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CSE 245: Computer Aided Circuit Simulation and Verification Matrix Computations: Iterative Methods (II) Chung-Kuan Cheng.

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Presentation on theme: "CSE 245: Computer Aided Circuit Simulation and Verification Matrix Computations: Iterative Methods (II) Chung-Kuan Cheng."— Presentation transcript:

1 CSE 245: Computer Aided Circuit Simulation and Verification Matrix Computations: Iterative Methods (II) Chung-Kuan Cheng

2 2 Outline  Introduction  Direct Methods  Iterative Methods Formulations Projection Methods Krylov Space Methods Preconditioned Iterations Multigrid Methods Domain Decomposition Methods

3 3 Introduction Direct Method LU Decomposition Iterative Methods Jacobi Gauss-Seidel Conjugate Gradient GMRES Multigrid Domain Decomposition Preconditioning General and Robust but can be complicated if N>= 1M Excellent choice for SPD matrices Remain an art for arbitrary matrices

4 4 Formulation  Error in A norm Matrix A is SPD (Symmetric and Positive Definite  Minimal Residue Matrix A can be arbitrary

5 5 Formulation: Error in A norm Min (x-x * ) T A(x-x * ), where Ax * =b, and A is SPD. Min E(x)=1/2 x T Ax - b T x Search space: x=x 0 +Vy, where x 0 is an initial solution, matrix V nxm has m bases of subspace K vector y m contains the m variables.

6 6 Solution: Error in A norm Min E(x)=1/2x T Ax - b T x, Search space: x=x 0 +Vy, r=b-Ax 1.V T AV is nonsingular if A is SPD, V is full ranked 2.y=(V T AV) -1 V T r 0 3.x=x 0 +V(V T AV) -1 V T r 0 4.V T r=0 5.E(x)=E(x 0 )-1/2 r 0 T V(V T AV) -1 V T r 0

7 7 Solution: Error in A norm For any V’=VW, where V is nxm and W is a nonsingular mxm matrix, the solution x remains the same. Search space: x=x 0 +V’y, r=b-Ax 1.V’ T AV’ is nonsingular if A is SPD & V is full ranked 2.x=x 0 +V’(V’ T AV’) -1 V’ T r 0 3. =x 0 +V(V T AV) -1 V T r 0 4.V’ T r=W T V T r=0 5.E(x)=E(x 0 )-1/2 r 0 T V(V T AV) -1 V T r 0

8 8 Steepest Descent: Error in A norm Min E(x)=1/2 x T Ax - b T x, Gradient: Ax-b= -r Set x=x 0 +yr 0 1.r 0 T Ar 0 is nonsingular if A is SPD 2.y=(r 0 T Ar 0 ) -1 r 0 T r 0 3.x=x 0 +r 0 (r 0 T Ar 0 ) -1 r 0 T r 0 4.r 0 T r=0 5.E(x)=E(x 0 )-1/2 (r 0 T r 0 ) 2 /(r 0 T Ar 0 )

9 9 Lanczos: Error in A norm Min E(x)=1/2 x T Ax-b T x, Set x=x 0 +Vy 1.v 1 =r 0 2.v i is in K{r 0, A, i} 3.V=[v 1,v 2, …,v m ] is orthogonal 4.AV=VH m +v m+1 e m T 5.V T AV=H m Note since A is SPD, H m is T m Tridiagonal

10 10 Conjugate Gradient: Error in A norm Min E(x)=1/2 x T Ax-b T x, Set x=x 0 +Vy 1.v 1 =r 0 2.v i is in K{r 0, A, i} 3.V=[v 1,v 2, …,v m ] is orthogonal in A norm, i.e. V T AV= [diag(v i T Av i )] 4.y i = (v i T Av i ) -1

11 11 Formulation: Residual Min |r| 2 =|b-Ax| 2, for an arbitrary square matrix A Min R(x)=(b-Ax) T (b-Ax) Search space: x=x 0 +Vy where x 0 is an initial solution, matrix V nxm has m bases of subspace K vector y m contains the m variables.

12 12 Solution: Residual Min R(x)=(b-Ax) T (b-Ax) Search space: x=x 0 +Vy 1.V T A T AV is nonsingular if A is nonsingular and V is full ranked. 2.y=(V T A T AV) -1 V T A T r 0 3.x=x 0 +V(V T A T AV) -1 V T A T r 0 4.V T A T r= 0 5.R(x)=R(x 0 )-r 0 T AV(V T A T AV) -1 V T A T r 0

13 13 Steepest Descent: Residual Min R(x)=(b-Ax) T (b-Ax) Gradient: -2A T (b-Ax)=-2A T r Let x=x 0 +yA T r 0 1.V T A T AV is nonsingular if A is nonsingular where V=A T r 0. 2.y=(V T A T AV) -1 V T A T r 0 3.x=x 0 +V(V T A T AV) -1 V T A T r 0 4.V T A T r= 0 5.R(x)=R(x 0 )-r 0 T AV(V T A T AV) -1 V T A T r 0

