Fast Inductance Extraction of Large VLSI Circuits

Transcription

1 Fast Inductance Extacton of Lage VLSI Ccuts Hemant Mahawa Vvek San Texas A&M Unvesty College Staton, TX 83 Wepng Sh Abstact Accuate estmaton of sgnal delay s ctcal to the desgn and vefcaton of VLSI ccuts. At vey hgh feuences, sgnal delay n ccuts wth small featue szes s domnated by the paastc nductance and capactance of the nteconnect. Inductance extacton nvolves the soluton of lage, dense, complex lnea systems of euatons by pecondtoned teatve methods. Fast nductance extacton eues effectve, paallelzable pecondtones fo the system matx whch s avalable only mplctly va appoxmate heachcal matxvecto poducts. Ths pape pesents a novel algothm to solve these lnea systems by estctng cuent to a dscete solenodal subspace and solvng the educed system va an teatve method. A pecondtone based on the Geen s functon s suggested fo acceleatng the convegence of the teatve method. The pape outlnes a paallelzaton scheme fo matxvecto poducts wth the system matx as well as the pecondtone. Expemental esults ae pesented to show the advantages of the pecondtonng scheme ove exstng appoaches. The expements also llustate the paallel effcency acheved on the SGI Ogn2000 multpocesso. 1. Intoducton Wth nceasng clock speed and deceasng featue sze, the delays n VLSI ccuts ae playng a cucal ole. Fo the next geneaton mcopocessos wth moe than a bllon tansstos and hundeds of mllons of segment nteconnect, sgnal delays wll be domnated by the paastc esstance (R), capactance (C) and nductance (L) of the nteconnects. Moeove, as the opeaton feuency eaches the teahetz ange, nductance wll play an mpotant ole n tmng vefcaton. Ths wok has been suppoted n pat by NSF unde the gants NSF CCR 99800, NSFCCR , and NSFCCR Computatonal esouces wee povded by NCSA, Unvesty of Illnos at Ubana Champagn. Whle thee ae a numbe of algothms and softwae packages avalable fo estmatng the paastc R and C, thee s only one softwae package called FASTHENRY [5] fo nductance extacton. Snce most of these softwae packages ae avalable only fo unpocesso wokstatons, the sze of poblems that can be solved s seveely lmted. Thee s sgnfcant nteest n developng fast, paallel algothms fo nductance extacton of lage VLSI ccuts. The geneal technue fo nductance extacton dscetzes the conductos wth a mesh n whch edges epesent cuent cayng flaments. Potental s defned at each node and the dffeence between potental at adjacent nodes epesents the potental dop acoss the coespondng flament. The potental dop acoss a flament s due to ts own mpedance as well as pawse mutual nductance fom all othe flaments. The lnea systems epesentng ths elaton ae lage, complex and dense. These systems ae solved by teatve methods such as Genealzed Mnmum Resdual (GMRES) method [10]. At each teaton one needs to compute the poduct of a dense matx wth a vecto. These matxvecto poducts ae often computed by fast appoxmate heachcal methods such as Fast Multpole Method (FMM) and ts vaants [2, 8, 9, 11] and the sngula value decomposton method [6]. The success of the appoach depends on the ate of convegence of the teatve method whch can be acceleated by pecondtonng the system. In ths pape, we popose a novel appoach to solve the lnea systems asng n nductance extacton. Secton 2 descbes the dscetzaton of the ntegal euaton fomulaton fo magnetouasstatc analyss of a multconducto system. In secton 3, we popose a novel scheme called the solenodal bass method, to solve the lnea system and descbe an effectve pecondtonng scheme. Secton outlnes the paallel mplementaton of the poposed algothm and secton 5 pesents expemental esults.

