Introduction to Multigrid Method

Transcription

1 Introduction to Multigrid Metod Presented by: Bogojeska Jasmina /08/005 JASS, 005, St. Petersburg 1

2 Te ultimate upsot of MLAT Te amount of computational work sould be proportional to te amount of real pysical canges in te computed system! In fully developped Multigrid processes te amount of computations sould be determined only by te amount of real pysical information /08/005 JASS, 005, St. Petersburg

3 Content Model Problems Basic Iterative Scemes Te Multigrid Metod Is everyting really tat simple??? /08/005 JASS, 005, St. Petersburg 3

4 Testing Ground One-dimensional boundary value problem describing te steady-state temperature distribution in a long uniform rod Grid: ''( x) + u( x) = f ( x) 0 < x < 1 u(0) = u(1) = 0 σ 0 1 = n x j j = 0,1 x x 1 = j,..., n 0 n /08/005 JASS, 005, St. Petersburg 4 x

5 Approximation wit te finite difference metod v + v v v 0 j 1 j j+ 1 = v = 0 n + σ v = f j = 1,,..., n 1 j j Were v is te approximation to te exact solution u( x ) j and f is f( x ) for j = 1,,..., n 1 j j j /08/005 JASS, 005, St. Petersburg 5

6 Matrix Form f T T Defining = (f 1,..., fn 1) and = ( v1,..., vn 1) we get te v matrix form of te system of linear equations A v= f: + σ 1 v1 f1 1 1 v + σ f = σ vn 1 fn 1 were A is ( n 1) ( n 1) symmetric, positive, definite matrix /08/005 JASS, 005, St. Petersburg 6

7 Testing Ground II Two-dimensional boundary value problem u u + σ u = f( x, y), 0 < x< 1, 0 < y< 1, σ > 0 xx yy were u = 0 on te boundary of te unit square. Te grid is defined wit te points: ( x, y ) = ( i, j ), i j x y 1 1 were i = 1,..., m 1, j = 1,..., m 1, x = and y = m n /08/005 JASS, 005, St. Petersburg 7

8 Approximation wit te finite difference metod Replacing te derivatives by te second-order finite differences leads to te system of linear equations v + v v v + v v + + σ v = i0 i 1, j ij i+ 1, j i, j 1 ij i, j+ 1 x y v = v = 0, i= 1,,..., m 1, j = 1,,..., n 1 in were v is an approximation of te exact solution u( x, y ) ij i j and f = f( x, y ) ij i j ij f ij /08/005 JASS, 005, St. Petersburg 8

9 Matrix Form Considering te lexicograpic ordering by lines we get T T i = ( i1,..., in, 1) and i=( i1,..., in, 1) for = 1,..., 1 v v v f f f i m Te block-tridiagonal matrix form of te system is A v= f A1 a I v1 f1 a I A a I v f..... =..... a I An a I v n f n a I An 1 vn 1 fn 1 /08/005 JASS, 005, St. Petersburg 9

10 y Matrix Form II 1 were a =, I is an ( n 1) ( n 1) identity matrix and Ai is an ( n 1) ( n 1) tridiagonal matrix A i σ x + y x σ x x + y x = σ x x + y /08/005 JASS, 005, St. Petersburg 10

11 Content Model Problems Basic Iterative Scemes Te Multigrid Metod Is everyting really tat simple??? /08/005 JASS, 005, St. Petersburg 11

12 Some Notations and Definitions Notation for model problems is Au = f, uis te exact solution and v is te approxiamtion Te error is given by e= u v and measured wit vector norms Te residual is given by r = f A v Te residual equation Ae = r Te residual correction u= v+ e /08/005 JASS, 005, St. Petersburg 1

13 Stationary Linear Iterations 1 Using u v= A r, v v and u v ( old ) ( new) ( new) ( old ) ( old ) an iteration is formed by v = v + B r were B is an approximation to A 1 ( new) ( old ) ( old ) = + = + Anoter form of te iteration v R v B f R v were R = I B A Te exact solution u is a fixed point i.e. u= R u+ g ( new) ( old ) Te error is or after relaxation e = R e sweeps e = R e ( m) m (0) ( m) m (0) e R e if R < 1 te error is forced towards zero wit te repeated iterations m g /08/005 JASS, 005, St. Petersburg 13

14 Assymptotic Convergence Factor ( R) = { } convergence factor ( R) 1 ( ) m R 0 as m ( R is convergent) ρ R < 1, were ρ max λ,..., λ is te spectral radius or n (0) ( m) For any given initial vector v, e 0 as asymptotic ρ < 1, so tis condition ensures te convergence of te iteration m /08/005 JASS, 005, St. Petersburg 14

