An Multigrid Tutorial

Transcription

1 An Multgrd Tutoral Van Emden Henson Center for Appled Scentfc Computng Lawrence Lvermore Natonal Laboratory Aprl 0, 999

2 AMG: What s Algebrac Multgrd?? Any multlevel method where geometry s not used (and may not be avalable) to buld coarse grds, nterpolaton and restrcton, or coarse-grd operators. Classcal AMG was ntroduced by Brandt, McCormck and Ruge n 982. It was explored early on by Stueben n 983, and popularzed by Ruge and Stuben n 987. Ths tutoral wll descrbe only the classcal AMG algorthm. veh 2

3 AMG: What s Algebrac Multgrd?? Many other algorthms qualfy under the defnton gven. Some whose approaches are closely related to classcal AMG : Chang Grebel, Neunhoeffer, Regler Huang Krechel, Stueben Zaslavsky Work close to the orgnal, but usng dfferent approaches to coarsenng or nterpolaton: Fuhrmann Kcknger veh 3

4 AMG: What s Algebrac Multgrd?? Other approaches that are mportant, novel, hstorcal, or werd: Multgraph methods (Bank & Smth) Aggregaton methods (Braess; Chan & Zkatanov & Xu ) Smoothed Aggregaton methods (Mandel & Brezna & Vanek) Black Box Multgrd (Dendy, Dendy & Bandy) Algebrac Multlevel Recursve Solver (Saad) Element based algebrac multgrd (Charter; Cleary et al) MultCoarse correcton wth Suboptmal Operators (Sokol) Multlevel block ILU methods (Jang & Saad; Bank & Smth & Wagner; Reusken) AMG based on Element Agglomeraton (Jones & Vasslevsk) Sparse Approxmate Inverse Smoothers (Tang & Wan) Algebrac Schur-Complement approaches (Axelsson & Vasslevsk & Neytcheva) veh 4

5 veh 5 Hghlghts of Multgrd: The -d Model Problem Posson s equaton: n [0,], wth boundary condtons. Dscretzed as: Leads to the Matrx equaton, where = f u u u = ) ( = ) ( 0 0 u u 0 = = N 0 f u A = = + f h u u u f f f f f f u u u u u u h A =, =, = N N N N

6 u Hghlghts of Multgrd: Weghted Jacob Relaxaton Consder the teraton: u new u old u old u old ( ) ( ) ω ( ω) + ( ( ) ( ) f 2 ) 2h Lettng A = D+L+U, the matrx form s: ( new) ( old) = ( ω) I ωd ( L + U) u + ωd f ( old) = G u + ωd f ω. ( exact) ( approx) It s easy to see that f e u u, then e ( n ew) = G e( old) ω veh 6

7 Hghlghts of Multgrd: Relaxaton Typcally Stalls The egenvectors of G ω are the same as those of, and are Fourer Modes: = sn ( kπ / N), k =, 2,, N v A The egenvalues of G ω are 2 ω sn2 ( k π / 2N ), so the effect of relaxaton on the modes s: N = 400 ω = / 3 ω = / 2 ω = 2/ 3 ω = Note: No value of ωwll damp out the low frequency waves veh 7

8 Hghlghts of Multgrd: Relaxaton Smooths the Error Intal error, Error after several teraton sweeps: Many relaxaton schemes have the smoothng property, where oscllatory modes of the error are elmnated effectvely, but smooth modes are damped very slowly veh 8

9 Hghlghts of Multgrd: Smooth error can be represented on a coarse grd A smooth functon: Can be represented by lnear nterpolaton from a coarser grd: On the coarse grd, the smooth error appears to be relatvely hgher n frequency: n the example t s the 4-mode, out of a possble 6, on the fne grd, /4 the way up the spectrum. On the coarse grd, t s the 4-mode out of a possble 8, hence t s /2 the way up the spectrum. Relaxaton wll be more effectve on ths mode f done on the coarser grd!! veh 9

10 Hghlghts of Multgrd: Coarse-grd Correcton Perform relaxaton on A u = f on fne grd untl error s smooth. h h Compute resdual, r = f A u and transfer to the coarse grd r2 h 2h = I rh. h Solve the coarse-grd resdual equaton to obtan the error: 2h A e2h 2 r2h e2h h =, = ( A ) r2h Interpolate the error to the fne grd and correct the fne-grd soluton: u h uh + I e2h h h h h h h 2 h veh 0