2 S ^ _ ) S m N ~ ~ j j L B B l 2. Backgound and Poblem Defnton The nteconnect of VLSI ccuts s modeled as a mesh whch s subdvded nto smalle flaments fo moe accuate epesentaton. Wth the complex aangement of components n moden chps, ths epesentaton essentally maps to two dmensonal stuctue wth seveal layes of nteconnect stacked on top of each othe. The mpedance of an conducto geomety can be summazed by an mpedance matx. The column of the mpedance matx s detemned as follows: a unt cuent s sent though conducto l, and zeo cuent s specfed n othe conductos. The numecal value of the potental dffeence between the two ends of conducto k gves values of the matx element. The above pocedue s epeated s tmes to compute all columns of. The cuent densty at a pont s elated to the potental by the followng ntegal euaton [5] whee "! #" $&% '(*)+,./ ),1032 )5.6 (*8 (1) s the magnetc pemeablty, s a theedmensonal poston vecto, s the feuency,,./9), s the Eucldean dstance between and 1), and ;: =. The volume of conductos s denoted by 2 and ncemental volume wth espect to ) s denoted by > 2. A numecal soluton of (1) can be obtaned by dscetzng the conductos nto flaments 25A 25F. Assum CBDBEBE ng cuent flows along the length of the flament and cuent densty s constant wthn each flament, a lnea system of the followng fom s obtaned: G H IKMLCN PO whee R s an Q R dagonal matx of self esstances wth HTSUS WVYX SZ*[1S n whch V s the esstvty, X S s the length of the \ th flament, []S s the cosssectonal aea of the \ th flament, I s the vecto of banch cuents, and V s the vecto of banch potentals. Let ^ denote the unt vecto along the \ th flament 2. The elements of the nductance matx M ae gven by S`_ " $&% [1S*[3_ S B 1acbUdeCb'YagfEdehf ^, S / _g,!0y2 The cuent satsfes Kchoff s cuent law at each node accodng to the constants kj j N whee s the mn o banch ndex matx of flaments and nodes and N*l s the known banch cuent vecto of length NCl 032 _ B (2) wth nonzeo values coespondng to the souce cuents. Assumng O l s the unknown node potental vecto of length m, the followng lnea system must be solved p H K j p N O l pts N l A staghtfowad appoach to solve ths system nvolves emoval of N by a blockstep of Gaussan elmnaton. The esultng system s defned n tems of the unknowns O kj only, and can be expessed as G H O l When solvng ths system by an teatve method, each step nvolves the soluton of G H QKuLyx {z va an nne teatve scheme, esultng n expensve oute teatons. Futhemoe, wth ths knd of a complcated system matx, t s dffcult to fnd effcent pecondtones fo the oute solve. 3. An Iteatve Algothm fo Computng Inductance 3.1. The Solenodal Bass Appoach The solenodal bass method s used to solve the poblem p j p N O l whee H tk s the mpedance matx, s an p} s t I dense, symmetc and postve defnte matx, an t Im spase matx, and H s a dagonal matx. The flament cuent vecto N has sze, the node potental vecto O l has sze m, and the ght hand sde vecto has sze. The oveall system s complex and symmetc, wth the complex j pat estcted to Z. The vecto F s the esdual of the patcula soluton G N that satsfes the constants N l, and s gven by N ~ Such a patcula soluton, N ~, can easly be found by numbe of technues. The smplest way s to assgn 9 eƒ cuent to flaments on a path fom node wth nput souce cuent to the node wth output souce cuent (see, e.g., Fg. 1). Ths appoach gves a spase vecto N&. Snce a unt j cuent flowng n a closed loop n the mesh satsfes Kchoff s law, t also satsfes the constants mposed by. The solenod bass method uses these mesh cuents to epesent the unknown cuent I. Ths scheme s smla to the mesh cuent appoach poposed n [3, ] wth the excepton of the teatment of souce cuent. In addton, the pecondtonng poposed n secton 3.2 s moe poweful than those suggested n []. N8l&B s

