimensiona perormance mode with about 3 simuation sampes. It, however, remains i-equipped to address the muti-mode/muticorner modeing probem that we study in this paper. For instance, i more than perormance modes are required to cover a modes and corners, over 5 samping points must be generated by SPICE simuation to buid a these modes, resuting in unaordabe computationa cost. We wi deveop a new sparse regression agorithm in this paper that is particuary tuned or muti-mode/muti-corner appications. 3. UI-ODE/UI-CORNER PERFORANCE ODEING Our proposed muti-mode/muti-corner sparse regression (SR is aciitated by a nove Bayesian inerence that is derived rom advanced statistica theories [], [3]. In this section, we describe its mathematica ormuation and highight the noveties. 3.. Probem Deinition he objective o SR is to generate dierent perormance modes or dierent working modes and environmenta corners. oday s integrated circuits oten operate at mutipe modes/corners, e.g., high perormance vs. ow power, high V DD vs. ow V DD, high temperature vs. ow temperature, etc. It is diicut to accuratey capture the perormance variabiity at a these modes/corners by a uniied mode. Instead, mutipe modes must be generated to cover a modes/corners: 9 ΔY α g ΔY =,, (9 (( (, m m( (, where represents the tota number o modes/corners. Simiar to 898

3 5. (4-(7, simuation sampes can be generated or { ( (Y; =,,...,} and a number o inear equations can be created to sove the mode coeicients { (,m ; =,,...,; m =,,...,}: G α = F =,, ( where 3 G ( α g g = g ( ( ( (, [ ] ( α(, α(, α (, ( ( ( F( [ ( ( ( ] ( ( ( ΔY( g ΔY( g ( ( ( ΔY( g ΔY( g = ( = ( ( ( ( ΔY( ( ( ( ΔY( ( ( ( ( ΔY( g( ΔY( g ( ΔY(. (3 In (-(3, Y ( (k and ( (k are the vaues o Y and ( (Y at the k-th samping point o the -th perormance unction. One straightorward way to sove { (,m ; =,,...,; m =,,...,} is to consider each ( (Y independenty and sove a inear equations in ( one by one. his simpe approach, however, is not optima, since it competey ignores the correation among the perormance unctions { ( (Y; =,,...,}. Such correation exists, because these perormance unctions capture the perormance variabiity o the same circuit. It, in turn, motivates us to deveop a new muti-mode/muti-corner sparse regression (SR agorithm that expores the extra correation inormation to improve modeing accuracy and/or reduce modeing cost. his goa is achieved by adapting a nove Bayesian inerence recenty deveoped by statistics community [] to buid a shared sparse mode tempate or a modes/corners. In what oows, we irst present the Bayesian ramework or sparse regression, and then show how to adapt and appy such a Bayesian inerence to muti-mode/muti-corner perormance modeing probems. 3.. Bayesian Inerence Given a set o simuation sampes {G (,F ( ; =,,...,}, most traditiona regression methods []-[7] sove the inear equations in ( and ind the deterministic vaues o a mode coeicients { ( ; =,,...,}. Bayesian inerence, however, takes a competey dierent strategy. It considers a unknown mode coeicients as random variabes and appies Bayes theorem to ind the probabiity density unction {pd( ( G (,F ( ; =,,...,}. he distribution pd( ( G (, F ( contains important inormation about ( that is earned rom the simuation sampes {G (,F ( ; =,,...,}. It tes us: ( the vaue o ( that is most ikey to occur (i.e., determined by the mode o the PDF, and ( the uncertainty o ( that corresponds to perormance modeing error (i.e., determined by the covariance matrix o the PDF. A. Prior Deinition o ormuate a Bayesian inerence, we irst need to deine a so-caed prior distribution or { ( ; =,,...,}. Intuitivey, the prior distribution represents our prior knowedge about { ( ; =,,...,} without seeing any simuation sampes. In genera, i we do not have any prior inormation, a uniorm distribution over (, + can be used to mode the prior distribution {pd( ( ; =,,...,}, impying that each coeicient (,m can possiby take any vaue with equa probabiity. However, in our appication o sparse regression, we know that the coeicient vector ( in ( is sparse. Hence, we shoud deine an appropriate prior distribution to carry this unique inormation o sparsity. o this end, we borrow the idea o hierarchica Bayesian inerence described in []. We irst mode the regression error as a zero-mean Norma distribution. Eq. (9 is