An Improved AMG-based Method for Fast Power Grid Analysis

Transcription

1 An Improved AMG-based Metod for Fast Power Grd Anayss Ceng Zuo, Jang u 1 and Kangseng Cen Department of Informaton Scence & Eectronc Engneerng, Zejang Unversty, angzou, , Cna 1 Department of Eectrca Engneerng, Texas A&M Unversty, Coege Staton, TX {zuoceng, 1 jangu@ece.tamu.edu Abstract Te contnung VLSI tecnoogy scang eads to ncreasngy sgnfcant power suppy fuctuatons, wc need to be modeed accuratey n crcut desgn and verfcaton. Meanwe, te uge sze of power grd requres ts anayss to be fast and gy scaabe. Agebrac mutgrd (AMG) as been recognzed as a promsng approac for fast power grd anayss. We propose severa tecnques to mprove AMG-based power grd anayss: (1) dynamc reducton tresod, (2) wegted nterpoaton and (3) a new error smootng sceme. Expermenta resuts on power grd wt up to 1.6 mon nodes sow tat tese tecnques can mprove accuracy by over 10 tmes compared to a reported ndustra metod we retanng te same fast speed. 1. Introducton As VLSI tecnoogy enters utra-deep sub-mcron regme, decreasng power suppy eve and ncreasng devce densty resut n ncreasngy sgnfcant IR drop, eectro-mgraton and smutaneous swtcng nose. A of tese penomena may severey affect bot performance and reabty of ntegrated crcuts. Terefore, accurate power grd anayss s of paramount mportance for crcut desgn and verfcaton. In practce, a power grd may contan mons of nodes and suc uge sze demands g computaton effcency n te anayss. Tus, t s mperatve to ave a power grd anayss metod wt bot g accuracy and fast runtme. Te mportance of power grd anayss as attracted many researc attentons n bot academa and ndustry. Cen, et a, empoyed a precondtoned conjugate gradent (PCG) teratve sover for power grd anayss [1]. Zao, et a, proposed a erarcca macro-mode metod to ande te uge sze of power grd [2]. Qan, et a, deveoped a random wak based anayss metod [3]. Mutgrd s a popuar tecnque for sovng arge-scae dfferenta equatons. Kozaya, et a, apped standard (geometrc) mutgrd tecnque on power grd anayss [4]. In order to ande rreguar power grd structures, agebrac mutgrd (AMG) based tecnques are deveoped n [5, 6]. Te Su-Acar-Nassf metod [5] empaszed on te system reducton part of AMG so tat fast computaton speed s aceved. owever, t negected te smootng steps and resuted n some accuracy degradaton. Te AMG based power grd anayss of [6] foowed te compete AMG procedure wt smootng operaton at every eve. Ts approac can aceve better souton accuracy but te runtme mprovement becomes ess. Te work of [6] aso proposed an adaptve coarsenng sceme based on crcut actvtes to reduce computaton runtme. In ts paper, we propose severa mprovements to AMG-based power grd anayss to obtan a better tradeoff between accuracy and runtme. Te mprovements ncude: (1) Dynamc reducton tresod. Te reducton (coarsenng) operaton n AMG can reduce probem sze to a manageabe eve. Te reducton rate at eac eve depends on a certan tresod. Prevous works [5, 6] use a constant tresod troug a eves of reductons. We propose a dynamc reducton tresod sceme suc tat te probem sze can be furter reduced wtout accuracy degradaton. (2) Wegted nterpoaton. We mprove te nterpoaton of [5] by ncudng more, but not a of, strongconnected nodes. Suc nterpoaton can aceve a better compromse between accuracy and runtme. (3) New error smootng sceme. In contrast to [5], wc does not perform smootng at a, and [6] wc performs smootng at every eve, we propose to perform one teraton of pre-smootng at te begnnng and perform a mut-eve post-smootng at te end so tat te tradeoff between runtme and accuracy s furter mproved.