11 Hghlghts of Multgrd: Coarse-grd Correcton Relax on A h Compute u r h h = = f h h f h A u h u Correct h uh + eh r Restrct 2 h 2h = I rh h e Interpolate h h I e2h 2 h 2h Solve A e 2 h = r2h e2 h 2h A ) = ( r2h veh

12 veh 2 Hghlghts of Multgrd: Tools Needed Interpolaton and restrcton operators: Lnear Injecton Full-weghtng Interpolaton Coarse-grd Operator. Two methods: () Dscretze equaton at larger spacng (2) Use Galerkn Formula: I I I h h h h h h =, =, = A 2h I A I A h h h h h h =

13 Hghlghts of Multgrd: Recurson: the ( ν,0) V-cycle f Major queston: How do we solve the coarse-grd resdual equaton? Answer: recurson! u h G ν ( A, f ) 2h 2h h h I f A uh h ( u u h 4 4h 4h h G ν ( A, f ) u ) h 2 2h 2h h G ν ( A, f ) 4h 4h 2h 2h h 2h f I ( f A u2 8h 8h 4h 4h f I f A u4h 4h ( 8 8h 8h h G ν ( A, f ) ) ) u h u 2h + e2 2 h e h I u 8 u e e I u 4 4 4h 8h h u h u8h + e8 u 8 h h h + e h h I u2h 2 h 2 h 2h h 4h u h u4h + e4 h 4 h H e H = ( A ) f H veh 3

14 Algebrac multgrd: for unstructured-grds Automatcally defnes coarse grd AMG has two dstnct phases: setup phase: defne MG components soluton phase: perform MG cycles AMG approach s opposte of geometrc MG fx relaxaton (pont Gauss-Sedel) choose coarse grds and prolongaton, P, so that error not reduced by relaxaton s n range(p) defne other MG components so that coarse-grd correcton elmnates error n range(p) (.e., use Galerkn prncple) (n contrast, geometrc MG fxes coarse grds, then defnes sutable operators and smoothers) veh 4

15 AMG has two phases: Setup Phase Select Coarse grds, Ω m +, m =, 2,... Defne nterpolaton, Im m +, m =, 2,... Defne restrcton and coarse-grd operators I + m m m + T = ( I ) A I A I m + = + m m m m + Solve Phase Standard multgrd operatons, e.g., V-cycle, W- cycle, FMG, etc veh 5

16 In AMG, we choose relaxaton frst: Typcally, pontwse Gauss-Sedel s used A = ( D + L + U) The teraton s developed: A x = Add and subtract ( D + L) ( D + L) x ol d to get: b ( D + L) x = b Ux x n ew = ( D+ L) b ( D+ L ) Ux old x n ew = x old + ( D + L ) r old veh 6

17 Gauss-Sedel relaxaton error propagaton: The teraton s: x n ew = x old + ( D + L ) r old Subtractng both sdes from the exact soluton: exact new exact old x x = x ( x + ( D + L) r old ) e n ew = e old ( D + L ) r old r = A e Usng ths can be wrtten as: e n ew = I ( D + L ) A e old veh 7

18 An observaton: error that s slow to converge small resduals Consder the teratve method error recurrence ek + = ( I Q A ) ek Error that s slow to converge satsfes ( I Q A) e e Q A e 0 r 0 Perhaps a better vewpont s ( I Q A) e e Q A e, Ae «e, Ae veh 8

19 Some mplcatons of slow convergence For most teratons (e.g., Jacob or Gauss-Sedel) ths last holds f D Ae, Ae «e, Ae. () N r2 N a «= = r e Hence mplyng that, on average, r a e «An mplcaton s that, f s an error slow to converge, then locally at least, e can be wellapproxmated by an average of ts neghbors: e veh 9

20 In Multgrd, error that s slow to converge s geometrcally smooth Combnng the algebrac property that slow convergence mples small resduals wth the observaton above, n AMG we DEFINE smooth error: Smooth error s that error whch s slow to converge under relaxaton, that s, ( I Q A) e e or, more precsely, Q A) e A ( I e A veh 20

