Parallel VLSI CAD Algorithms. Lecture 7 Iterative methods for IC thermal analysis Zhuo Feng

Transcription

1 Parallel VLSI CAD Algortms Lecture 7 Iteratve metods for IC termal analyss Zuo Feng 7.

2 We ve taled about usng LU to solve a sparse lnear matr t problem Anoter way of dealng wt sparse matr problems s to solve tem teratvely ere are no fll-ns ns wt teratve soluton metods! But te callenge s ensurng effcent convergence 7.

3 Iteratve metods are great for sparse system storage f tey converge A b Most popular teratve metods are: Gauss-Jacob Gauss-Sedel We wll tal about some more complcated metods later 7.3

4 A b A b A b A b Iteratvely solve for A b Defne: and B t t rue soluton 7.4

5 B b B b B t For an ntal guess B t It follows tat a necessary and suffcent condton for convergence s: lm 0 B y for all y 7.5

6 OR: lm B 00 e egenvalues of egenvalues of B B are te -t powers of te Anoter necessary and suffcent condton for convergence s tat all of te egenvalues of ave a magntude less tan all egenvalues le wtn te unt crcle B B We also want converge rapdly to be small so tat teratons 7.6

7 Calculatng egenvalues s more dffcult tan solvng orgnal problem! Matr Norms: m n B F b n B ma n B n ma n b b Eucldean norm s farly g complety terefore -norm s most commonly used But -norm s suffcent but NO necessary 7.7

8 -norm follows from dagonal domnance condton Suffcent but not necessary: A b A? n a a for n 7.8

9 Normalze eac row of A by dagonal al element e a a a a a a 3 a a 3 7.9

10 B A a a3 0 a a a a3 0 a a 7.0 n a B f a

11 Gauss - Jacob 0; 0 guess repeat for all ; A b Only maes sense for parallel processng applcatons oterwse Gauss-Sedel s faster n n b a a ; a untl n ; 7.

12 Gauss - Sedel 0; 0 guess repeat ; for all n n b a a ; a untl n ; 7.

13 Successve overrelaaton SOR can mprove te rate of convergence sometmes Use G-S to calculate ~ -- K+ t G-S teraton But use wegted average of t and for actual update ~ ~ ~ Selecton of s nontrval!! 7.3

14 Matr nterpretatons A D L U Dagonal Lower rangular Upper rangular Guass-Jacob D L U b D L U D b Guass-Sedel D L U b D L U D L b 7.4

15 How would te teratve soluton metods apply to some problems oter tan crcuts? ermal analyss of ICs s becomng etremely mportant emperature gradents on a cp can substantally t mpact te performance 7.5

16 Heat conducton s governed by te followng PDE: Materal densty C p y z t tt Specfc eat [ y z t] ermal conductvty p y z t Power densty of eat sources A control volume e PDE can be numercally solved usng fnte element/dfference dscretzaton 7.6

17 A control volume z y z y z Dscretze te PDE over all control volumes ] [ t p z t y t t t t C p t t t t Z. Z. Feng Feng MU MU EE5900 EE

18 Rewrte te fnte dfference dscretzaton G G dt d C I G G G G dt z z y y were: z y G y z G z y G z y z y p I z y C C z y p Z. Z. Feng Feng MU MU EE5900 EE

19 ranslates nto a lnear crcut of termal ranslates nto a lnear crcut of termal resstance capactance and eat sources + + G z G G G y + - G G G C - - G y G z I G G G G dt d C y y Z. Z. Feng Feng MU MU EE5900 EE I G G z z y y

20 Creates an etremely large crcut problem Generally nterested only n te steady state -- termal capactance s not consdered Drect soluton of tese large problems can be mpractcal due to problem sze Iteratve metods are very appealng due to sparsty y z t C p [ y z t ] p y z t t 7.0

21 Gauss Jacob/Sedel can often be appled to tese lnear problems due to specal matr propertes Dagonally domnant or symmetrc postve defnte SPD We prevously dscussed te matr condton of dagonal domnance A b A b Necessary and suffcent condton for convergence s: lm 0 B B y for all y 7.