2 Experments are performed on bencmark power grds wt sze of up to 1.6 mon nodes. Te resuts sow tat our metod can mprove accuracy by over 10 tmes compared to Su-Acar-Nassf metod [5] we retanng te same fast speed. 2. Crcut Mode for Power Grd In modern VLSI crcut desgns, power grd s usuay a meta mes connectng te power pads and actve devces. Te power pads can be treated as dea votage sources. Te actve devces (or te oads) are usuay modeed as dstrbuted tme-varyng current sources n parae wt grounded capactance [7]. Te meta mes s usuay modeed as a resstve mes wt RL eements nked to te package. Appyng Modfed Noda Anayss to ts crcut mode, one can reac te foowng equaton [1]: T G A () 0 '() ( ) xt C x t u t A 0 b() t + = 0 b'() t L 0 (1) were x() t s te node votage vector, b ( t) s te branc current vector and ut () ndcates te votage sources and current drans. By usng Backward Euer metod, we ave: T G A ( ) 0 ( ( ) ( ))/ ( ) xt+ C xt+ xt u t+ A 0 b( t ) + = 0 L ( b( t ) b( t))/ From (2), we can obtan T 1 T ( G+ C/ + A L A) x( t+ ) = x( t) C/ + u( t+ ) + A b( t) (3) 1 and b( t+ ) = L Ax ( t+ ) + b( t) (4) We may rewrte (3) as Ax= b, were A s defned as T 1 G+ C/ + A L A. Snce L 1 (or K ) s symmetrc postve defnte (S.P.D.) [8], matrx A s aso S.P.D. Terefore, te core part of power grd anayss s to sove te near system Ax = b. If ony RC s consdered, equaton (2) can aso be smpfed as Ax= b, were A =G + C/, b = ut ( + ) + xtc ( ) /. 3. Revew of Mutgrd Metod Mutgrd s orgnay deveoped for acceeratng convergence rate wen sovng dfferenta equatons numercay. It ncudes two key operatons [9, 10]: (1) Smootng: usng tradtona teratve metods to smoot g frequency errors; (2) Coarse grd correcton: emnatng ow frequency errors on coarse grds. Te orgna probem s successvey reduced (coarsened) to coarse grds troug restrcton operatons and te soutons on te coarse grds are ten mapped back to fne grds troug nterpoatons [9, 10]. (2) 4. An Improved AMG-based Metod for Power Grd Anayss 4.1. Overvew Drecty appyng a genera-purpose AMG sover to te power grd anayss s not computatonay effcent because te pre-smootng and post-smootng at eac coarse grd correcton eve are very tme consumng. Instead, we propose a customzed AMG metod for power grd anayss, wc s summarzed n Fgure 1. In ts new approac, a wegted Jacob ( w Jacob ) based presmootng s performed ony once at te begnnng. Ts pre-smootng can sgnfcanty mprove souton accuracy we ony one teraton of pre-smootng as mted mpact on runtme. Te oter customzed mprovements w be descrbed as foows. Step1:Intazaton. Use MNA and reformuate te dscretzed equatons to get A x=b Step2: Pre-smooter. One w-jacob teraton to obtan an nta souton x 0. Step3:AMG-based mut-eve reducton to obtan system matrx A on te coarsest grd. Step4: Reorder te system matrx A, and sove A x =b usng Coesky factorzaton based metod. Step5:Interpoaton.Map te souton to te orgna grd: x=px and x=x+x0 Step6:Optona post-smootng for DC anayss Fgure 1. Proposed AMG-based power grd anayss 4.2. Interpoaton Operator Interpoaton s a crtca step n AMG metods. In ts subsecton, we w ntroduce an nterpoaton operator tat can provde desred compromse between accuracy and runtme for power grd anayss. Snce te system matrx A s S.P.D., te restrcton operator s smpy te transpose of te nterpoaton operator. For te convenence of te presentaton, we defne te foowng notatons for reducton (coarsenng) from one eve to a coarser grd eve: C={te set of nodes remaned n coarse grd}, F ={te set of nodes removed from fne grd n coarsenng}, N ={negborng nodes for node }, S ={node 's strong connected node set}, P ={node 's nterpoaton node set}. In te process of reducton or coarsenng, some nodes (or varabes) are removed we te oters are retaned n te coarsened grd. At te begnnng, C = F =. Te nodes w be vsted n an arbtrary and pre-fxed order. Usuay, crtca nodes suc as power pads, crtca oads

