Multigrid Methods for Discretized PDE Problems

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1 Towards Metods for Discretized PDE Problems Institute for Applied Matematics University of Heidelberg Feb 1-5, 2010

2 Towards Outline A model problem Solution of very large linear systems Iterative Metods Two-Grid and -Metods Preconditioning and wat to do, wen te problem gets a problem

3 Towards Summary, Basic Metods

4 Towards Direct Metods LR-Decomposition A = LU, L =... R = Direct Solver for te System Ax = b Very slow Large Memory usage E LR = O(N 3 ) M LR = O(N 2 )

5 Towards Basic Iterative Metods Wit initial guess x 0 R N x t+1 = x t + θb (b Ax t ) }{{} d t Convergence depends on Iteration Matrix B: Analyze Spectral Norm e t+1 I BA e t I BA 2 = λ max(i BA) Problem, very bad convergence rate (depends on mes size ) I BA Many steps necessary so solve system up to ɛ Overall complexity bad N iter = O(N) E Jacobi = O(N 2 ), M Jacobi = O(N)

6 Towards Gradient-Metods Instead of solving Ax = b minimize functional E(x) = 1 x, x b, x 2 In every step t, set searc direction r t, make optimal step: Optimal searc direction would be Since x is not available, use min E(x t + sr t ) s = d t, r t s R Ar t, r t r t = x x t r t = b Ax t = Ax Ax t = A(x x t ) Problem, searc directions are bad, many steps necessary ( ) 1 1/κ t e t+1 e 0 1 1/κ, 1 + 1/κ 1 + 1/κ 1 1 κ = 1 2 Bad overall complexity (due to many steps) N iter = O(N), E GM = O(N 2 ), M GM = O(N)

7 Towards Metod of Conjugate Gradients Use better searc directions d 3 d 2 d 4 d 2 d 1 d 1 Ortogonalize r k, k = 0,..., t r k, r l A = Ar k, r l = 0 (k l) Since A symmetric, ortogonalization can be done in one step: d t+1 = b Ax t, r t+1 = d t+1 d t+1, r t A r t Convergence rate depends on square root of condition number: e t+1 ( ) t 1 1/ κ 1 + 1/ e 0, κ 1 1/ κ 1 + 1/ κ 1 1 κ = 1

8 Towards Comparison Solve 2D-Laplace Equation, problem size N = M 2, = 1 M, 2 = 1 N Metod E iter conv. N iter E all N = 10 6 LR N N 3 30 years Jacobi 5N 1 2 N 5N 2 2 ours Gradient 10N 1 2 N 10N 2 3 ours 3 Conjugate Gradients 20N 1 N 20N 2 1 min

9 Towards Towards Metods

10 Towards Wat can we do better Optimal Numerical Complexity Numerical complexity in every step E step = O(N) Is given by Jacobi, Gradient, Conjugate Gradient Convergence Rate, Number of steps To reduce error by ɛ N iter = O(1) Not te case for Jacobi (O(N)), Gradient (O(N)), Conjugate Gradients (O( N))

11 Towards Dependence of convergence rate on mes-size e t+1 ρ e t Jacobi Gradient Conjugate Gradient ρ = λ max(i BA) = λ max(d 1 (L + R)) 1 2 ρ = ρ = ( ) 1 1/κ = /κ κ 1 2 ( ) 1 1/ κ 1 + 1/ = κ κ Oter interpretation On a fixed Mes Ω were is a constant we already ave an optimal solver!

12 Towards Wy does convergence depend on te mes-size? A detailed analysis of te Jacobi Iteration, 1D-Laplace Start wit random initial guess x 0 Error in te beginning is randomly distribute: Perform some steps of Jacobi

13 Towards Jacobi Solution - 0

14 Towards Jacobi Solution - 1

15 Towards Jacobi Solution - 2

16 Towards Jacobi Solution - 3

17 Towards Jacobi Solution - 4

18 Towards Jacobi Solution - 5

19 Towards Jacobi Solution - 6

20 Towards Jacobi Solution - 7

21 Towards Jacobi Solution - 8

22 Towards Jacobi Solution - 9

23 Towards Solution Beavior Solution canges very rapidly in te beginning (te first two or tree steps) Ten, nearly noting appens Solution in step 0 2 and step 2 9