14 14 GMRES: Residual Min R(x)=(b-Ax) T (b-Ax) Gradient: -2A T (b-Ax)=-2A T r Let x=x 0 +yA T r 0 1.v 1 =r 0 2.v i is in K{r 0, A, i} 3.V=[v 1,v 2, …,v m ] is orthogonal H 4.AV=VH m +v m+1 e m T =V m+1 H m 5.x=x 0 +V(V T A T AV) -1 V T A T r 0 HHH 6.=x 0 +V(H m T H m ) -1 H m T e 1 |r 0 | 2

15 15 Conjugate Residual: Residual Min R(x)=(b-Ax) T (b-Ax) Gradient: -2A T (b-Ax)=-2A T r Let x=x 0 +yA T r 0 1.v 1 =r 0 2.v i is in K{r 0, A, i} 3.(AV) T AV= D Diagonal Matrix 4.x=x 0 +V(V T A T AV) -1 V T A T r 0 5.=x 0 +VD -1 V T A T r 0

16 16 Conjugate Gradient Method  Steepest Descent Repeat search direction  Why take exact one step for each direction? Search direction of Steepest descent method

17 Orthogonal Direction  We don’t know !!! Pick orthogonal search direction: We like to leave the error of x i+1 orthogonal to the search direction d i, i.e. 17

18 18 Orthogonal  A-orthogonal  Instead of orthogonal search direction, we make search direction A –orthogonal (conjugate)

19 Search Step Size 19

20 Iteration finish in n steps Initial error: A-orthogonal The error component at direction dj is eliminated at step j. After n steps, all errors are eliminated. 20

21 Conjugate Search Direction  How to construct A-orthogonal search directions, given a set of n linear independent vectors.  Since the residue vector in steepest descent method is orthogonal, a good candidate to start with 21

22 Construct Search Direction -1  In Steepest Descent Method New residue is just a linear combination of previous residue and  Let We have Krylov SubSpace: repeatedly applying a matrix to a vector 22

23 Construct Search Direction -2 let For i > 0 23

24 Construct Search Direction -3 let then  can get next direction from the previous one, without saving them all. 24

25 Conjugate Gradient Algorithm Given x0, iterate until residue is smaller than error tolerance 25

26 26 Conjugate gradient: Convergence  In exact arithmetic, CG converges in n steps (completely unrealistic!!)  Accuracy after k steps of CG is related to: consider polynomials of degree k that is equal to 1 at step 0. how small can such a polynomial be at all the eigenvalues of A?  Eigenvalues close together are good.  Condition number: κ(A) = ||A|| 2 ||A -1 || 2 = λ max (A) / λ min (A)  Residual is reduced by a constant factor by O(κ 1/2 (A)) iterations of CG.

27 27 Other Krylov subspace methods  Nonsymmetric linear systems: GMRES: for i = 1, 2, 3,... find x i  K i (A, b) such that r i = (Ax i – b)  K i (A, b) But, no short recurrence => save old vectors => lots more space (Usually “restarted” every k iterations to use less space.) BiCGStab, QMR, etc.: Two spaces K i (A, b) and K i (A T, b) w/ mutually orthogonal bases Short recurrences => O(n) space, but less robust Convergence and preconditioning more delicate than CG Active area of current research  Eigenvalues: Lanczos (symmetric), Arnoldi (nonsymmetric)

28 28 Preconditioners  Suppose you had a matrix B such that: 1.condition number κ (B -1 A) is small 2.By = z is easy to solve  Then you could solve (B -1 A)x = B -1 b instead of Ax = b  B = A is great for (1), not for (2)  B = I is great for (2), not for (1)  Domain-specific approximations sometimes work  B = diagonal of A sometimes works  Better: blend in some direct-methods ideas...

29 29 Preconditioned conjugate gradient iteration x 0 = 0, r 0 = b, d 0 = B -1 r 0, y 0 = B -1 r 0 for k = 1, 2, 3,... α k = (y T k-1 r k-1 ) / (d T k-1 Ad k-1 ) step length x k = x k-1 + α k d k-1 approx solution r k = r k-1 – α k Ad k-1 residual y k = B -1 r k preconditioning solve β k = (y T k r k ) / (y T k-1 r k-1 ) improvement d k = y k + β k d k-1 search direction  One matrix-vector multiplication per iteration  One solve with preconditioner per iteration

30 Outline  Iterative Method Stationary Iterative Method (SOR, GS,Jacob) Krylov Method (CG, GMRES) Multigrid Method 30

31 What is the multigrid  A multilevel iterative method to solve Ax=b  Originated in PDEs on geometric grids  Expend the multigrid idea to unstructured problem – Algebraic MG  Geometric multigrid for presenting the basic ideas of the multigrid method. 31