21 But sometmes, smooth error sn t! (example from Klaus Stueben) Consder the problem on the unt square, usng a regular Cartesan grd, wth fnte dfference stencls and values for a,b,and c: 2 = h [ 2 a= b=000 c=0 a= b= c=0 ( a u x ) x A= b= c=2 a=000 b= c=0 ( b u y ) y + c u xy = f ( x, y) u x x ] u xy = u yy 2h 2 = h veh 2

22 But sometmes, smooth error sn t! ( a u x ) x ( b u y ) y + c u xy = f ( x, y) Usng a zero rght-hand sde and a random ntal guess, after 8 sweeps of Gauss-Sedel teraton the error s unchangng n norm. By our defnton, the error s smooth. And t looks lke ths: veh 22

23 ( a u Smooth error for x ) x ( b u y ) y + c u xy = f ( x, y) veh 23

24 AMG uses dependence (nfluence) to determne MG components We need to choose a subset of the grdponts (coarse grd) that can be used ) to represent smooth errors, and 2) to nterpolate these errors to the fne grd. u j Intutvely, a pont s a good canddate for a C- pont f ts value s mportant n determnng the value of another pont, u n the th equaton. If the a j coeffcent s large compared to the other off-dagonal coeffcents n the th equaton then u j nfluences u (or u depends on u j ). veh 24

25 Dependence and smooth error For M-matrces, we defne depends on j by aj θ max {ak}, 0 < θ k alternatvely, j nfluences. It s easy to show from () that smooth error A e, e «De, satsfes (2) e veh 25

26 Dependence and smooth error For M-matrces, we have from (2) a j e ej 2 «2 2 j a e If e does not depend on e j then the nequalty may be satsfed because a j s small. If e does depend on e j, then a j need not be small, and the nequalty must be satsfed by Ths mples that smooth error vares slowly n the drecton of dependence. veh 26

27 Some useful defntons The set of dependences of a varable u, that s, the varables upon whose values the value of u depends, s defned as S = j : a j > max k { a k } The set of ponts that u nfluences s denoted: T S { j : Sj } veh 27

28 More useful defntons The set of coarse-grd varables s denoted C. The set of fne-grd varables s denoted F. The set of coarse-grd varables used to nterpolate the value of the fne-grd varable u, called the coarse nterpolatory set for, s denoted C. veh 28

29 Two Crtera for Choosng the Coarse Grd Ponts Frst Crteron: F - F dependence j F S C C (C) For each, each pont should ether be n tself or should depend on at least one pont n. j S k C n C F u Snce the value of depends on the value of u j, the value of u j must be represented on the coarsegrd for good nterpolaton. If j sn t a C-pont, t should depend on a pont n C so ts value s represented n the nterpolaton. veh 29

30 Two Crtera for Choosng the Coarse Grd Ponts Second Crteron: Maxmal Subset C (C2) should be a maxmal subset wth the property that no -pont depends on another. C (C) tends to ncrease the number of -ponts. In general, the more C-ponts on Ω H, the better the h-level convergence. But more C-ponts means more work for relaxaton and nterpolaton. (C2) s desgned to lmt the sze (and work) of the coarse grd. C veh 30

31 Two Crtera for Choosng the Coarse Grd Ponts It s sometmes not possble to satsfy both crtera smultaneously (an example wll be seen shortly). In those cases, we choose to satsfy (C), the requrement that F-F dependences be represented n the coarse-nterpolatory set, whle usng (C2) as a gude. Ths choce leads to somewhat larger coarse grds, but tends to preserve good convergence propertes. veh 3

32 Choosng the Coarse Grd Ponts Assgn to each grdpont k a value equal to the number of ponts that depend on k. Choose the frst pont wth global maxmum value as a C-pont. The new C-pont can be used to nterpolate values of ponts t nfluences. Assgn them all as F- ponts. Other ponts nfluencng these new F-ponts can be used n ther nterpolaton. Increment ther value. Repeat untl all ponts are C- or F-ponts. veh 32