24 Towards Jacobi Error - 0

25 Towards Jacobi Error - 1

26 Towards Jacobi Error - 2

27 Towards Jacobi Error - 3

28 Towards Jacobi Error - 4

29 Towards Jacobi Error - 5

30 Towards Jacobi Error - 6

31 Towards Jacobi Error - 7

32 Towards Jacobi Error - 8

33 Towards Jacobi Error - 9

34 Towards Error Beavior Overall error amplitude does not cange a lot Error frequencies cange!!!

35 Towards Te Concept of Smooting Smooting Jacobi is very bad and slow as a solver, but ig frequencies are smooted quickly

36 Towards Smooting Property of Ricardson Iteration - I x (t+1) = x (t) + θ(b Ax (t) ) e (t+1) = e t + θ(ax Ax (t) ), e (t) := x (t) x = e (t) θae (t) = [I θa]e (t) Eigenvalues & Eigenvectors. Let: A ω i = λ i ω i, i = 1,..., N, λ min(a ) = λ 1 λ N = λ max(a ) Eigenvectors are ortogonal ω i, ωj = { 0 if i j 1 if i == j We know Eigenvalues and Eigenvectors of te Iteration Matrix [I θa ]ω i = ωi θa ω i = ωi θλ i ω i = (1 θλ i )ω i

37 Towards Smooting Property of Ricardson-Iteration - II We ave Develop Error propagation e (t+1) = [I θa]e (t) = [I θa] t+1 e (0) = N e (0) = ɛ i ω i i=1 N e (t) = [I θa ] t e (0) = ɛ i [I θa ] t ω i = N ɛ i (1 θλ i ) t ω i i=1 i=1 Since Eigenvectors are ortogonal e (t) N 2 = e (t), e(t) = i,j=1 = N ɛ 2 i (1 θλ i ) 2t i=1 ɛ i ɛ j (1 θλ i ) t (1 θλ j ) t ω i, ωj }{{} =δ ij

38 Towards Smooting Property of Ricardson-Iteration - III We ave Convergence, if e (t) 2 N ɛ 2 i (1 θλ i ) 2t i=1 1 θλ i < 1 0 < θ < 1 λ max(a )

39 Towards Smooting Property of Ricardson-Iteration - IV Eigenvalues A closer look at te Eigenvalues. 1D-Laplace i = 1,..., N ( ) iπ λ i = 2 2 cos N + 1 ( )) ikπ N ω (sin i = N + 1 k=1 Distribution of Eigenvalues

40 Towards Smooting Property of Ricardson-Iteration - V Eigenvectors λ λ λ λ

41 Towards Smooting Property of Ricardson-Iteration - V Eigenvalues Eigenvalues of te Smooting Operator For θ = λ max(a ) 1 : ρ := λ(i θa ) = (1 θλ i ) For i = N/2:...

42 Towards Smooting Property of Ricardson-Iteration - VI For i = N/2 ρ N/2 = 1 1 ( ( )) pin/2 2 2 cos 4 N ( ) pi 2 cos 2 = 1 2 For i > N/2: ρ i < 1 2 Error contributions for large Eigenvalues: ẽ (0) := N i=n/2 ɛ i ω i ẽ (t) 2 = N ɛ 2 i i=n/2 ( 1 1 ) 2t 4 λ i ẽ (t) N 2 ɛ 2 i i=n/2 ( ) 1 2t 2

43 Towards Smooting Property of Ricardson-Iteration - Summary Ricardson Iteration is very bad and slow Solver Ricardson Iteration is very good Smooter for ig frequencies Hig frequencies are all ω i, i > N 2 i > 1 2 Hig frequencies are only visible on finest Mes Ω

44 Towards Hierarcical Approac Smoot ig frequencies on fine mes N = 40 Ω (some few steps): Transfer to coarse Mes N = 20 Treat coarse problem on mes Ω 2