3 p H H B ^ m " m s ^ Flament Node Cuent Souce whee. Ths educed system can be solved va a sutable teatve scheme such as the GMRES method. Once s obtaned, cuent s computed as N, and banch potental vecto s ecoveed by solvng the least suaes poblem O l šn B Fgue 1. Dscetzaton of a gound plane wth a mesh of flaments. Cuent flowng though the flaments must satsfy Kchoff s law at each node n the mesh. The bold lne ndcates a path fo cuent that satsfes bounday condtons. j j The null space of epesents a bass fo cuent that obeys Kchoff s law. Any fullank matx P T F5 ]ˆDF v9 Š, that satsfy can be used to compute I va the matxvecto poduct: N fo abtay Ž ˆDF v9 Š. A puely algebac appoach such as QR factozaton of B cannot be used to compute P due to the pohbtve cost of computaton and stoage. We defne a unt cuent flow n a closed loop as a local solenodal functon. Each such mesh cuent s epesented as a vecto and the set of these vectos foms the columns of P. The esultng system s spase due to localzed stuctue of the solenodal functons. Futhemoe, opeatons such as computaton and stoage of P and matxvecto poducts wth P, can be mplemented effcently n paallel. Restctng the cuent vecto to the subspace of P automatcally satsfes the k constant k Thus, we only need to solve N O j Afte elmnatng the banch potental unknowns O multplyng ths euaton wth, we get B B l by Euatng the eal and magnay components of (3), we get the followng = ealvalued = system = = = p p = H NU N (3) () The potental dffeence between two nodes s computed by addng the potental dops acoss flaments on any path connectng the nodes. The ght hand sde vecto N gves the potental dop acoss the flaments. Thus, t s not necessay to solve the above least suaes poblem to compute potental dffeence between any two nodes Pecondtonng the Lnea System To acceleate the convegence of the teatve method, one must devse obust and effectve pecondtonng schemes fo the lnea system (). Matxvecto poducts wth the system matx ae computed as a seuence of thee poducts: G H 8KuL œ Of these, the most tme consumng one s the poduct wth the dense matx whch s computed effcently va appoxmate heachcal methods such as those outlned n secton 3.3. The task of desgnng effectve pecondtones s made especally challengng due to the unavalablty of. Popula technues based on ncomplete factozatons of the system matx compute pats of the system matx selectvely, followed by ncomplete factozatons. These schemes tend to be expensve and lmted n the effectveness []. Accuate pecondtonng s ctcal to the success of the solenodal bass method. A pecondtone s sad to be optmal when the condton numbe of the pecondtoned system,.e., the ato of the lagest and the smallest sngula values, s bounded by a constant. In such a case, the numbe of teatons of GMRES ae fxed egadless of the feuency Ÿ and mesh wdth ž. The system matx can be analyzed usng the obsevaton that the matces and ae euvalent to dscete cul opeatos. Snce the system matx s domnated by the nductance matx at hgh feuency opeaton, we use the followng pecondtone fo (): whee s an $% p s B (5) matx defned as follows:, /, 0Y2 ) (6)

4 l l whee and ae the centes of loops and n the mesh. The pecondtonng step conssts of the matxvecto poduct x whch can be computed usng appoxmate heachcal technues dentcal to those used fo. Thee ae seveal advantages of ou pecondtonng appoach. The pecondtonng step eues a matxvecto poduct whch s elatvely nexpensve compaed to ncomplete factozaton based pecondtones. The latte nvolve ncomplete factozatons of patally computed and tangula solves whch ae expensve, especally on paallel platfoms. In addton, expemental evdence suggests that the pecondtone s obust and vey effectve fo the hgh feuency egme. Expements n secton 5 demonstate the numecal and computatonal supeoty of the pecondtone ove exstng technues. Ths pecondtonng scheme s beng extended to low feuency egme as well. Detals of the pecondtonng appoach ae pesented n a fothcomng pape [] Appoxmate Heachcal Methods fo Matx Vecto Poducts Accuate matxvecto poducts wth the dense matces and eue A opeatons whee s the sze of these matces. Thee ae a numbe of technues that can be used to explot the apd decay of the kenel wth dstance n (2) and (6) to compute appoxmate matxvecto poducts n D & o opeatons. These nclude heachcal technues such as BanesHut [1] and Fast Multpole Method (FMM) [2] whch use a tuncated sees appoxmaton of flament cuents wthn a localzed egon to estmate mpact on wellsepaated sets of flaments. The method of Banes and Hut eles only on flamentcluste nteactons to acheve an D &K computatonal bound fo unfom flament dstbutons. The Fast Multpole Method uses both flamentcluste