15 Jacobi Relaxation Te one-dimensional model problem wit σ =0 is u + u u = f, j = 1,,..., n 1, u = u = 0 j 1 j j+ 1 j 0 1 ( new) 1 ( ( old ) ( old ) Jacobi relaxation v ) j = vj 1 + vj+ 1 + f j, j = 1,..., n 1 1 Matrix form, were R = D L+ U v R v D f ( ) ( new) ( old ) 1 = J + and A= D L U Weigted Jacobi relaxation ( new) ( old ) ω ( old ) ( old ) vj = vj + vj + vj + f j = + ( 1 ω) ( ) ( new) ( old ) 1 Matrix form v Rω v ω D f, ( ω) were R = 1 I+ ωr ω J /08/005 JASS, 005, St. Petersburg 15 J

16 Gauss-Seidel Relaxation Components of te new approximation are used as soon as tey are calculated reduced storage requirements ( new) 1 ( ( new) ( old ) Gaus-Seidl relaxation v ) j = vj 1 + vj+ 1 + f j, j = 1,..., n 1 v R v D L f R D L U ( ), were ( ) ( new) ( old ) 1 1 Matrix form = G + G = and A= D L U /08/005 JASS, 005, St. Petersburg 16

17 Fourier Modes We work wit te omogeneous linear system Au = 0 exact solution is u= 0, error is v j k p Fourier modes vj = sin, 0 j n, 1 k n 1 n were k is frequency or wavenumber indicating te number of alf sine waves tat constitute v on te domain Low frequency( smoot) modes wavenumbers in n range 1 k Hig frequency( oscillatory) modes wavenumbers in n range k n 1 /08/005 JASS, 005, St. Petersburg 17

18 Fourier Modes I /08/005 JASS, 005, St. Petersburg 18

19 Numerical Example Te error in weigted Jacobi iteration ω= for 100 iterations 3 wit Fourier modes k = 1, 3 and 6 as initial iteration /08/005 JASS, 005, St. Petersburg 19

20 Numerical Example I Te error in weigted Jacobi iteration ω= for 100 iterations 3 1 j π 6 j π 3 j π and initial guess v j = sin sin sin n n n /08/005 JASS, 005, St. Petersburg 0

21 Observation Standard iterations converge quickly as long as te error as ig-frequency components BUT te slow elimination of te low frequency components of te error degrades te performance /08/005 JASS, 005, St. Petersburg 1

22 Wy? ω ω Rω = I A λ( Rω) = 1 λ( A) k π λ ( A) = k n n Eigenvalues of A: k 4 sin, 1 1 j k π Eigenvectors of A and Rω : w k,j = sin, 1 k n 1 n k π λ ( R ) = ω k n n Error in an initial guess: Eigenvalues of Rω: k ω 1 sin, 1 1 n 1 n 1 n 1 (0) m = m c ( ) k = m ck = m e wk e Rω wk ck λk Rω wk k= 1 k= 1 k= 1 /08/005 JASS, 005, St. Petersburg

23 k ( ) Wy? If 0 < ω 1 ten λ < 1 convergent Jacobi relaxation For all ω, 0 < ω 1: R ω π π ω π λ1 = 1 ω sin = 1 ω sin 1 n /08/005 JASS, 005, St. Petersburg 3

24 Conclusion Te eigenvalue associated wit te smootest mode will always be close to 1 (esspecially for smaller grid spacing) No value of ω can reduce te smoot components of te error effectively Wat value of ω damps best te oscillatory components of te error? n/ ( R ) λ ( R ) Solving λ = for weigted Jacobi leads to ω 1 n ω= and λk, for k n n ω /08/005 JASS, 005, St. Petersburg 4

25 Smooting Factor Smooting factor - te largest absolute value among te eigenvalues in te upper alf of te spectrum (te oscillatory modes) of te iteration matrix: λ max k n k n Smooting property for weigted Jacobi after 35 iteration sweeps: ( ) R ω /08/005 JASS, 005, St. Petersburg 5

26 Content Model Problems Basic Iterative Scemes Te Multigrid Metod Is everyting really tat simple??? /08/005 JASS, 005, St. Petersburg 6

27 Elements of Multigrid Coarse Grids Nested Iteration Correction Sceme Interpolation Operator Restriction Operator Two-Grid Correction Sceme V-Cycle Sceme Full Multigrid V-Cycle - FMG /08/005 JASS, 005, St. Petersburg 7