22 OR: lmb 0 B n ma n b Suffcent but not necessary dagonal domnance: A b A? n a a for n B We also want converge rapdly to be small so tat teratons 7.

23 Strct dagonal domnance s not a necessary condton Anoter suffcent condton for convergence s a near dagonally domnant M-matr: A s an M-matr f. a 0 n. a 0... a 0... n 3. A s nonsngular 4. A 0 every element of te nverse s nonnegatve 7.3

24 It can be sown tat A s an M-matr f t satsfes te followng condtons:. a 0... n. a 0... n n 3. a a n... n 4. a a for at least one e matr from our termal problem -- wt boundary condtons -- s an M-matr 7.4

25 Even wt dagonal domnance or an M-matr G-S and G-J metods do not scale well wt problem sze Ecessve smulaton runtme Mult-level approaces suc as multgrd can be an deal opton W. L. Brggs A multgrd tutoral SIAM Press 987 Metod for solvng large lnear matr problems teratvely 7.5

26 A smple termal problem eample -d Posson equaton to descrbe te steady-state temperature dstrbuton along a unform rod u u0 u 0 f C p y z t t [ y z t] p y z t Steady state G u f 0 7.6

27 Appromate te nd order dervatve usng fnte dfference 3-pont stencl G u f 0 G -t f u u u f G N u 0 u N 0 step sze 7.7

28 e lnear system s: Au u f G A s an M-matr terefore Gauss Jacob/Sedel wll converge for any ntal guess Effcent drect soluton does ests for ts trdagonal system But we use ts model problem to study te converge rates of teratve metods 7.8

29 Note: not strctly dagonally domnant but an M-matr wt boundary condtons Au u f G 7.9

30 For our d termal problem G-J/G-S wll converge snce underlyng matr s a M-matr t Drect applcaton of GJ/GS can be slow due to te geometrc dstrbuton of te soluton error Consder te cange n te error dstrbuton along te rod as we teratve wt basc GJ/GS u -t u 0 u 0 f 7.30

31 Eact temperature dstrbuton gven a eat source of f along te lengt of te rod u u 0 u 0 u Assume tat te ntal guess s: uˆ 7.3

32 Error can ave dfferent frequency components e u uˆ Low frequency vs. g frequency Fast varyng Slowly varyng 7.3

33 GJ/GS acts dfferently on te error components of dfferent frequences Hg frequency errors can be removed rater qucly wle low frequency ones cannot Fourer mode analyss can elp us to understand te convergence rate of te model problem 7.33

34 Consder te followng modes of te termal response for our d model problem: w sn N 0 N and N K= K=3 K=6 7.34

35 K s referred to as te wavenumber e Fourer modes wt a g wavenumber represent g frequency components of waveform Gauss Jacob/Sedel remove te error components of te ntal guess at dfferent rates 7.35

36 As an eample consder te case for wc te RHS s set to zero f=0 and we solve: u u u 0 N u 0 u N 0 N s te number of dscretzaton ponts along te rod Snce te eact soluton s zero for ts case ten te error s equvalent to te ntal guess We can decompose our ntal guess u nt as a lnear combnaton of dfferent Fourer modes 7.36

37 We would epect te error to decay muc more qucly for te g frequency ntal t guesses We ll consder te error beavor for a wegted GJ teraton u n n u u n GJ Soluton obtaned n te standard Gauss Jacob e use of relaaton parameter can mprove te convergence We wll see tat affects te dampng rates for errors of dfferent frequences 7.37

38 Consder 3 dfferent cases of ntal guesses namely w w 3 w 6 respectvely u n n n u u GJ N

39 Let te ntal guess consst of all modes w 63 Run te wegted Gauss Jacob wt untl te norm of eac mode error s reduced by a factor of