and flamentflament nteactons to acheve an bound fo unfom dstbutons. In each case, educton n computatonal complexty s assocated wth decease n accuacy of the matxvecto poduct. We use a vaant of the BanesHut algothm developed specfcally fo the nductance extacton poblem. A heachcal uadtee s used to patton the flaments n the mesh. Intenal nodes of the tee epesent hypothetcal flaments cayng cuent eual to the sum of cuents n that subtee. The coodnates of these hypothetcal flaments ae computed as a weghted sum of coodnates of the subtee flaments n whch the weghts coespond to the magntude of cuent flowng n those flaments. Ths stategy explots the fact that the combned nductance effect of a cluste of flaments can be appoxmated by the nductance effect of a sngle flament placed at the cente of cuent of that goup. Ths appoxmaton s used to compute nductance on a flament whch s wellsepaated fom the cluste. Ths s analogous to the cente of mass concept n BanesHut algothm. The leaf nodes of the uadtee ae flaments of the mesh. Inductance on a leaf node s computed by tavesng the tee n a top down manne. A subtee s tavesed only f the coespondng box s not wellsepaated fom the leaf node. Wellsepaatedness s establshed by usng a dstance metc to detemnes f the leaf node s suffcently fa off such that the eo n the assocated tuncated expanson s below a theshold. The nductance due to flaments n a wellsepaated subtee s computed by dect nteacton wth the coespondng hypothetcal flament. Remanng nteactons consst of leafleaf nteactons whch consst of mutual nductance computaton between pas of flaments. Cae should be taken to ensue that the algothm doesn t check the wellsepaatedness ctea fo the box contanng the leaf node tself.. Paallel Fomulaton The man components of the algothm ae matxvecto poducts wth and, vecto addtons, nnepoducts and heachcal appoxmate matxvecto poducts wth the dense matces and. Paallelzaton of these opeatons eue that the undelyng mesh to be dstbuted acoss pocessos. Snce the computaton n each component s a functon of the submesh sze on each pocesso, a load balancng scheme should ceate almost eual szed pattons. On a pocesso machne, the mesh s dvded nto pattons and the th patton s gven to pocesso. The unknowns fo flament cuents, loop cuents, and potental defned on the edges, cells, and nodes, espectvely, wthn a patton esde on pocesso that owns the patton. Paallel matxvecto poducts wth and eue ntepocesso communcaton among pocessos wth neghbong submeshes. The choce of local solenodal functons leads to aj spase ª wth nonzeo patten smla to the adjacency matx of the mesh. The matxvecto poducts ª «and ª can be computed concuently n a manne smla to that of a poduct wth the dscete Laplace matx. One can patton a mesh acoss pocessos n seveal ways. The amount of ntepocesso communcaton needed n computng the matxvecto poducts vaes lagely on the mesh pattonng scheme. Fo unfom gds, pattonng acoss any one dmenson s the smplest appoach. One can acheve ths by a smple loop paallelzaton scheme by dstbutng the loop contol ndex whch coesponds to the dmenson along whch the gd s pattoned. The esultng communcaton fo a ddmensonal unfom gd nodes along each dmenson. Fo a 2D unfom gd A nodes along each

5 ³ Z ± A dmenson, the ntepocesso communcaton s. An optmal pattonng fo such a gd s obtaned by pattonng the gd acoss all the dmensons eually. Paallelzaton of the heachcal algothm fo matxvecto poducts wth and s the most challengng step. Seveal appoaches have been poposed fo these heachcal algothms and the pefomance analyzed extensvely. One can patton the uadtee acoss the pocessos by allocatng the subtees at an appopate level of the uadtee to ndvdual pocessos. The level can be chosen to ealze tasks of specfc ganulaty. Each pocesso ntalzes ts subtee, followed by a coopeatve dstbuton of tasks to ntalze the emanng nodes. Once the uadtee data stuctue has been ntalzed, pocesso s esponsble fo computng nductance fo all the flaments n ts subtee. In ths manne, the paallel fomulaton explots the concuency avalable n ndependent tee tavesals fo each flament. The flaments can be soted n a poxmtypesevng ode (a PeanoHlbet odeng) and mutual nductance computaton fo appopate flament clustes aggegated nto a sngle task. The mplementaton can be optmzed to mpove datalocalty acoss pocessos and elmnate false shang. 