3 and corner nodes, are kept n te coarse grd or added nto set C [5]. Weter or not a non-crtca node s kept depends on connecton strengt between nodes [5]. Same as n [5], te connecton strengt between node j and node s defned as: Strengt = aj / a + aj / a jj /2 (5) were a j s te eement at te t row and te j t coumn of te system matrx. For a node, ts negborng nodes wt connecton strengt greater tan a certan tresod φ (usuay n te range ) [10] form te set S. Te node remova s based on eac S and fewer nodes w be removed f φ s very g. If a node s removed durng coarsenng, ts smoot error can be nterpoated from ts strongy connected nodes [10]. Usng too few negborng nodes for nterpoaton may degrade accuracy we usng too many nodes may cost arge runtme and memory overead. In [5] ony one negborng node, wc as te strongest connecton, s utzed for nterpoaton. Te cassca AMG metod uses a te nodes n S [10]. In order to aceve better compromse between accuracy and computaton cost, we propose to perform nterpoaton based on a subset of S defned as: P = { j Strengt max{ Strengt} ε, max{ Strengt} } S S (6) were ε s an emprcay cosen constant wt vaue of about 0.001~ We just coose te nodes wose connecton strengt s cose to te max{ Strengt} and S terefore ave te most sgnfcant nfuence to te removed node. Ts works especay we wen many nodes ave smar strengt. Snce ε s sma, te nterpoaton wegt can be smpy set as 1/ P, were P s te number of nodes n te set P. Ten F = F {}, C = C N. For a removed node, a of ts negborng nodes w be kept n te coarse grd. Fgure 2. A smpe exampe of RC mode Ts nterpoaton s ustrated n a smpe exampe n Fgure 2. Node s at te center and ts four negborng nodes are j 1, j, j and j so tat N = { j1, j2, j3, j4}. Assumng tat a of te four negborng nodes j1, j2, j3, j 4 ave strong connecton wt node, we can obtan te nterpoaton operator n Fgure 3. Based on te nterpoaton operaton obtaned as above, we can obtan te nterpoaton operator P as we as te coarsened T system matrx ( P) A P. Ts procedure s performed teratvey t te matrx s sma enoug for drect sove. j1 j2 j3 j4 j j /4 1/4 1/4 1/4 j j Fgure 3. Te nterpoaton matrx 4.3. Dynamc Reducton Tresod Te reducton rate of coarsenng eavy depends on te tresodφ. Te reducton rate can be quantfed as te reducton rato between two consecutve teratons: number of nodes at prevous teraton rato=. If te tresod number of nodes at current teraton φ s too ow, te reducton rato s very g durng te frst few teratons and te accuracy s degraded. In ater teratons, te node degree ncreases rapdy due to te aggressve reducton. Consequenty, te matrx qucky becomes very dense and te reducton dramatcay sows down. Terefore, a too ow tresod may urt bot accuracy and convergence rate. If te tresod s too g, t s qute key tat te sets S for some nodes are empty and very few nodes are removed. Weter a tresod s too ow or too g depends on te densty of te system matrx durng te mut-eve coarsenng. Prevous AMG based power grd anayss works [5, 6] use a constant tresod trougout a eves of coarsenng. Wen te system matrx canges durng coarsenng, te constant tresod may sometmes be too ow or too g. In order to sove ts probem, we propose a dynamc tresod sceme suc tat a stabe reducton rate s retaned n a eves of coarsenng. At te begnnng, te tresod s set to 0.2 wc s an emprcay good vaue empoyed n te prevous work [5]. After te frst eve of coarsenng, te tresod φ s determned by an emprca functon: φter = f ( ratoter 1, φter 1) = 0.15, φter < , φter > 0.25 (7) [ urato ( ter 1-1.5) - 0.5]*( 2* φter ) * k2 ratoter k1 *( e -1), oterwse were ut () s unt step functon, k 1 and k 2 are emprcay cosen as and 12. By dong so, te reducton rato can be stabzed n range of trougout a eves of coarsenng.