28 Coarse Grids Te smooting property becomes an advantage Relaxation on a coarse grid is less expensive ( ) Because of te convergence factor 1 Ο te coarse grid as a marginally improved convergence rate A smoot wave on a fine grid Ω looks more oscillatory on a coarse grid Ω /08/005 JASS, 005, St. Petersburg 8

29 Coarse Grids /08/005 JASS, 005, St. Petersburg 9

30 Coarse Grids Te kt mode on Ω becomes te kt mode on Ω n for 1 k < : j k π j k π wk, j = sin = sin = wk, j n n/ n For k >, te kt mode on Ω becomes te ( n k)t Ω ing- so te oscilatory modes mode on - because of alias are misinterpreted as relatively smoot: w k, j ( ) ( ) j k π j π n k = sin = sin = n n j n k π = sin = w n / /08/005 JASS, 005, St. Petersburgn k, j 30

31 Nested Iteration Compute an improved initial guess for te fine-grid relaxation Relax on Au = fon a very coarse grid to obtain an initial guess for te next finer grid... Au = f Ω for 4 Relax on on to obtain an initial guess Ω Relax on A u= f on Ω to obtain an initial guess for Ω = Ω to te solution Relax on Au fon to obtain a final approximation /08/005 JASS, 005, St. Petersburg 31

32 Correction Sceme We can relax directly on te error using te residual equation Ae = rwit a specific initial guess e= 0 Relax on Au = fon Ω to obtain an approximation v Compute te residual r = f A v Ae = r Ω Relax on te residual equation on to obtain an approximation to te error Correct te approximation obtained on Ω wit te error e estimate obtained on Ω : v v + e /08/005 JASS, 005, St. Petersburg 3

33 Interpolation Operator (1D) Mapping I : Ω Ω I v = v, were 1 ( ) n v j = vj, v j+ 1 = vj + vj+ 1, 0 j 1 /08/005 JASS, 005, St. Petersburg 33

34 Interpolation Operator (1D) If te error on Ω is smoot, an interpolant of te coarse-grid error gives a good representation of te error - te interpolant is smoot /08/005 JASS, 005, St. Petersburg 34

35 Interpolation Operator (1D) If te error on Ω is oscillatory, an interpolant of te coarse-grid error may give a poor representation of te error - te interpolant is smoot /08/005 JASS, 005, St. Petersburg 35

36 Interpolation Operator (1D) n 1 n 1 I is a liner operator from to, wit a full rank and a trivial nullspace If te error is oscillatory even a very good coarse-grid approximation may produce a not very accurate interpolant because te interpolant is smoot nested iteration is effective wen te error is smoot Example for n = v v v = v4 = v I = 1 1 v /08/005 JASS, 005, St. Petersburg 36 v v v v v v 7

37 Restriction Operator (1D) Mapping I : Ω Ω Injection: I v = v, were v = v j j Full weigting: I v = v, were 1 ( = ) n vj v j 1+ v j + v j+ 1, 1 j -1 4 /08/005 JASS, 005, St. Petersburg 37

38 Full Weigting n 1 n 1 I is a liner operator from to, wit a full n rank and a nullspace of dimension Example for n = 8: v1 v 1 1 v v v = v 4 = v = v I v v v6 v /08/005 JASS, 005, St. Petersburg 38

39 Two-Grid Correction Sceme 1 Solve v MG( v, f ) Relax ν times on A u = f on Ω wit initial guess v Compute te fine-grid residual r = f A v and restrict it to te coarse grid by r A e = r on Ω = I r Interpolate te coarse-grid error to te fine-grid by e = I e and correct te fine-grid approxiamtion by v v + e Relax ν times on A u = f on Ω wit initial guess v /08/005 JASS, 005, St. Petersburg 39

40 Two-Grid Correction Sceme /08/005 JASS, 005, St. Petersburg 40

41 V-Cycle /08/005 JASS, 005, St. Petersburg 41

42 V-Cycle - Recursive 1 (, ) 1. Relax ν times on A u = f wit a given initial guess. If Ω coarsest grid, ten go to 4. Else 3. Correct v MV v f v ( ) f I f A v v 0 (, ) v MV v f v v + I v 4. Relax ν times on A u = f wit a given initial guess v /08/005 JASS, 005, St. Petersburg 4

43 v and f d-dimensions as te finer grid Storage Costs must be stored at eac level -d coarse grid as te number of points d ( d d Md) n Storage costs: n < 1 d d /08/005 JASS, 005, St. Petersburg 43