32 The model problem + v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 v7v7 v8v8 vsvs Ax = b 32

33 Simple iterative method  x (0) -> x (1) -> … -> x (k)  Jacobi iteration  Matrix form : x (k) = R j x (k-1) + C j  General form: x (k) = Rx (k-1) + C (1)  Stationary: x* = Rx* + C (2) 33

34 Error and Convergence Definition: error e = x* - x (3) residual r = b – Ax (4) e, r relation: Ae = r (5) ((3)+(4)) e (1) = x*-x (1) = Rx* + C – Rx (0) – C =Re (0) Error equation e (k) = R k e (0) (6) ((1)+(2)+(3)) Convergence: 34

35 Error of diffenent frequency  Wavenumber k and frequency  = k/n High frequency error is more oscillatory between points k= 1 k= 4 k= 2 35

36 Iteration reduce low frequency error efficiently  Smoothing iteration reduce high frequency error efficiently, but not low frequency error Error Iterations k = 1 k = 2 k = 4 36

37 Multigrid – a first glance  Two levels : coarse and fine grid A 2h x 2h =b 2h A h x h =b h hh Ax=b  2h 37

38 Idea 1: the V-cycle iteration  Also called the nested iteration A 2h x 2h = b 2h hh  2h A h x h = b h Iterate => Start with Prolongation:  Restriction:  Iterate to get Question 1: Why we need the coarse grid ? 38

39 Prolongation  Prolongation (interpolation) operator x h = x 2h

40 Restriction  Restriction operator x h = x 2h

41 Smoothing  The basic iterations in each level In  ph : x ph old  x ph new  Iteration reduces the error, makes the error smooth geometrically. So the iteration is called smoothing. 41

42 Why multilevel ?  Coarse lever iteration is cheap.  More than this… Coarse level smoothing reduces the error more efficiently than fine level in some way. Why ? ( Question 2 ) 42

43 Error restriction  Map error to coarse grid will make the error more oscillatory K = 4,  =  /2 K = 4,  =  43

44 Idea 2: Residual correction  Known current solution x  Solve Ax=b eq. to  MG do NOT map x directly between levels Map residual equation to coarse level 1.Calculate r h 2.b 2h= I h 2h r h ( Restriction ) 3.e h = I h 2h x 2h ( Prolongation ) 4.x h = x h + e h 44

45 Why residual correction ?  Error is smooth at fine level, but the actual solution may not be.  Prolongation results in a smooth error in fine level, which is suppose to be a good evaluation of the fine level error.  If the solution is not smooth in fine level, prolongation will introduce more high frequency error. 45

46 ` Revised V-cycle with idea 2  Smoothing on x h  Calculate r h  b 2h= I h 2h r h  Smoothing on x 2h  e h = I h 2h x 2h  Correct: x h = x h + e h Restriction Prolongation  2h  h 46

47 What is A 2h  Galerkin condition 47

48 Going to multilevels  V-cycle and W-cycle  Full Multigrid V-cycle h 2h 4h h 2h 4h 8h 48

49 Performance of Multigrid  Complexity comparison Gaussian eliminationO(N 2 ) Jacobi iteration O(N 2 log) Gauss-Seidel O(N 2 log) SOR O(N 3/2 log) Conjugate gradient O(N 3/2 log) Multigrid ( iterative ) O(Nlog) Multigrid ( FMG )O(N) 49

50 Summary of MG ideas Important ideas of MG 1.Hierarchical iteration 2.Residual correction 3.Galerkin condition 4.Smoothing the error: high frequency : fine grid low frequency : coarse grid 50

51 AMG :for unstructured grids  Ax=b, no regular grid structure  Fine grid defined from A

52 Three questions for AMG  How to choose coarse grid  How to define the smoothness of errors  How are interpolation and prolongation done 52

53 How to choose coarse grid  Idea: C/F splitting As few coarse grid point as possible For each F-node, at least one of its neighbor is a C-node Choose node with strong coupling to other nodes as C-node

54 How to define the smoothness of error  AMG fundamental concept: Smooth error = small residuals  ||r|| << ||e|| 54

55 How are Prolongation and Restriction done  Prolongation is based on smooth error and strong connections  Common practice: I 55

56 AMG Prolongation (2) 56

57 AMG Prolongation (3)  Restriction : 57

58 Summary  Multigrid is a multilevel iterative method.  Advantage: scalable  If no geometrical grid is available, try Algebraic multigrid method 58

59 59 Direct A = LU Iterative y’ = Ay Non- symmetric Symmetric positive definite More RobustLess Storage (if sparse) More Robust More General The landscape of Solvers

60 60 References  G.H. Golub and C.F. Van Loan, Matrix Computataions, Third Edition, Johns Hopkins, 1996  Y. Saad, Iterative Methods for Sparse Linear Systems, Second Edition, SIAM, 2003.


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