45 Towards Hierarcical Approac Low frequencies on fine mes Ω are ig frequencies on coarse mes Ω 2 Treat coarse problem on mes Ω 2

46 Towards Possible Benefits Only few (fixed number O(1)?) operations on fine mes Ω Coarse mes problem Ω 2 is smaller (2D-model problem by a factor 4). Peraps direct solver possible? Nested approac possible: Ω Ω 2 Ω 4... Ω H

47 Towards Analysis of te Smooting Property Ricardson Error contributions for large Eigenvalues: ẽ (0) := N i=n/2 ɛ i ω i ẽ (t) 2 = ẽ (t) Reduce ig frequency error by ɛ N 2 ɛ 2 i i=n/2 N ɛ 2 i i=n/2 ( ) 1 2t 2 ( ) 1 t ɛ t = log(ɛ) 2 Fixed number of iterations, independent of of N! ( 1 1 ) 2t 4 λ i

48 Towards Te Two-Grid Iteration

49 Towards Te Basic 2-Grid Iteration 2-Grid Metod for solving A x = b Initial guess x (0), iterate t 0 1 Smoot: y (t) 2 Defect: d (t) 3 Restrict: d (t) H := S(A, b, x (t) ) := b A y (t) := R d (t) 4 Coarse Mes Solution: x (t) H 5 Prolongate: x (t+1) := A 1 H d (t) H := x (t) + P x (t) H

50 Towards Te Basic 2-Grid Iteration - Smooting Smooting 1 Smoot: y (t) := S(A, b, x (t) ) Easy and fast iterative sceme (Jacobi, Ricardson, Gauss-Seidel) Just a few steps N iter = 2 5 Smoot ig frequencies ẽ (t) ρt ẽ (0), ρ < 1, i > N 2

51 Towards Te Basic 2-Grid Iteration - Restriction Smooting 3 Restrict: d (t) H... see later... := R d (t)

52 Towards Te Basic 2-Grid Iteration - Prolongation Prolongation 5 Prolongate: x (t+1) := x (t) + P x (t) H Transfer Solution x H from Ω H to x on Ω. Simple Interpolation Matrix Notation H P =

53 Towards Te Basic 2-Grid Iteration - Restriction Restriction 3 Restrict: d (t) H := R d (t) Transfer Solution x from Ω to x H on Ω H. Use adjoint of Prolongation: R = P = Example Matrix Notation H

54 Towards Te Basic 2-Grid Iteration - Coarse Mes Solution Coarse Mes Solution 4 Coarse Mes Solution: x (t) H := A 1 H d (t) H Direct solution of te system A H x H = d H For 2D-Model Problem: N H = 1 4 N Solution Wit LR: E LR (N H ) = NH 3 = 1 64 E LR(N ) Better, but still too muc!

55 Towards Two-Grid-Iteration: Compact Notation I Coarse Mes Problem: A H x H = d H Write as iterative Metod on fine mes Ω x (t+1) = x (t) + P A 1 H R (b A x (t) ) = [I P A 1 H R (t) A ] x }{{} B CM + P A 1 H R b For te error e (t+1) e (t+1) = x (t+1) = B CM e (t) x = B CM x (t) + P A 1 H R A x x

56 Towards Two-Grid-Iteration: Compact Notation I Coarse Mes Problem: A H x H = d H Write as iterative Metod on fine mes Ω x (t+1) = x (t) + P A 1 H R (b A x (t) ) = [I P A 1 H R (t) A ] x }{{} B CM + P A 1 H R b For te error e (t+1) e (t+1) = x (t+1) = B CM e (t) x = B CM x (t) + P A 1 H R A x x

57 Towards Two-Grid-Iteration: Compact Notation I Coarse Mes Problem: A H x H = d H Write as iterative Metod on fine mes Ω x (t+1) = x (t) + P A 1 H R (b A x (t) ) = [I P A 1 H R (t) A ] x }{{} B CM + P A 1 H R b For te error e (t+1) e (t+1) = x (t+1) = B CM e (t) x = B CM x (t) + P A 1 H R A x x