4 Tabe 1. Comparson between dynamc and constant reducton tresod #nodes after Crcut #nodes NNZ #ter. reducton NNZ after reducton Reducton rate Dynamc Const. Dynamc Const. Dynamc Const. 1 10K 49K 3 3.7K 4.7K 44K 42K 63.40% 52.83% 2 32K 155K 6 6.2K 14.8K 89K 132K 80.97% 54.23% 3 112K 557K K 54.1K 231K 483K 86.49% 51.72% 4 450K 2.25M K 222.1K 689K 1.98M 93.31% 50.67% 5 1M 4.97M K 493.0K 983K 4.40M 91.18% 50.71% 6 1.6M 8.10M K 803.6K 1.09M 7.17M 94.94% 43.98% 4.4. Post-smootng n DC Anayss If a crcut as a severey uneven current dstrbuton, ony one pre-smootng s not adequate for emnatng g frequency errors n DC anayss. Terefore, we propose an optona post-smootng sceme to sove ts probem as ndcated by step 6 n Fgure 1. Te postsmootng s not necessary for transent anayss snce pre-smootng s performed at every tme step and te errors are normay smoot. Moreover, performng postsmootng at every tme step of transent anayss may greaty sow down runtme. A naïve approac for post-smootng s to drecty perform a genera teratve metod suc as Jacob. In order to mprove te convergence rate, we propose a new post-smootng sceme nspred by te W-cyce stye mutgrd metod [9]. Our proposed post-smootng sceme s outned n Fgure 4. k s defned as te kt teraton of te sceme. Usuay, ts procedure can be carred out 5-7 tmes n DC anayss to obtan a resdua norm around 1e-4. In fact, t can reac any requred accuracy f more teratons are performed. Step1: Obtan resdua on te orgna grd b =b -A *xk-1. Step2: Map te resdua to coarse grd and sove te system on coarse grd to obtan ek. Step3: Map te souton back to te orgna grd and correct te souton xk-1,x k-1 =x k-1 +P *ek Step4: One w-jacob teraton to get a corrected souton xk Fgure 4. Proposed post-smootng fow We ave te foowng emma about te convergence of te proposed post-smootng fow. Lemma1: System matrx A s S.P.D. and te smooter error operator s S e = I MA were M s te smooter. * * 2 2 If Pe P e Pe s satsfed, te proposed postsmootng fow converges faster tan ony performng a c c c genera teratve sover. Proof: omtted due to page mt. P s te deay perfect nterpoaton operator [10] and * ec s te correspondng error at te coarsest eve. P s te actua nterpoaton operator empoyed n practce and e s ts correspondng error at te coarsest eve. As for * c * * 2 2 te condton * * Pe P e Pe, notcng tat e c c c c, P are just te approxmaton of e c and P respectvey. And * * Pe c P ecs actuay very cose to zero foowng our proposed metod, so t s easy to satsfy ts condton n practce Drect Lnear System Sover Te near system on te coarsest grd s soved drecty as ndcated by step 4 of Fgure 1. A supernode based Coesky factorzaton metod [11] s empoyed for te drect sove. Te supernode tecnque can mprove te effcency of factorzaton due to te foowng advantages [11]: (1) Reduce te cance of ndrect addressng wc s usuay neffcent; (2) Utze cace effcenty; (3) Represent te factorzed matrx s sparse structure n a smpe manner. We aso use te reorderng tecnque (METIS) [12] to reorder A to mprove te computaton speed before performng Coesky factorzaton. 