re-written as: 4 ΔY = α g ΔY + ε =,, (4 (( (, m m( ( (, where ( represents the regression error and its PDF is: 5 θ(, θ(, ε ( pd ( ε ( θ(, = exp ( =,,,. (5 π In (5, (, is the precision o the Norma distribution. It is equa to the inverse o the variance. In our perormance modeing appication, the precision (, represents the accuracy o the -th perormance mode. he vaue o (, can be estimated by a maximum ikeihood method, as wi be discussed in detai in Section 3..C. Next, we deine the prior distribution o each (,m as a parameterized, zero-mean Norma distribution: θ (, m θ (, m α (, m pd ( α θ = exp π (6 6 (, m (, m ( =,,, ; m =,,, where (,m is the precision o the Norma distribution. he key idea o the hierarchica Bayesian inerence in [] is to assign appropriate vaues to the parameters { (,m ; m =,,...,} so that the coeicient vector ( has a sparse soution. Intuitivey, i the precision (,m is arge (i.e., variance is sma, the corresponding mode coeicient (,m is ikey to be zero. Otherwise, i the precision (,m is sma (i.e., variance is arge, the corresponding (,m can be ar away rom zero. However, we ony know that the coeicient vector ( is sparse, but we do not know the exact ocation o zeros. For this reason, it is not trivia to determine the optima vaues o { (,m ; m =,,...,}. In Section 3..C, we wi present a maximum ikeihood method to address this issue. o compete the deinition o the prior distribution, we assume that the random variabes ( and { (,m ; m =,,...,} are mutuay independent. Hence, the joint PDF o { (, ( } is: 7 pd ( ε (, α( θ(,,, = pd ( ε ( θ(, pd ( α(, m θ(, m. (7 ( =,,, he independence assumption we made simpy impies that we do not know the correation between ( and { (,m ; m =,,...,} in advance. Such correation inormation wi be considered, once the simuation sampes {G (,F ( ; =,,...,} are avaiabe to cacuate the posterior distribution. B. Perormance Correation he prior distributions deined in (5-(7 promote a sparse soution o { ( ; =,,...,}, i the vaues o { (,m ; =,,...,, m =,,...,} are propery seected. For our proposed mutimode/muti-corner sparse regression (SR, we expect that a perormance modes corresponding to dierent working modes and environmenta corners are simiar and, hence, strongy correated. Such correation inormation shoud be expicity expored to improve modeing accuracy and/or reduce modeing cost. In this sub-section, we urther present a systematic methodoogy to take into account the strong correation among a perormance modes. A cose ook at the prior distributions (5-(7 motivates severa additiona assumptions that we can make to expore the simiarity among { ( (Y; =,,...,}. First, we assume that 899 3

4 5. the vaues o { (, ; =,,...,} in (5 are identica: 8 θ θ = = θ =. (8 (, = (, (, θ Eq. (8 impies that the error o a perormance modes oows the same Norma distribution with identica variance. (Note that even with the same prior distribution {pd( ( ; =,,...,}, the regression error ( can sti vary or dierent perormance modes. his assumption is typicay vaid, especiay i a proper scaing is appied to a perormance unctions (e.g., dividing each perormance unction by its nomina vaue. Next, we urther assume that the vaues o { (,m ; =,,...,} in (6 are identica: 9 θ (, m = θ (, m = = θ(, m = θ m ( m =,,,. (9 Remember that the vaues o { (,m ; =,,...,, m =,,...,} indicate the importance o the corresponding basis unction g m (Y or the -th perormance unction ( (Y. Hence, Eq. (9 simpy impies that a particuar basis unction g m (Y may either simutaneousy impact a perormance unctions or aect no perormance unction at a. In other words, a perormance unctions share the same mode tempate, because they represent the behavior o the same circuit at dierent modes/corners. It is important to note that { (,m ; =,,...,, m =,,...,} are not the unknown mode coeicients { (,m ; =,,...,, m =,,...,} and, hence, Eq. (9 does not necessariy ead to identica mode coeicients. he soution o { (,m ; =,,...,, m =,,...,} wi be determined by cacuating the posterior distribution o the proposed Bayesian inerence, as wi be discussed in detai in Section 3..C. he equaity constraints deined by (8-(9 oer a systematic approach to expore the correation among a perormance unctions. In other words, the same prior distribution {pd( (, (,, ; =,,...,} is now shared by a modes/corners, since they are