33 Ruge AMG: start select C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 33

34 Ruge AMG: select C-pt select next C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 34

35 Ruge AMG: select F-pt select C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 35

36 Ruge AMG: update F-pt neghbors select C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 36

37 Ruge AMG: select C-pt select next C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 37

38 Ruge AMG: select F-pt select next C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors 5 3 veh 38

39 Ruge AMG: update F-pt neghbors select next C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors 6 3 veh 39

40 Ruge AMG: select C-pt, F-pts, update neghbors select next C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 40

41 Ruge AMG: select C-pt, F-pts, update neghbors select next C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 4

42 Ruge AMG: select C-pt, F-pts, update neghbors select next C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 42

43 Ruge AMG: select C-pt, F-pts, update neghbors 6,7,8,9 select next C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 43

44 veh 44 Examples: Laplacan Operator 5-pt FD, 9-pt FE (quads), and 9-pt FE (stretched quads) pt FD 9-pt FE (quads)

45 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 45

46 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 46

47 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors veh 47

48 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of F-pt neghbors 8 8 veh 48

49 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of new F-pt neghbors veh 49

50 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of new F-pt neghbors veh 50

51 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of new F-pt neghbors veh 5

52 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of new F-pt neghbors veh 52

53 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of new F-pt neghbors veh 53

54 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of new F-pt neghbors veh 54

55 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons select C-pt wth maxmal measure select neghbors as F-pts update measures of new F-pt neghbors veh 55

56 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons Modulo perodcty, t s the same coarsenng as n the Drchlet case. However, t has many F-F connectons that do not share a common C-pont veh 56

57 Coarse-grd selecton: 9pt Laplacan, perodc boundary condtons A second pass s made n whch some F-ponts are made nto C-ponts to enforce (C). Goals of the second pass nclude mnmzng C-C connectons, and mnmzng the number of C-ponts converted to F-ponts. veh 57

58 How well does AMG coarsen: ( a u x ) x ( b u y ) y + c u xy = f ( x, y) a= b=000 c=0 A= b= c=2 a= b= c=0 a=000 b= c=0 In each regon, AMG coarsens only n the drecton of dependence! veh 58

59 How well does AMG coarsen: ( a u x ) x ( b u y ) y + c u xy = f ( x, y) a= b=000 c=0 In each regon, AMG coarsens only n the drecton of dependence! veh 59

60 How well does AMG coarsen: ( a u x ) x ( b u y ) y + c u xy = f ( x, y) a=000 b= c=0 In each regon, AMG coarsens only n the drecton of dependence! veh 60

61 How well does AMG coarsen: ( a u x ) x ( b u y ) y + c u xy = f ( x, y) a= b= c=0 In each regon, AMG coarsens only n the drecton of dependence! veh 6

62 How well does AMG coarsen: ( a u x ) x ( b u y ) y + c u xy = f ( x, y) A= b= c=2 In each regon, AMG coarsens only n the drecton of dependence! veh 62

63 Prolongaton C F C F C F e, C ( Pe) = ωk ek, F k C The nterpolated value at pont s just e f s a C- pont. If s an F-pont, the value s a weghted sum of the values of the ponts n the coarse nterpolatory set C. veh 63

64 To defne prolongaton at, we must examne the types of connectons of. u Sets of connecton types: C F C C F C D s s dependent on these coarse nterpolatory C- ponts. s dependent on these F- ponts. F D w does not depend on these weakly connected ponts, whch may be C- or F- ponts. veh 64

65 Prolongaton s based on smooth error and dependences (from M-matrces) Recall that smooth error s characterzed by small resduals: r = a e + aj ej 0 a e j N whch we can rewrte as: j We base prolongaton on ths formula by solvng for e and makng some approxmatng substtutons. a j e j veh 65

66 Prolongaton s based on smooth error and dependences (from M-matrces) We begn by wrtng the smooth-error relaton: a e a e j j j Identfyng ts component sums: a e j C a j Coarse nterpolatory set e j s j D a j e F-pont dependences j w j D a j e j Weak connectons We must approxmate e n each of the last j two sums n terms of e or of e for j C j. veh 66

67 veh 67 C C C F F F For the weak connectons: let. e e j Coarse nterpolatory set F-pont dependences Weak connectons e a j j D j j j D j j j C j w s e a e a e a Ths approxmaton can t hurt too much: Snce the connecton s weak. If depended on ponts n, smooth error vares slowly n the drecton of dependence D w Effectvely, ths throws the weak connectons onto the dagonal: + e a e a e a a j j D j j j C j j D j s w