58 Towards Two-Grid-Iteration: Compact Notation II Coarse Mes Solution Smooting Operation e (t+1) e (t+1) = B CM e (t) = B SM e (t) Two-Grid-Iteration Initial guess x (0), iterate e (t+1) = B ν 2 SM B CMB ν 1 SM e(t)

59 Towards Two-Grid Iteration: Convergence I Analyze iteration matrix B TG = B ν SM B CM Coarse Mes Solution Ricardson-Iteration B CM = I P A 1 H R A B SM = I 1 λ max A Te Two-Grid-Iteration B TG = [I P A 1 H R A ][I 1 A ] = [A 1 P A 1 H λ R ] max }{{} Approximation A [I 1 λ max A ] ν } {{ } Smooting Sow B TG ρ(ν) = cν 1 < 1 for ν > ν 0

60 Towards Two-Grid Iteration: Convergence I Analyze iteration matrix B TG = B ν SM B CM Coarse Mes Solution Ricardson-Iteration B CM = I P A 1 H R A B SM = I 1 λ max A Te Two-Grid-Iteration B TG = [I P A 1 H R A ][I 1 A ] = [A 1 P A 1 H λ R ] max }{{} Approximation A [I 1 λ max A ] ν } {{ } Smooting Sow B TG ρ(ν) = cν 1 < 1 for ν > ν 0

61 Towards Two-Grid Iteration: Convergence I Analyze iteration matrix B TG = B ν SM B CM Coarse Mes Solution Ricardson-Iteration B CM = I P A 1 H R A B SM = I 1 λ max A Te Two-Grid-Iteration B TG = [I P A 1 H R A ][I 1 A ] = [A 1 P A 1 H λ R ] max }{{} Approximation A [I 1 λ max A ] ν } {{ } Smooting Sow B TG ρ(ν) = cν 1 < 1 for ν > ν 0

62 Towards Two-Grid Iteration: Convergence I Analyze iteration matrix B TG = B ν SM B CM Coarse Mes Solution Ricardson-Iteration B CM = I P A 1 H R A B SM = I 1 λ max A Te Two-Grid-Iteration B TG = [I P A 1 H R A ][I 1 A ] = [A 1 P A 1 H λ R ] max }{{} Approximation A [I 1 λ max A ] ν } {{ } Smooting Sow B TG ρ(ν) = cν 1 < 1 for ν > ν 0

63 Towards Two-Grid Iteration: Convergence Proof Proof: 1 Split error influences B TG e (t+1) [A 1 P A 1 H R ] A [I 1/λ maxa ] ν 2 Sow, tat for te Smooting Operator it olds A [I 1/λ maxa ] ν v c sν 1 2 v 3 Sow, tat for te Coarse Mes Approximation it olds (A 1 P A 1 H R )v c a 2 v 4 Put it all togeter, wit ρ(ν) := c sc aν 1 < 1

64 Towards Two-Grid Iteration: Convergence Proof Proof: 1 Split error influences B TG e (t+1) [A 1 P A 1 H R ] A [I 1/λ maxa ] ν 2 Sow, tat for te Smooting Operator it olds A [I 1/λ maxa ] ν v c sν 1 2 v 3 Sow, tat for te Coarse Mes Approximation it olds (A 1 P A 1 H R )v c a 2 v 4 Put it all togeter, wit ρ(ν) := c sc aν 1 < 1

65 Towards Two-Grid Iteration: Convergence Proof Proof: 1 Split error influences B TG e (t+1) [A 1 P A 1 H R ] A [I 1/λ maxa ] ν 2 Sow, tat for te Smooting Operator it olds A [I 1/λ maxa ] ν v c sν 1 2 v 3 Sow, tat for te Coarse Mes Approximation it olds (A 1 P A 1 H R )v c a 2 v 4 Put it all togeter, wit ρ(ν) := c sc aν 1 < 1

66 Towards Two-Grid Iteration: Convergence Proof Proof: 1 Split error influences B TG e (t+1) [A 1 P A 1 H R ] A [I 1/λ maxa ] ν 2 Sow, tat for te Smooting Operator it olds A [I 1/λ maxa ] ν v c sν 1 2 v 3 Sow, tat for te Coarse Mes Approximation it olds (A 1 P A 1 H R )v c a 2 v 4 Put it all togeter, wit ρ(ν) := c sc aν 1 < 1