deined by using the same parameters { m ; m =,,...,}. his unique modeing structure or SR enabes us to achieve superior modeing accuracy and/or cheap modeing cost over the traditiona sparse regression agorithms (e.g., east-ange regression [7], as wi be demonstrated by severa industria circuit exampes in Section 4. In addition, Eq. (8-(9 aso reduce the compexity o the proposed prior distribution. Namey, they reduce the number o independent parameters that deine the prior distribution. In what oows, we wi describe an eicient agorithm to determine the vaues o { m ; m =,,...,} rom a imited number o simuation sampes. Once { m ; m =,,...,} are ound, the prior distribution in (5-(9 is uniquey determined. C. Posterior Cacuation A critica component o the aorementioned Bayesian inerence is to determine the vaues o { m ; m =,,...,} so that the prior distribution o { ( ; =,,...,} is accurate. In this subsection, we wi irst present an eicient maximum ikeihood estimation (E method to sove { m ; m =,,...,}. Next, we wi urther show a maximum posterior method (AP to determine the vaues o the unknown mode coeicients { ( ; =,,...,}. Given a set o simuation sampes {G (, F ( ; =,,...,} coected or a perormance unctions, the key idea o E is to determine { m ; m =,,...,} or the shared prior distribution so that the samping points {G (, F ( ; =,,...,} are most ikey to occur. Namey, we aim to maximize the oowing conditiona probabiity: maximize θ,, θ pd ( F θ (,,,, ( I a simuation sampes {G (, F ( ; =,,...,} are created independenty, pd(f (,,,, can be represented as: pd F θ = pd F θ. ( ( (,,,, (,, = ( aking the ogarithm o ( resuts in the og-ikeihood: og pd F θ = og pd F θ. ( [ ( (,,,, ] ( = [ ( ],, Since og( monotonicay increases, maximizing the PDF in ( is equivaent to maximizing the og-ikeihood in (, yieding: 3 maximize og pd θ. (3 θ,, θ = [ ( ] F(,, o sove (3, we need to ind an eicient way to cacuate the og-ikeihood unction. owards this goa, we appy the Bayes theorem [3]: 4 pd α pd ( α ( θ,, pd ( F( α (, θ,, ( ( F(, θ,, = pd ( F θ (,,. (4 By simpy rearranging (4, we have: 5 pd ( ( ( ( α θ,, pd ( F( α(, θ,, pd F θ,, =. (5 pd ( α( F(, θ,, Eq. (5 impies that we can easiy cacuate pd(f (,, i we know pd( (,,, pd(f ( (,,, and pd( ( F (,,,. First, we note that pd( (,, is the prior distribution or (, as deined in (6-(7 and (9: 6 θ θ ( m α m, m pd ( α ( = θ,, exp. (6 π Second, pd(f ( (,,, is reerred to as the ikeihood unction. Based on the perormance mode in (4-(5, we know that once ( and { m ; m =,,...,} are ixed, pd(f ( (,,, is a Norma distribution determined by the modeing error [], [3]: 7 θ θ G( ( ( ( ( ( α F = (7 pd F α, θ,, exp π where is the number o simuation sampes or the -th perormance unction. Finay, pd( ( F (,,, is the posterior distribution or (. It modes the uncertainty o ( ater we observe the simuation sampes {G (, F ( }. Given the prior distribution in (6 and the ikeihood unction in (7, the posterior distribution pd( ( F (,,, is Norma and its covariance Σ ( and mean μ ( are respectivey equa to [], [3]: 8 Σ( ( ( ( = θ G G + A (8 9 μ ( = θ Σ( G( F (9 ( where θ 3 (3 A = θ is a diagona matrix. Combining (5-(9, it can be shown that the optimization in (3 is equivaent to: 3 og[ det( θ ( (] I + G A G + (3 minimize θ,, θ = F( ( + ( ( ( θ I G A G F where det( stands or the determinant o a matrix. Eq. (3 presents an anaytica orm o the cost unction that we shoud 9 4

5 5. minimize. Given this compicated cost unction, the optimization in (3 is not convex. An eicient greedy agorithm has been deveoped in [] to sove { m ; m =,,...,} rom (3. Even though goba convergence is not guaranteed, the greedy agorithm in [] has been demonstrated as a robust numerica sover or many practica probems. Due to imited space, we wi not discuss the numerica sover here. ore detais on the greedy agorithm can be ound in []. Once the parameters { m ; m =,,...,} are determined, the posterior distribution o ( is Norma and its mean μ ( and covariance Σ ( are speciied by (8-(9. Since a Norma distribution is peaked at the mean vaue μ (, the maximum posterior soution o ( (i.e., the vaue o ( that is most ikey to occur is exacty equa to μ ( : 3 α ( = μ( = θ Σ( G( F. (3 ( Eq. (3 soves the unknown mode coeicients { ( ; =,,...,} or our proposed muti-mode/muti-corner perormance modeing probem Summary As shown