68 For the F-pont dependences: use a weghted avg. of errors n C C. j C F C F C F a Weak connectons Approxmate e j by a weghted average of the e k n the coarse nterpolatory set C C. + j j D w a j e j C a j Coarse nterpolatory set e e j j F-pont dependences k C j D s a k C a j jk a e j e jk k It s for ths reason that the ntersecton of the coarse nterpolatory sets of two F-ponts wth a dependence relatonshp must be nonempty (C). veh 68

69 Fnally, the prolongaton weghts are defned Makng the prevous substtuton, and wth a bt of messy algebra, the smooth error relaton can be solved for to yeld the nterpolaton formula: e e j C ω j e j where the prolongaton weghts are gven: ω j = a j a + j D + n D a s m C w k a a kj a n km veh 69

70 Hghlghts of Multgrd: Storage: f h, u h must be stored each level In -d, each coarse grd has about half the number of ponts as the fner grd. In 2-d, each coarse grd has about onefourth the number of ponts as the fner grd. In d-dmensons, each coarse grd has about 2 d the number of ponts as the fner grd. d d 2d 3d Storage cost: 2N ( Md 2 less than 2, 4/3, 8/7 the cost of storage on the fne grd for, 2, and 3-d problems, respectvely. ) < 2N d d veh 70

71 AMG storage: grd complexty For AMG there s no smple predctor for total storage costs. u m, f m, and m A = m m Im A I must be stored on all levels. m σ Ω Defne, the grd complexty, as the total number of unknowns (grdponts) on all levels, dvded by the number of unknowns on the fnest level. Total u f 2σ Ω storage of the vectors and occupy storage locatons. veh 7

72 AMG storage: operator complexty σ A Defne, the operator complexty, as the total number of nonzero coeffcents of all operators A m dvded by the number of nonzero coeffcents n the fne-level operator A 0. Total storage of the operators occupes storage locatons. σ A veh 72

73 AMG storage: nterpolaton σ I We could defne, an nterpolaton complexty, as the total number of nonzero coeffcents of all operators Im + dvded by the number of nonzero coeffcents n the operator I0. Ths measure s not generally cted, however (lke most multgrdders, the AMG crowd tends to gnore the cost of ntergrd transfers). Two measures that occasonally appear are κ A, the average stencl sze, and κ I, the average number of nterpolaton ponts per F-pont. veh 73

74 AMG Setup Costs: flops Flops n the setup phase are only a small porton of the work, whch ncludes sortng, mantanng lnked-lsts, keepng counters, storage manpulaton, and garbage collecton. Estmates of the total flop count to defne nterpolaton weghts ( ω I ) and the coarse-grd operators ( ω A ) are: ωa = N κ I ( 2κ I ( κ A κ I ) + 3κ and ωi = N κ I ( 3( κa κ I) 2) I + κ A ) veh 74

75 AMG setup costs: a bad rap Many geometrc MG methods need to compute prolongaton and coarse-grd operators The only addtonal expense n the AMG setup phase s the coarse grd selecton algorthm AMG setup phase s only 0-25% more expensve than n geometrc MG and may be consderably less than that! veh 75

76 Hghlghts of Multgrd: Computaton Costs Let Work Unt (WU) be the cost of one relaxaton sweep on the fne-grd. Ignore the cost of restrcton and nterpolaton (typcally about 20% of the total cost). (See?) Consder a V-cycle wth pre-coarse-grd correcton relaxaton sweep and post-coarse- Grd correcton relaxaton sweep. Cost of V-cycle (n WU): 2 ( + 2 d 2d 3d Md ) < 2 2 Cost s about 4, 8/3, 6/7 WU per V-cycle n, 2, and 3 dmensons. d veh 76

77 AMG Solve Costs: flops per cycle The approxmate number of flops n on level m for one relaxaton sweep, resdual transfer, and nterpolaton are (respectvely) 2N A A m 2N 2 I F C m + κ Nm N I F m + 2κ Nm where Nm A s the number of coeffcents n A m C F and N N are the numbers of C-, F-ponts on Ω m. m, m ( ν, ν2) N N ν = ν + ν2 The total flop count for a V-cycle, notng that m F and lettng s approxmately m N 2 A Ω I ( ( ν + ) κ σ + 4κ + σ ) Ω veh 77