5. Expemental Results In ths secton, we show the effectveness of ou pecondtoned solenodal bass method fo the goundplane poblem. Ths benchmak poblem allows staghtfowad dscetzaton by unfom twodmensonal meshes wth vayng mesh wdth ž. Ths poblem exhbts the dependence of the condton numbe of the system matces on ž va the gowth n teatons of the unpecondtoned GMRES algothm. Such convegence behavo povdes an deal poblem to test the effectveness of the pecondtonng appoach fo vayng levels of efnement as well as opeatng feuences. The expements epoted hee nvolve successve efnng of the mesh fom &± &± flaments to ±² flaments n the feuency ange of 1GHz 1THz. A two dmensonal gound plane of sze 1cm 1cm s dscetzed by a mesh of \/ Q\ flaments, whee \ 32, Z 6 and 128. Each flament has length ³ \, beadth v, and thckness (all lengths ae measued n cm). Table 1 shows the numbe of teatons needed by the pecondtoned GMRES to solve the lnea system (). A toleance of v9 was specfed on the elatve esdual nom. The table shows that the ate of convegence s ndependent of the feuency and the mesh wdth ž ndcatng that the pecondtone s optmal fo the benchmak poblem. ±² Mesh Flament Feuency (n GHz) Sze Length v] ¹3º v]» v]½ T & ¹&º ± ±¼ ±¼ ± Table 1. Iteatons fo convegence of pecondtoned GMRES Compason wth FASTHENRY The benchmak poblem was used to compae the pefomance of solenodal bass method wth that of FAS THENRY, the only publc doman softwae fo nductance extacton. FASTHENRY uses mesh cuents to geneate a educed system whch s then solved by the pecondtoned GMRES method. Fast Multpole Method s used to compute matxvecto poducts wth dense matces. Pecondtones ae deved fom ncomplete factozatons of appoxmatons to the educed system. These appoxmatons ae computed by vaous technues to spasfy the nductance matx. The softwae allows pecondtones such as CUBE and SHELL, n whch the offdagonal nonzeos n each column of ae estcted to those esultng fom mutual nductance between flaments n the same box at a specfc level n the uad tee and mutual nductance between flaments wthn a specfc adus, espectvely. The DIAG pecondtone coesponds to the case when all mutual nductances ae gnoed. Fo the solenodal bass method, the BanesHut vaant descbed n secton 3.3 was used to compute poducts wth the system matx and pecondtone dectly wthout explctly computng these matces. The esultng mplementaton s a matxfee code n whch nethe the system matx no the pecondtone matx s eve computed. Ths educes the stoage euement consdeably, theeby allowng lage poblems to be solved. FASTHENRY Solenodal Mesh DIAG CUBE Method ¾ ¹&º ¹&º ±¼ ±¼ Table 2. Compason of the numbe of teatons needed fo convegence of the pecondtoned GMRES. Table 2 shows the numbe of teatons needed by pecondtoned GMRES to solve the lnea system. The fe

6 FASTHENRYDIAG FASTHENRYCUBE Solenodal Method Mesh Pec GMRES Total Pec GMRES Total GMRES Total ¹&º ¹&º ±¼ ±¼ Table 3. Compason of the executon tme (n seconds). GMRES Matvec Pecondtonng Total Pocessos Tme Speedup Tme Speedup Tme Speed up Tme Speedup 3± ¹ ² $ ± Table. Executon tme and speed mpovement fo a poblem wth mesh sze 6 the SGI Ogn obtaned on GMRES Matvec Pecondtonng Total Pocessos Tme Speedup Tme Speedup Tme Speedup Tme Speedup 3± ¹ ² $ ± Table 5. Executon tme and speed mpovement fo a poblem wth mesh sze obtaned on the SGI Ogn2000. uency of opeaton was 10GHz. A toleance of v9 was specfed on the elatve esdual nom of both the methods. FASTHENRY was allowed to use default values fo all othe paametes. A gowth n the numbe of teatons wth mesh sze ndcates a suboptmal pecondtonng scheme whch contbutes an addtonal facto towads the ncease n cost of solvng these systems as the mesh sze s nceased. In contast, the solenodal bass method s able to solve the systems n almost fxed numbe of teatons. Table 3 shows the tme taken to solve the same lnea systems. The table shows a beakup of tme nto the man steps: constucton of the pecondtone (Pec), soluton of the lnea system va pecondtoned GMRES (GMRES), and the total tme. In the case of solenodal method, no addtonal computaton s eued to constuct the pecondtone. All paametes ae