in Figure, the key idea o the proposed mutimode/muti-corner spare regression (SR is to appy the same prior distribution (i.e., mode tempate to a perormance unctions. Agorithm summarizes the major steps o the SR method. Starting rom a number o random samping points {G (, F ( ; =,,...,}, Agorithm irst soves the parameters { m ; m =,,...,} o the shared prior distribution by using the greedy agorithm described in []. Next, the maximum posterior soution o the unknown mode coeicients { ( ; =,,...,} is determined by Bayesian inerence. Due to the shared prior distribution appied by SR, the number o required simuation sampes or each perormance mode can be signiicanty reduced, as wi be demonstrated by industria circuit exampes in Section 4. Simuation Sampes Shared Prior Distribution ode ode ode G (, F ( G (, F ( G (, F ( pd( (, (,,, ode ( ( Coeicients ( Figure. he same prior distribution (i.e., mode tempate is shared by a perormance modes or muti-mode/muti-corner sparse regression (SR. Agorithm : uti-ode/uti-corner Sparse Regression. Randomy generate the samping points {G (, F ( ; =,,...,} or perormance unctions where samping points are associated with each perormance unction, as shown in ( and (3.. Deine the prior distribution or the mode coeicients { ( ; =,,...,} and the regression errors { ( ; =,,...,}, as shown in (5-(9. 3. Sove the optimization in (3 to determine the parameters { m ; m =,,...,} by using the greedy agorithm described in []. 4. Find the posterior distribution (which is Norma or the mode coeicients { ( ; =,,...,} using (8-(9. 5. Determine the maximum posterior soution o the mode coeicients { ( ; =,,...,} using (3. 4. NUERICA EXAPES In this section, we demonstrate the eicacy o SR using severa industria circuit exampes. A circuits are designed in a commercia COS process. For each circuit exampe, there are mutipe working modes and environmenta corners. One perormance mode is extracted or each o these modes/corners. owards this goa, two independent random samping sets, caed training set and testing set respectivey, are generated. he training set is used or coeicient itting (i.e., Agorithm, whie the testing set is used or mode vaidation. A numerica experiments are perormed on a 3GHz inux server. 4.. High-Speed Adder A ~A 7 B ~B 7 C Adder C 8 S ~S 7 Figure. Bock diagram o an 8-bit adder odeing Error (% SR # o raining Sampes per ode odeing Error (% Figure 3. inear perormance modeing error AR SR # o raining Sampes per ode abe. inear perormance modeing cost SR AR SR # Sampes per ode Simuation (Sec Fitting (Sec ota (Sec Figure shows the bock diagram o an 8-bit adder. In this exampe, we aim to mode the path deay rom the input A to the output S 7. Both goba variation and oca mismatch are taken into account. Ater PCA based on oundry data, 3876 independent random variabes are extracted to mode the variations. In addition, we consider the circuit to operate at 3 dierent V DD corners with 7 dierent input sopes at A and 8 dierent output oad capacitors at S 7, resuting in 68 dierent modes/corners. Such a mutimode/muti-corner modeing probem occurs in severa practica appications, e.g., statistica timing ibrary characterization [5]. Figure 3 shows the inear modeing error or three dierent techniques: east-squares regression (SR [4], east-ange regression (AR [7], and the proposed muti-mode/muti-corner sparse regression (SR. Given the same number o training sampes, SR yieds substantiay better accuracy than SR and AR. his is because SR expores the strong correation between dierent modes/corners by appying the same prior distribution (i.e., mode tempate, as shown in Agorithm. On the other hand, or a given accuracy requirement, SR requires much ess training sampes than SR and AR. Studying Figure 3, one woud notice that to achieve 5% modeing error, 9 5