78 AMG Solve Costs: flops per cycle, agan All that s very well, but n practce we fnd the solve phase s generally domnated by the cost of relaxaton and computng the resdual. Both of those operatons are proportonal to the number of nonzero entres n the operator matrx on any gven level. Thus the best measure of the rato of work done on all levels to the work done on the fnest level s operator complexty: σ A veh 78

79 Hghlghts of Multgrd: dffcultesansotropc operators and grds Consder the operator : α «β 2 u α x 2 If then the GSsmoothng factors n the x- and y- drectons are shown at rght. Note that GS relaxaton does not damp oscllatory components n the x-drecton. 2 u β y = f ( x, y) The same phenomenon occurs for grds wth much larger spacng n one drecton than the other: veh 79

80 Hghlghts of Multgrd: dffcultesdscontnuous or ansotropc coeffcents ( D( x, y) u) Consder the operator:, where D( x, y) = d d 2 ( x, y) ( x, y) d d 2 22 ( x, y) ( x, y) Agan, GS-smoothng factors n the x- and y- drectons can be hghly varable, and very often, GS relaxaton does not damp oscllatory components n the one or both drectons. Solutons: lne-relaxaton (where whole grdlnes of values are found smultaneously), and/or semcoarsenng (coarsenng only n the strongly coupled drecton). veh 80

81 AMG does sem-coarsenng automatcally! Consder the operator : 2 u α x 2 2 u β y2 = f ( x, y) In the lmt, as α, the stencl becomes: AMG automatcally produces a sem-coarsened grd!! veh 8

82 AMG Convergence: there s theory (some) There s some theory, although t s of lmted utlty. It generally looks lke: Theorem Let A m A be SPD, and let the nterpolaton operator Im m + be full rank, and let restrcton and coarse-grd operators be defned by + m T m + + m m Im = ( Im + ) and A = Im A Im + and let there be smoothng operators G m and coarse-grd correcton operators m m m m + + m T = I Im + ( A ) Im A veh 82

83 AMG Convergence: there s theory (some) Theorem (contnued) suppose that, for all, e m G em 2 em 2 m δ T em m 2 A A A holds for some δ > 0 ndependently for all and m. e m Then δ, and, provded the coarsest problem s solved and at least one smoothng step s performed after each coarse-grd correcton step, the V-cycle has a convergence factor wrt the energy norm bounded above by δ. veh 83

84 How s t perform (vol I)? Regular grds, plan, old, vanlla problems The Laplace Operator: Convergence Tme Setup Stencl per cycle Complexty per Cycle Tmes 5-pt pt skew pt (-,8) pt (-,-4,20) Ansotropc Laplacan: ε U xx U yy Epslon Convergence/cycle veh 84

85 How s t perform (vol II)? Structured Meshes, Rectangular Domans 5-pont Laplacan on regular rectangular grds Convergence factor (y-axs) plotted aganst number of nodes (x-axs) veh 85

86 How s t perform (vol III)? Unstructured Meshes, Rectangular Domans Laplacan on random unstructured grds (regular trangulatons, 5-20% nodes randomly collapsed nto neghborng nodes) Convergence factor (y-axs) plotted aganst number of nodes (x-axs) veh 86

87 How s t perform (vol IV)? Isotropc dffuson, Structured/Unstructured Grds ( d( x, y) u) on structured, unstructured grds Problems used: a means parameter c=0, b means c=000 6: d( x, y) =. 0 + c x y 7: d( x, y) N=6642 N=66049 N=3755 N=5458 Structured Structured Unstruct. Unstruct.. 0 = c x x 0. 5 > : 9: d( x, y) d( x, y). 0 = c =. 0 c max { x 0. 5 ( x 0. 5), otherwse 2 + ( y 0. 5) otherwse y a 6 b 7 a 7 b 8 a 8 b 9 a 9 b } veh 87

88 ( 0 How s t perform (vol IVa)? Isotropc dffuson, Structured/Unstructured Grds d( x, y) u) N=6642 N=66049 N=3755 N=5458 Structured Structured Unstruct. Unstruct. on structured, unstructured grds 0 a, n = 2 0 b, n = 2 a, n = 0 b, n = 0 2 a, n = b, n = 5 0 Problem used: a means parameter c=0, b means c=000 Checkerboard of coeffcents.0 and c, squares szed /n: + j j +. 0 n x < n, n y < n, + j even d( x, y) = + j j + c n x < n, n y < n, + j odd veh 88

