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

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

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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

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

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5 Formulation: Error in A norm Min E(x)=1/2 x T Ax – b T x, A is SPD Suppose that we knew Ax * =b E(x)=1/2 (x-x * ) T A(x-x * ). For Krylov space approach, search space: x=x 0 +Vy, where x 0 is an initial solution, matrix V nxm is given and full ranked vector y m contains the m variables.

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6 Solution and Error Min E(x)=1/2x T Ax - b T x, Search space: x=x 0 +Vy, r=b-Ax Derivation 1.V T AV is nonsingular (A is SPD & V is full ranked 2.The variable: y=(V T AV) -1 V T r 0 (derivation) 3.Thus, solution: x=x 0 +V(V T AV) -1 V T r 0 Property of the solution 1.Residue: V T r=0 (proof) 2.Error: E(x)=E(x 0 )-1/2 r 0 T V(V T AV) -1 V T r 0

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7 Solution and Error Min E(x)=1/2x T Ax - b T x, Search space: x=x 0 +Vy, r=b-Ax The variable: y=(V T AV) -1 V T r 0 Derivation of y: Use the condition that dE/dy=0 We have V T AVy+V T Ax 0 -V T b=0 Thus y=(V T AV) -1 V T (b-Ax 0 )

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8 Solution and Error Min E(x)=1/2x T Ax - b T x, Search space: x=x 0 +Vy, r=b-Ax Property of Solution 1.Residue: V T r=0 (proof) Proof: V T r=V T (b-Ax)=V T (b-Ax 0 -AV(V T AV) -1 V T r 0 ) =V T (r 0 -AV(V T AV) -1 V T r 0 )=0 The residue of the solution is orthogonal to the previous bases.

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9 Transformation of the bases For any V’=VW, where V nxm is given and W mxm is nonsigular, solution x remains the same for search space: x=x 0 +V’y Proof: 1.V’ T AV’ is nonsingular (A is SPD & V’ is full ranked) 2.x=x 0 +V’(V’ T AV’) -1 V’ T r 0 =x 0 +V(V T AV) -1 V T r 0 3.V’ T r=W T V T r=0 4.E(x)=E(x 0 )-1/2 r 0 T V(V T AV) -1 V T r 0

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10 Steepest Descent: Error in A norm Min E(x)=1/2 x T Ax - b T x, Gradient: -r = Ax-b Set x=x 0 +yr 0 1.r 0 T Ar 0 is nonsingular (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 )

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11 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

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12 Lanczos: Derive from Arnoldi Process Arnoldi Process: AV=VH+h m+1 v m+1 e m T Input A, v 1 =r 0 /|r 0 | Output V =[v 1,v 2, …, v m ], H and h m+1, v m+1 k=0, h 10 =1 While h k+1,k !=0, k<=m v k+1 =r k /h k+1,k k=k+1, r k =Av k For i=1, k h ik =v i T r k r k =r k -h ik v i End h k+1,k= |r k | 2 End

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13 Lanczos: Create V=[v 1,v 2, …,v m ] which is orthogonal Input A, r 0, Output V and T=V T AV Initial: k=0, β 0 =|r 0 | 2, v 0 =0, v 1 =r 0 /β 0 while k≤ K or β k != 0 v k+1 =r k /β k k=k+1 a k =v k T Av k r k =Av k -a k v k - β k-1 v k-1 β k =|r k | 2 End

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14 Lanczos: Create V=[v 1,v 2, …,v m ] which is orthogonal Input A, r 0, Output V and T=V T AV T ii =a i, T ij =b i (j=i+1), =b j (j=i-1), =0 (else) Proof: By induction that v j T Av j = 0, if i

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15 Conjugate Gradient: 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.Set V=[v 1,v 2, …,v m ] orthogonal in A norm, i.e. V T AV= [diag(v i T Av i )]=D 4.x=x 0 +VD -1 V T r 0, 5.x= x 0 +∑ i=1,m d i v i v i T r 0, where d i =(v i T Av i ) -1

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

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17 Orthogonal A-orthogonal Instead of orthogonal search direction, we make search direction A –orthogonal (conjugate)

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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 18

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19 Conjugate Gradient: Min E(x)=1/2 x T Ax-b T x, Set x=x 0 +Vy Lanczos Method to derive V and T x=x 0 +VT -1 V T r 0 Decompose T to LDU=T (U=L T ) Thus, we have x=x 0 +V(LDL T ) -1 V T r 0 =x 0 +VL -T D -1 L -1 V T r 0

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Conjugate Gradient Algorithm Given x0, iterate until residue is smaller than error tolerance 20

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21 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.

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22 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...

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23 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

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24 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)

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25 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.

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26 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

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27 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

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28 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

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29 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

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

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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

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

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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

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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

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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

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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

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Multigrid – a first glance Two levels : coarse and fine grid 1 2 3 4 56 7 8 1 2 3 4 A 2h x 2h =b 2h A h x h =b h hh Ax=b 2h 37

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

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Prolongation Prolongation (interpolation) operator x h = x 2h 1 2 3 4 56 7 8 1 2 3 4 39

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Restriction Restriction operator x h = x 2h 1 2 3 4 56 7 8 1 2 3 4 40

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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

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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

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

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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

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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

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` 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

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What is A 2h Galerkin condition 47

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

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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

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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

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AMG :for unstructured grids Ax=b, no regular grid structure Fine grid defined from A 1 2 3 4 5 6 51

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

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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 1 2 3 4 5 6 53

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

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

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AMG Prolongation (2) 56

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AMG Prolongation (3) Restriction : 57

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

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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

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60 References G.H. Golub and C.F. Van Loan, Matrix Computataions, 4th Edition, Johns Hopkins, 2013 Y. Saad, Iterative Methods for Sparse Linear Systems, Second Edition, SIAM, 2003.

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