44 Computational Costs 1 Working Unit cost of one relaxation sweep on te fine grid Cost of V-cycle wit one pre-coarse-grid correction relaxation sweep and one post-coarse-gris relaxation sweep ( d d 3d Md ) 1 + < 1 d /08/005 JASS, 005, St. Petersburg 44

45 Convergence Analysis Continuous problem: Au = f, u = u( x ) Discrete problem: A u = f, v u Discretization error: E = u( x ) u, E K Algebraic error: e = u v Error condition: u- v < ε i p i i i i i i i Tis is guarantied if: E + e < ε E e 1/ p ε * ε < K ε < determines te number of iterations /08/005 JASS, 005, St. Petersburg 45

46 Converging to Level of Truncation ε ( ) p * * If te iteration goes until e = K on Ω ten we ave converged to te level of truncation Te V-cycle sceme as a convergence factor γ <1 independent of 1 d-dimensional problem on a grid wit = n () ( p) ( p n ) Error sould be reduced from Ο 1 to Ο = Ο ( ( )) d n n ( n) ( ) Number of V-cycles ν = Ο log costs Ο log /08/005 JASS, 005, St. Petersburg 46

47 Full Multigrid V-Cycle v FMG ( f ) Initialize f I f, f I f, Solve or relax on coarsest grid Interpolate initial guess v I 4 4 ( ) Perform V-cycle MV, ν 0 times Interpolate initial guess ( ) Perform V-cycle MV, ν 0 times I v v v f v v v v f /08/005 JASS, 005, St. Petersburg 47

48 Full Multigrid /08/005 JASS, 005, St. Petersburg 48

49 Full Multigrid - Recursive ( f ) ( f ) 1. Initialize f I f, f I f, If Ω coarsest grid, set v 0, ten go to 3. Else I 3. Correct v I 4. v f v v MV ( f ) FMG FMG v ( v, f ) ν 0 times /08/005 JASS, 005, St. Petersburg 49

50 Costs of Full Multigrid jd Te size of te WU for te coarse-grid j is times of te size of te WU on te fine-grid Cost of FMG for ν = ν =... = 1 is LESS tan 0 1 ( d d ) 1... d = 1 1 ( d ) /08/005 JASS, 005, St. Petersburg 50

51 Building A ( I ) ( ) How does A act on Range I? e Range A I u = r odd rows in A I are zeros I A I u = I r A = I A I /08/005 JASS, 005, St. Petersburg 51

52 Variational Properties Galerkin Condition: A I A I = T I c I, c ( ) = /08/005 JASS, 005, St. Petersburg 5

53 Spectral Properties of te Restriction Operator How does I act upon te modes of A? kπ n w = w k n smoot modes: I k cos k, 1 kπ n w( n k) = w k < n oscillatory modes: I sin k, 1 { } { } If W = span w, w I : W span w k k n k k k /08/005 JASS, 005, St. Petersburg 53

54 Spectral Properties of te Interpolation Operator How does I act on te modes of A? kπ kπ n w = w w k < n n I k cos k sin n k, 1 Interpolation of smoot modes excites oscillatory modes on Ω Ω /08/005 JASS, 005, St. Petersburg 54

55 Two-Grid Correction Sceme ( ) ( ) ( ) 1 ( ν I ( )) I C + C + + ν v R v f A f A R v f e I ( A ) A R e e No relaxation ( ν =0): TGw = s w + s w 1 ν I I TG k k k k n k n TGwn k = ckwk + ckwn k, 1 k CG eliminates te smoot modes but does not damp te oscillatory modes of te error Wit relaxation: TGw = λ s w + λ s w ν ν k k k k k k n k ν ν n TGwn k = λn kckwk + λn kckwn k, 1 k /08/005 JASS, 005, St. Petersburg 55

56 Algebraic Analysis ( ) ( ) ( I ) ( I ) A Range ( T ) ( ) ( ) N Range N Range N I I I I A ( ) ( I ) N I Ω = Range A e = s + t /08/005 JASS, 005, St. Petersburg 56

57 Spectral and Algebraic Decompozition Spectral decompozition: Algebraic decompozition: /08/005 JASS, 005, St. Petersburg 57

58 How it works? /08/005 JASS, 005, St. Petersburg 58

59 How it works? /08/005 JASS, 005, St. Petersburg 59

60 How it works? /08/005 JASS, 005, St. Petersburg 60

61 Is everyting really so simple??? Anisotropic operators and grids Discontinuous or anisotropic coefficients Nonlinear problems Non-scalar PDE systems Hig order discretization Algebraic Turbulence models Cemicaly reacting flows Socks Small-scale singularities /08/005 JASS, 005, St. Petersburg 61