89 How s t perform (vol V)? Laplacan operator, unstructured Grds Convergence factor Grdponts veh 89

90 So, what could go wrong? Strong F-F connectons: weghts are dependent on each other e j For pont the value s nterpolated from,, and s needed to make the nterpolaton weghts for approxmatng e. For pont j the value e s nterpolated from k, k 2, and s needed to make the nterpolaton weghts for approxmatng e j. It s an mplct system! k k C C j k k 2 k 3 C k 4 C j j k k 2 2 C C j veh 90

91 Is there a fx? A Gauss-Sedel lke teratve approach to weght defnton s mplemented. Usually two passes suffce. But does t work? Frequently, t does: Convergence factors for Laplacan, stretched quadrlaterals x = 0 y theta Standard Iteratve x = 00 y veh 9

92 Another Fx: ndrect nterpolaton (see Stueben s text for detal) The 5-pont problem cannot gve full coarsenng because the F-pont n the mddle has no connecton to any of the 4 C-ponts. Hence, there s no way to nterpolate ts value. veh 92

93 Another Fx: ndrect nterpolaton (see Stueben s text for detal) Full coarsenng could be acheved by ndrect nterpolaton. Frst nterpolate the F-ponts from the C-ponts. veh 93

94 Another Fx: ndrect nterpolaton (see Stueben s text for detal) Full coarsenng could be acheved by ndrect nterpolaton. Frst nterpolate the F-ponts from the C-ponts. Then nterpolate the mddle from the F-ponts. Smlar treatment could be appled whenever F-F dependences arse. veh 94

95 AMG for systems How can we do AMG on systems? A A 2 A A 2 22 Naïve approach: Block approach (block Gauss-Sedel, usng scalar AMG to solve at each cycle) u v = f g u ( A ) ( f A 2 v) v ( A 22 ) ( g A 2 u ) Great Idea! Except that t doesn t work! (relaxaton does not evenly smooth errors n both unknowns) veh 95

96 AMG for systems: a soluton To solve the system problem, allow nteracton between the unknowns at all levels: A k = A A k k 2 A A k 2 k 22 and I k k + = ( I k k + 0 ) u ( I 0 k k + ) v Ths s called the unknown approach. Results: 2-D elastcty, unform quadrlateral mesh: mesh spacng Convergence factor veh 96

97 So, what else can go wrong? Ouch! Thn body elastcty! Elastcty, 3-d, thn bodes! u v w xx yy zz ν + ( u 2 ν + ( v 2 ν + ( w 2 yy xx xx u v w zz zz yy ) ) ) + ν + ( v 2 + ν + ( u 2 + ν + ( u 2 Slde surfaces, Lagrange multplers, force balance constrants: S U T V x x 2 S s generally postve defnte, V can be zero, U T T. = xy xy xz F F w w v xz yz yz ) ) ) = = = f f f 2 3 Wanted: Good soluton method for ths problem. REWARD veh 97

98 Needed: more robust methods for characterzng smooth error Consder quadrlateral fnte elements on a stretched 2D Cartesan grd (dx -> nfnty): A = Drecton of dependence s not apparent here Iteratve weght nterpolaton wll sometmes compensate for ms-dentfed dependence Elastcty problems are stll problematc veh 98

99 Scalablty s central for large-scale parallel computng A code s scalable f t can effectvely use addtonal computatonal resources to solve larger problems Many specfc factors contrbute to scalablty: archtecture of the parallel computer parallel mplementaton of the algorthms convergence rates of teratve lnear solvers Lnear solver convergence can can be be dscussed ndependent of of parallel computng, and and s s often overlooked as as a key key scalablty ssue. veh 99

100 In Concluson, AMG Rules! Interest n AMG methods s hgh, and probably stll rsng, because of the ncreasng mportance of terra-scale smulatons on unstructured grds. AMG has been shown to be a robust, effcent solver on a wde varety of problems of real-world nterest. Much research s underway to fnd effectve ways of parallelzng AMG, whch s essental to largescale computng. veh 00

101 Acknowledgment Ths work was performed under the auspces of the U. S. Department of Energy by Lawrence Lvermore Natonal Laboratory under contract number: W-7405-Eng-48. Document Release number UCRL-MI veh 0