6 5. SR ony requires 3 sampes per mode (68 modes in tota, whie SR and AR require 4 and 9 sampes per mode, respectivey. abe urther shows the computationa cost or the aorementioned three techniques. Note that SR is computationay cheaper than SR and AR in both simuation cost (i.e., to generate simuation sampes and itting cost (i.e., to sove mode coeicients. In this exampe, SR achieves 4 speedup over SR and 6.3 speedup over AR. 4.. Simpiied SRA Read Path Figure 4. Simpiied circuit schematic o an SRA read path odeing Error (% SR (inear # o raining Sampes per ode odeing Error (% AR (inear SR (inear SR (Quad # o raining Sampes per ode Figure 5. inear and quadratic perormance modeing error abe. inear perormance modeing cost SR AR SR # Sampes per ode 56 3 Simuation (Sec Fitting (Sec ota (Sec Shown in Figure 4 is the simpiied circuit schematic o an SRA read path. Ater PCA based on oundry data, 3345 independent random variabes are extracted to mode both goba variation and oca mismatch. Simiar to the previous exampe, we consider the circuit to operate at 3 dierent V DD corners with 7 dierent word ine sopes and 8 dierent output oad capacitors, resuting in 68 dierent modes/corners. In this exampe, we irst buid inear perormance modes or read path deay using three dierent techniques: SR [4], AR [7], and SR. Figure 5 shows the modeing error as a unction o the number o training sampes per mode (68 modes in tota. o achieve 3% modeing error, SR ony requires 3 sampes per mode, whie SR and AR require 56 and sampes per mode, respectivey. abe urther shows the computationa cost or these three modeing techniques. In this exampe, SR achieves 85 speedup over SR and 6.7 speedup over AR, as shown in abe. o urther improve accuracy, we seect a subset o important process parameters corresponding to non-zero inear mode coeicients. Next, we create quadratic perormance modes or these critica process parameters using SR. Figure 5 aso shows the quadratic modeing error or SR. Compared to simpe inear modeing, the proposed quadratic modeing scheme urther reduces modeing error rom % to % ( reduction, i a suicient number o simuation sampes (more than sampes per mode are avaiabe. 5. CONCUSIONS In this paper, we propose a nove muti-mode/muti-corner spare regression (SR method to eicienty generate arge-scae perormance modes o integrated circuits at mutipe working modes and environmenta corners. An eicient Bayesian inerence with shared prior distribution (i.e., mode tempate is appied to expore the strong perormance correation among dierent modes/corners to improve modeing accuracy and/or reduce modeing cost. Severa industria circuit exampes demonstrate that SR achieves up to 85 speedup over eastsquares regression (SR [4] and 6.7 speedup over east-ange regression (AR [7] without surrendering any accuracy. he proposed SR agorithm can be appied to a number o practica appications or both anaog and digita circuits such as statistica timing ibrary characterization and parametric yied estimation. 6. REFERENCES [] Semiconductor Industry Associate, Internationa echnoogy Roadmap or Semiconductors, 7. [] X. i, J. e,. Pieggi and A. Strojwas, Projection-based perormance modeing or inter/intra-die variations, IEEE ICCAD, pp. 7-77, 5. [3] Z. Feng and P. i, Perormance-oriented statistica parameter reduction o parameterized systems via reduced rank regression, IEEE ICCAD, pp , 6. [4] A. Singhee and R. Rutenbar, Beyond ow-order statistica response suraces: atent variabe regression or eicient, highy noninear itting, IEEE DAC, pp. 56-6, 7. [5] A. itev,. areat, D. a and J. Wang, Principe Hessian direction based parameter reduction or interconnect networks with process variation, IEEE ICCAD, pp , 7. [6] X. i and H. iu, Statistica regression or eicient highdimensiona modeing o anaog and mixed-signa perormance variations, IEEE DAC, pp , 8. [7] X. i, "Finding deterministic soution rom underdetermined equation: arge-scae perormance modeing by east ange regression," IEEE DAC, pp , 9. [8] F. Schenke,. Pronath, S. Zizaa, R. Schwencker, H. Graeb and. Antreich, ismatch anaysis and direct yied optimization by spec-wise inearization and easibiityguided search, IEEE DAC, pp ,. [9]. cconaghy and G. Gieen, empate-ree symboic perormance modeing o anaog circuits via canonica-orm unctions and genetic programming, IEEE rans. CAD, vo. 8, no. 8, pp. 6-75, Aug. 9. [] B. Foyd and D. Ozis, ow-noise ampiier comparison at GHz in.5-m and.8-m RF-COS and SiGe BiCOS, IEEE RFIC, pp , 4. [] S. Ji, D. Dunson and. Carin, uti-task compressive sensing, IEEE rans. Signa Processing, vo. 57, no., pp. 9-6, Jan. 9. [] G. Seber, utivariate Observations, Wiey Series, 984. [3] G. Casea and R. Berger, Statistica Inerence, Duxbury Press,. [4]. Hastie, R. ibshirani and J. Friedman, he Eements o Statistica earning, Springer, 3. [5] S. Sapatnekar, iming, Springer,