5. Expermenta Resuts Te proposed metod s mpemented n C anguage wt TAUCS [13]. Te experments are carred out on PC wt Pentum IV 2.6G CPU 1GB memory and WnNT operatng system. Sx bencmark power grds are empoyed for te experment. A of tem are n mes structure wt some oca rreguartes. Te sze (number of nodes) of eac power grd and te NNZ (number of non-zero eements) of correspondng system matrx are sted n coumn 2 and 3 of Tabe 1, respectvey. One can see tat te argest crcut as 1.6 mon nodes. Te nomna VDD eve s 1.8V. In our mpementaton of w Jacob, w s emprcay cosen as Dynamc vs. Constant Reducton Tresod In order to test te effect of te dynamc reducton tresod sceme ntroduced n Secton 4.3, we run experments to compare metods wt dynamc and constant reducton tresod (=0.2). Te resuts are sown n Tabe 1 and Tabe 2. In Tabe 1, te reductons

5 Tabe 3. Grd reducton resuts and CPU tme comparson on DC anayss Crcut #ter. # nodes after NNZ after CPU tme (sec) reducton reducton Our SAN Our SAN Our SAN SPICE PCG metod [5] metod [5] metod [5] K 4.3K 44K 32K K 12.7K 89K 76K K 30.1K 231K 209K 7.74 X K 116.0K 689K 802K X K 223.0K 983K 1.66M X K 396.3K 1.09M 2.67M X on te number of nodes and NNZ after te same number of teratons (coumn 4) are compared. Te resuts n coumn 5 and coumn 6 of Tabe 1 te tat usng dynamc reducton tresod can resut n greater node reducton, especay for arge crcuts. For te argest crcut (crcut 6), te reducton from usng dynamc tresod s amost ten tmes greater tan usng constant tresod. Smary, te data n coumn 7 and coumn 8 of Tabe 1 ndcate tat dynamc tresod may yed muc greater reducton on te number of non-zero eements (NNZ) for arge crcuts as we. Te computaton accuracy and runtme are compared n Tabe 2. Its data sows tat usng dynamc tresod can reac about te same accuracy as constant tresod wt remarkaby ess runtme for arge crcuts. Tabe 2. Accuracy and runtme comparson between dynamc and constant reducton tresod Maxmum CPU tme (sec) Crcut absoute error (V) Dynamc Const. Dynamc Const X X X X to SAN [5]. Te stogram n Fgure 5 aso sows tat te errors of our metod are concentrated around zero we te errors of SAN [5] are muc greater and spread out. Tabe 4. Accuracy comparson for DC anayss Maxmum Maxmum Crcut absoute error (V) reatve error Our SAN Our SAN metod [5] metod [5] Expermenta Resuts on DC Anayss For DC anayss, frst te node reducton rate s compared between our metod and Su-Acar-Nassf (SAN) metod [5] n te coumn 3-6 of Tabe 3. It s evdent tat our metod usuay as muc greater reducton rates on bot grd sze and NNZ. In te coumn 7-10 of Tabe 3, te runtme of our metod s compared wt SPICE and pre-condtoned conjugate gradent (PCG) metod n addton to SAN [5]. One can see tat bot our metod and SAN [5] run sgnfcanty faster tan PCG we SPICE cannot fns for many crcuts. Te man advantage of our metod s to aceve muc better accuracy at te same fast speed compared to SAN [5]. Ts s evdenced by te maxma error n Tabe 4 and error dstrbuton n Fgure 5. Te resuts n Tabe 4 ndcate tat our metod can mprove accuracy by ten tmes compared Fgure 5. Te error stograms of te proposed metod n DC anayss 5.3. Resuts on Transent Anayss We aso performed transent anayss usng our metod and compared te resuts wt SAN [5]. Te souton from a PCG sover s empoyed as a basene for evauatng te accuracy on crcut 1-4. We ran bot metods wt 1000 tme steps (step sze = 5ps) for te frst fve crcuts and 200 tme steps for te ast crcut. Tabe 5 compares te runtme and accuracy of te two metods and t s cear tat our metod can obtan a muc ger accuracy at about te same computaton speed. We recorded te maxma error at eac node durng te transent anayss for crcut 4. Fgure 6 dspays to