dentcal to the expements n Table 2. Although the solenodal method s slghtly moe expensve pe GMRES teaton, the fxed numbe of teatons allows t to be compettve wth FASTHENRY. The modest pefomance of the pecondtones n FASTHENRY come wth a sgnfcant cost of computng the pecondtones themselves. As a esult, the oveall tme spent eued by the solenodal method s consdeably less than FASTHENRY. Ths compaatve advantage wll gow sgnfcantly wth lage mesh szes Paallel Pefomance The esults on paallel pefomance of the solenodal method ae pesented n Table and 5. These expements wee conducted on a 32 pocesso SGI Ogn2000 at NCSA, Unvesty of Illnos. The code was paallelzed wth OpenMP dectves. Executon tme and speedup n the man steps of the solenodal method ae epoted. The speedup s computed as the ato of the untme of the theaded veson wth multple kenel theads to that of the sngle thead veson. The column fo GMRES epots tme and speedup fo all opeatons n GMRES outne othe than

7 the matxvecto poduct and pecondtonng. Consdeng the dvese set of computatons nvolved n the GMRES algothm ncludng vecto opeatons and heachcal matxvecto poducts, the paallel pefomance of the code s vey good on up to 8 pocessos. On moe pocessos, the pefomance degades due the collectve effect of nceased communcaton n the heachcal matvec outne, multple nne poducts, and the pecondtonng outne. As expected, the paallel pefomance s magnally bette fo the lage poblem n Table 5 due to hghe computaton to communcaton ato pe pocesso. Hghe effcency s expected on lage numbe of pocessos fo ealstc poblems wth bllons of flaments. Pefomance can be futhe mpoved by usng the Fast Multpole Method nstead of the BanesHut vaant n whch cachefendly computaton can be nceased by selectng a hghe multpole degee paamete. Ths s accompaned by elatvely smalle ncease n communcaton whch leads to much bette paallel pefomance. In addton, ths has the desable effect of mpovng accuacy as well. 6. Conclusons [1]. Banes and P. Hut. A heachcal O(n log n) foce calculaton algothm. Natue, Vol. 32():6 9, Decembe [2] L. Geengad. The Rapd Evaluaton of Potental Felds n Patcle Systems. The MIT Pess, Cambdge, Massachusetts, [3] M. Kamon. Fast Paastc Extacton and Smulaton of Theedmensonal Inteconnect va Quasstatc Analyss. PhD thess, EE & CS Dept., MIT, [] M. Kamon and. R. Phlps. Pecondtonng technues fo constaned vecto potental ntegal euatons, wth applcaton to 3D magnetouasstatc analyss of electonc packages. In Poceedngs of the Coloado Confeence on Iteatve Methods, Apl 199. [5] M. Kamon, M.. Tsuk, and. Whte. FASTHENRY: A multpoleacceleated 3D nductance extacton pogam. IEEE Tansacton on Mcowave Theoy and Technues, Vol. 2(9): , Septembe 199. [6] S. Kapu and D. Long. A fast ntegal euaton solve fo effcent 3D extacton. In Poc. IEEE Intenatonal Confeence on Compute Aded Desgn, pages 8 55, Novembe 199. [] H. Mahawa and V. San. A class of pecondtoned teatve methods fo fast nductance extacton. Tech. Repot, Dept. of Compute Scence, Texas A&M Unvesty (n pepaaton). [8] K. Nabos and. Whte. Fastcap: A multpole acceleated 3D capactance extacton pogam. IEEE Tanscatons on Compute Aded Desgn, Vol. 10(11):1 159, Novembe [9] V. Rokhln. Rapd soluton of ntegal euatons of classcal potental theoy. ounal of Computatonal Physcs, Vol. 60(2):18 20, Septembe [10] Y. Saad. Iteatve methods fo spase lnea systems. PWS publshng company, Boston, massachusetts, [11] W. Sh,. Lu, N. Kakan, and T. Yu. A fast heachcal algothm fo 3D capactance extacton. In Desgn Automaton Confeence, pages , une Ths pape pesents a solenodal bass method fo nductance extacton of VLSI ccuts. The poposed appoach solves a lnea system of euatons to compute the mutual nductance effect on a flament mesh usng eddy cuents defned on the mesh loops. The esultng educed system s solved teatvely by the pecondtoned GMRES method. The pecondtone suggested fo the system matx exhbts convegence n a fxed numbe of teatons espectve of the mesh efnement and opeatonal feuency. Expemental esults ndcate that the seal pefomance of the algothm s supeo to exstng methods. Futhemoe, the paallel mplementaton demonstates good paallel effcency on modest numbe of pocessos. Sgnfcantly hghe pefomance s expected on lage, ealstc poblems. Refeences