67 Towards Two-Grid Iteration: Convergence Proof I, te Smooting Operator Step 2: sow, tat A [I 1/λ maxa ] ν v c sν 1 2 v. Done for Ricardson and Laplace-Matrix Must be done separately for every matrix (partial differential equation)! Separate analysis necessary for different smooters (Jacobi, Gauss-Seidel, Ricardson, ILU) Difficult for complex equations and complex smooting operators

68 Towards Two-Grid Iteration: Convergence Proof II, te Coarse Mes Approximation Operator Step 3: sow, tat (A 1 P A 1 H R )v c a 2 v We must use properties of te partial differential equation and te discretization! Use a priori error estimates: u u c 2 f, Ten, using triangle inequality < C H u H u ch 2 f u u H u u + u H u c( 2 + H 2 ) 2 f Assume geometric mes growt Use special v above: Finally < C H, v := A u = f, A 1 f P A 1 u u H c(1 + C 2 )2 f A 1 v = u, P A 1 H R v = u H H R f = u u H c(1 + C 2 }{{} ) c a 2 f

69 Towards Two-Grid Iteration: Convergence Proof II, te Coarse Mes Approximation Operator Step 3: sow, tat (A 1 P A 1 H R )v c a 2 v We must use properties of te partial differential equation and te discretization! Use a priori error estimates: u u c 2 f, Ten, using triangle inequality < C H u H u ch 2 f u u H u u + u H u c( 2 + H 2 ) 2 f Assume geometric mes growt Use special v above: Finally < C H, v := A u = f, A 1 f P A 1 u u H c(1 + C 2 )2 f A 1 v = u, P A 1 H R v = u H H R f = u u H c(1 + C 2 }{{} ) c a 2 f

70 Towards Two-Grid Iteration: Convergence Proof II, te Coarse Mes Approximation Operator Step 3: sow, tat (A 1 P A 1 H R )v c a 2 v We must use properties of te partial differential equation and te discretization! Use a priori error estimates: u u c 2 f, Ten, using triangle inequality < C H u H u ch 2 f u u H u u + u H u c( 2 + H 2 ) 2 f Assume geometric mes growt Use special v above: Finally < C H, v := A u = f, A 1 f P A 1 u u H c(1 + C 2 )2 f A 1 v = u, P A 1 H R v = u H H R f = u u H c(1 + C 2 }{{} ) c a 2 f

71 Towards Two-Grid Iteration: Convergence Proof II, te Coarse Mes Approximation Operator Step 3: sow, tat (A 1 P A 1 H R )v c a 2 v We must use properties of te partial differential equation and te discretization! Use a priori error estimates: u u c 2 f, Ten, using triangle inequality < C H u H u ch 2 f u u H u u + u H u c( 2 + H 2 ) 2 f Assume geometric mes growt Use special v above: Finally < C H, v := A u = f, A 1 f P A 1 u u H c(1 + C 2 )2 f A 1 v = u, P A 1 H R v = u H H R f = u u H c(1 + C 2 }{{} ) c a 2 f

72 Towards Two-Grid Iteration: Convergence Proof III, Putting it togeter Step 4: Step 2 and Step 3: For ν > ν 0 : B TG A [I 1/λ maxa ] ν A 1 P A 1 H R c sν 1 2 c(1 + C 2 c sc(1 + C 2 )ν 1 0 < 1 )2 = c sc(1 + C 2 )ν 1 q.e.d.

73 Towards Two-Grid Iteration: Numerical Complexity Good convergence. To reduce error by ɛ: t = log(ɛ) Large effort: E step = E smoot + E coarse mes ( ) 3 1 = 5νN + C 2 N 3 = O(N 3 )

Introduction to Multigrid Method

Introduction to Multigrid Method Introduction to Multigrid Metod Presented by: Bogojeska Jasmina /08/005 JASS, 005, St. Petersburg 1 Te ultimate upsot of MLAT Te amount of computational work sould be proportional to te amount of real