40 e result gets better f we cange to / 3 However convergence s stll slow for smoot low frequency modes 7.40

41 Rapd decrease for early teratons s due to te fast elmnaton for g frequency modes of error Convergence starts to stall once te g frequency modes are removed e smootng property of te basc teraton scemes s a severe lmtaton An effectve remedy s multgrd metods 7.4

42 Snce te basc teraton metods struggle wt smoot errors alter te dscretzaton to ncrease g freq errors f 7.4

43 s suggest tat we sould move to a coarser grd once te relaaton on te fne grd stalls Relaatons on a coarser grd can also be muc ceaper snce te problem sze s smaller s dea s systematcally eplored n multgrd by solvng te problem at a erarcy of grds wo grd cycle for a D problem: Orgnal Problem Coarser grd problem 7.43

44 wo grd cycle for our D rod problem Incomplete relaaton of Au=b on fne grd to obtan an appromaton of u v -t f u b b f G 7.44

45 Relaaton of Au=b on fne grd to obtan ntermedate varable v Compute te resdue r = b Av Relaaton of te resdual equaton Ae = r on coarser grd to get an appromaton of te error e M Map e bac to e on Correct te soluton on by: v v + e Restart te frst step untl convergence Could etend ts to more tan te two levels descrbed ere 7.45

46 ypcally te grd spacng of a coarse grd s te twce of tat t of te mmedate fne grd ransfer of e from to s done by nterpolaton: I Interpolaton operator v I v maps a coarse grd operator to a fne grd operator: v v v v v N 0 Lnear nterpolaton among grd ponts for any number of levels l of erarcy 7.46

47 Lnear nterpolaton Average of two negborng ponts ae te pont drectly from te coarse grd

48 Can epress te nterpolaton operator n a matr form F f d l bl t N 6 d d t For our specfc model problem wt N=6 and zero endpont boundary condtons: v v v v I v v v v v I v Z. Z. Feng Feng MU MU EE5900 EE

49 Usng nterpolaton mples tat te error on te fne grd must be smoot enoug Interpolaton t tat t produces a good appromaton of error on te fne grd Interpolaton tat produces a rater poor appromaton Coarse grd correcton can only be used after g frequency errors are suffcently damped on te fne grd 7.49

50 We also need to defne te transfer from te fne grd to te coarse one I Restrcton operator defnes te mappng from to : v I v e smplest restrcton operator s necton Ponts wc are not on te coarse grd are smply removed 7.50

51 Inecton Dscard ts pont from te fne grd Retan ts pont from te fne grd

52 A more popular restrcton operator s full wegtng Full wegtng n D For specfc model problem wt N=: pont wegtn 0 g

53 Can epress te full wegtng operator n a matr form: 4 N v v v v v v v I v 4 3 v v 4 I N= v v v v v v v Full wegtng operator s proportonal to te transpose of 0 v v Z. Z. Feng Feng MU MU EE5900 EE Full wegtng operator s proportonal to te transpose of lnear nterpolator a factor of dfference n ts case

54 We want to terate on te resdue equaton on te coarse grd to adust te fne grd soluton Recall tat f ˆ s an appromate soluton for : A b en resdue s defned as suc tat : A ˆ e Ae r r b Aˆ We defne an appromate coarse grd problem for te resdue equaton b A e error we want to evaluate r coarse grd operator resdue vector appromated from fne grd soluton 7.54

55 r s obtaned by restrctng mappng te resdue of te fne grd to te coarse grd: r I r Note tat resdue from fne grd s te only nput to coarse grd problem A must be a good appromaton of te orgnal matr on te coarse grd can be obtaned by dscretzng te PDE usng a step sze We can also appromate te coarse grd problem from te fne grd for better robustness: Galern coarse grd appromaton 7.55