6 te stogram of te errors for bot metods. It s easy see tat te errors from our metod are amost around zero and are about one order of magntude ower tan te errors from SAN [5]. Te error dstrbuton of our metod as a mean vaue of 7.38*10e-4 and a standard devaton of 2.97*10e-4, we te error dstrbuton of SAN [5] as a mean of and a standard devaton of Te waveform of a node n crcut 4 s smuated by bot our metod and SAN [5]. Te smuated waveforms are depcted n Fgure 7. One can see tat te waveform obtaned from our metod matces te exact waveform muc better tan SAN [5]. Tabe 5. Transent anayss resuts on CPU tme and accuracy Crcut CPU tme (sec) Max error of our metod Max error of SAN [5] Our metod SAN [5] X X X X Fgure 6. Te error stograms of te two metods Fgure 7. Comparson of votage waveforms 6. Concuson In ts paper, we proposed an mproved AMG based fast power grd anayss metod. It can ande bot RC and RLC modes. A dynamc reducton tresod tecnque s deveoped to mprove te effcency of coarsenng n AMG. Te nterpoaton operator s mproved compared to prevous power grd anayss work. We aso ntroduced a new post-smootng sceme to mprove te computaton effcency for DC anayss. Compared to a recenty reported ndustra metod, our metod can mprove te accuracy by over ten tmes wt te same fast computaton speed. 7. Acknowedgement Te autors woud ke to tank Dr. Peng L for epfu dscussons. 8. References [1] T.. Cen and C. C-P. Cen, Effcent arge-scae power grd anayss based on precondtoned Kryovsubspace teratve metods, n Proc. DAC, 2001, pp [2] M. Zao, R.V. Panda, S. S. Sapatnekar, T. Edwards, R. Caudry and D. Baauw, erarcca anayss of power dstrbuton networks, n Proc. DAC, 2000, pp [3]. F. Qan, S. R. Nassf and S. S. Sapatnekar, Random waks n a suppy network, n Proc. DAC, 2003, pp [4] J. N. Kozaya, S. R. Nassf and F. N. Najm, A mutgrd-ke tecnque for power grd anayss, IEEE Trans. CAD, Vo. 21, No. 10, Oct. 2002, pp [5]. Su, E. Acar and S. R. Nassf, Power grd reducton based on agebrac mutgrd prncpes, In Proc. DAC, 2003, pp [6] Z. Zu, B. Yao and C. K. Ceng, Power network anayss usng an adaptve agebrac mutgrd, In Proc. DAC, 2003, pp [7].. Cen and J. S. Neey, Interconnect and crcut modeng tecnques for fu-cp power nose anayss, IEEE Trans. Components Packagng II, Vo. 21, Issue 3, Aug. 1998, pp [8] A. Devgan,. J, and W. Da, ow to effcenty capture on-cp nductance effects: ntroducng a new crcut eement K, n Proc. ICCAD, 2000, pp [9] W. ackbusc, Mutgrd Metods, Scence Press, Bejng, Cna, [10] K. Stüben, Agebrac Mutgrd (AMG): An Introducton wt Appcatons, Guest appendx, n Mutgrd Metods, Academc Press, New York, 2000, aso avaabe as GMD Report 70, Nov [11] J. W.. Lu, E. Ng and B. W. Payton, On fndng supernodes for sparse matrx computatons, SIAM J. Matrx Ana. App., Vo.14, Issue 1, Jan. 1993, pp [12] ttp://www-users.cs.umn.edu/~karyps/mets/ [13] ttp://