56 Recall on te fne grd te resdue equaton s: A e r r b A Based on te above we can say tat te followng nner product relatonsp must be satsfed: Ae v r v ˆ For any vector on We can also relate te coarse grd errors and appromatons to tose of te fne gran model as follows: e I e v I v Note tat tese nterpolaton operators can be dfferent n some cases but are te same for us n tese eamples 7.56

57 We get te appromate resdue equaton: A I e I v r I v I A Ie v I r v For our eample we can use: I I s suggests te Galern metod to defne te coarse grd problem: A I A I r I r A e r Maps fne grd resdue to coarse grd resdue 7.57

58 Multgrd combnes te benefts of fne grd and coarse grd relaatons rater ncely Fne grd teraton taes care of te g frequency errors Coarse grd one removes te low frequency errors Fast convergence s aceved by a proper nterplay between ee dfferent e grds 7.58

59 e basc two grd cycle can be appled recursvely to lead to te mult-level level teratons V-cycle W-cycle fner coarser 4 8 sequence sequence e multgrd sequences to te coarsest grd for wc drect solve s effcently appled to a very small problem 7.59

60 Recursve procedure untl we reac a coarse grd sze for wc we can solve te problem va drect metods Pre-smootng: Post-smootng: n G-S teratons to obtan v n G-S teratons wt corrected u A u b A e r r I r u v Ie = = Solvng for error snce nput fct s ust resdue from prevous level 4 e4 I8 e8 Coarse grd correcton 4 =3 8 =4 A8 e8 r8 7.60

61 Multgrd cycle: u 0 MG u A b m n n : grd level = ndcates te fnest level 0 u A b m n n : ntal guess for -t level based on boundary condtons : lnear problem matr at te t level : RHS at te -t level : teraton control parameters 7.6

62 0 Multgrd cycle flow: u MG u A b m n n. Pre-smootng: n v - Compute by performng smootng steps wt ntal guess v smoot u 0 A b n 0 u. Coarse grd correcton - Compute te resdue: r b A v - Restrct te resdue: r I r - Compute te soluton of resdue equaton on te +-t grd: A e r f at coarsest level drect solve oterwse: apply multgrd cycle m tmes wt zero ntal guess e m MG n 0 A r m n 7.6

63 - Interpolate te correcton: e - Correct te appromaton on te t level: u v I e I e 3. Post-smootng Perform smootng steps n tmes wt ntal guess u u smoot n u A b V-cycle corresponds to m= and W-cycle m= 7.63

64 Interpolaton n D coarse grd to fne grd Ponts on te coarse grd 7.64

65 Full wegtng n D Fne grd to coarse grd e center pont n te coarse grd s a nne-pont wegted average 4 6 Ponts on te boundary are slgtly dfferent fewer negborng gponts n te average 7.65

66 Eample: termal problem n D Heat solaton along z drecton and tree sdewalls Kept at fed temperature 7.66

67 We are dealng wt a D dscretzaton of te termal PDE C p y z t t [ y z t] p y z t 3D Dscretzaton n D g g g g g g p y y y g ermal conductance n g y y ermal conductance n y p power densty n a bo 7.67

68 ere are fewer eat conducton pats at a reflectve boundares g g y g g y p 7.68

69 e multgrd tecnque we dscussed s under te category of geometrc multgrd Algebrac multgrd AMG follows a dfferent plosopy were multgrd teraton s carred out completely based on a gven matr e prncple of multgrd as also been adopted to analyze bot IC termal problems and dscrete problems suc as power grd J. Kozaya S. Nassf and F. Nam A multgrd-le tecnque for power grd analyss IEEE rans. on CAD vol. no. 0 pp Oct. 00. L P.; Plegg L..; Aseg M.; Candra R. IC termal smulaton and modelng va effcent multgrd-based approaces IEEE ransactons on CAD Volume 5 Issue 9 Sept. 006 Pages: