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**CSE 245: Computer Aided Circuit Simulation and Verification**

Matrix Computations: Iterative Methods (II) Chung-Kuan Cheng

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**Outline Introduction Direct Methods Iterative Methods Formulations**

Projection Methods Krylov Space Methods Preconditioned Iterations Multigrid Methods Domain Decomposition Methods

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**Introduction Iterative Methods Direct Method LU Decomposition**

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

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**Formulation Error in A norm Minimal Residue**

Matrix A is SPD (Symmetric and Positive Definite Minimal Residue Matrix A can be arbitrary

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**Formulation: Error in A norm**

Min (x-x*)TA(x-x*), where Ax* =b, and A is SPD. Min E(x)=1/2 xTAx - bTx Search space: x=x0+Vy, where x0 is an initial solution, matrix Vnxm has m bases of subspace K vector ym contains the m variables.

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**Solution: Error in A norm**

Min E(x)=1/2xTAx - bTx, Search space: x=x0+Vy, r=b-Ax VTAV is nonsingular if A is SPD, V is full ranked y=(VTAV)-1VTr0 x=x0+V(VTAV)-1VTr0 VTr=0 E(x)=E(x0)-1/2 r0TV(VTAV)-1VTr0

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**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=x0+V’y, r=b-Ax V’TAV’ is nonsingular if A is SPD & V is full ranked x=x0+V’(V’TAV’)-1V’Tr0 =x0+V(VTAV)-1VTr0 V’Tr=WTVTr=0 E(x)=E(x0)-1/2 r0TV(VTAV)-1VTr0

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**Steepest Descent: Error in A norm**

Min E(x)=1/2 xTAx - bTx, Gradient: Ax-b= -r Set x=x0+yr0 r0TAr0 is nonsingular if A is SPD y=(r0TAr0)-1r0Tr0 x=x0+r0(r0TAr0)-1r0Tr0 r0Tr=0 E(x)=E(x0)-1/2 (r0Tr0)2/(r0TAr0)

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**Lanczos: Error in A norm**

Min E(x)=1/2 xTAx-bTx, Set x=x0+Vy v1=r0 vi is in K{r0, A, i} V=[v1,v2, …,vm] is orthogonal AV=VHm+vm+1 emT VTAV=Hm Note since A is SPD, Hm is Tm Tridiagonal

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**Conjugate Gradient: Error in A norm**

Min E(x)=1/2 xTAx-bTx, Set x=x0+Vy v1=r0 vi is in K{r0, A, i} V=[v1,v2, …,vm] is orthogonal in A norm, i.e. VTAV= [diag(viTAvi)] yi= (viTAvi)-1

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**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=x0+Vy where x0 is an initial solution, matrix Vnxm has m bases of subspace K vector ym contains the m variables.

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**Solution: Residual Min R(x)=(b-Ax)T(b-Ax) Search space: x=x0+Vy**

VTATAV is nonsingular if A is nonsingular and V is full ranked. y=(VTATAV)-1VTATr0 x=x0+V(VTATAV)-1VTATr0 VTATr= 0 R(x)=R(x0)-r0TAV(VTATAV)-1VTATr0

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**Steepest Descent: Residual**

Min R(x)=(b-Ax)T(b-Ax) Gradient: -2AT(b-Ax)=-2ATr Let x=x0+yATr0 VTATAV is nonsingular if A is nonsingular where V=ATr0. y=(VTATAV)-1VTATr0 x=x0+V(VTATAV)-1VTATr0 VTATr= 0 R(x)=R(x0)-r0TAV(VTATAV)-1VTATr0

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**GMRES: Residual Min R(x)=(b-Ax)T(b-Ax) Gradient: -2AT(b-Ax)=-2ATr**

Let x=x0+yATr0 v1=r0 vi is in K{r0, A, i} V=[v1,v2, …,vm] is orthogonal AV=VHm+vm+1 emT=Vm+1Hm x=x0+V(VTATAV)-1VTATr0 =x0+V(HmTHm)-1 HmTe1|r0|2

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**Conjugate Residual: Residual**

Min R(x)=(b-Ax)T(b-Ax) Gradient: -2AT(b-Ax)=-2ATr Let x=x0+yATr0 v1=r0 vi is in K{r0, A, i} (AV)TAV= D Diagonal Matrix x=x0+V(VTATAV)-1VTATr0 =x0+VD-1VTATr0

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**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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**Orthogonal Direction We don’t know !!!**

Pick orthogonal search direction: We like to leave the error of xi+1 orthogonal to the search direction di, i.e. We don’t know !!!

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**Orthogonal A-orthogonal**

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

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Search Step Size

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**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.

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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

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**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

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**Construct Search Direction -2**

let For i > 0

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**Construct Search Direction -3**

can get next direction from the previous one, without saving them all. let then

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**Conjugate Gradient Algorithm**

Given x0, iterate until residue is smaller than error tolerance

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**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 = λmax(A) / λmin(A) Residual is reduced by a constant factor by O(κ1/2(A)) iterations of CG.

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**Other Krylov subspace methods**

Nonsymmetric linear systems: GMRES: for i = 1, 2, 3, find xi Ki (A, b) such that ri = (Axi – b) Ki (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 Ki (A, b) and Ki (AT, 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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**Preconditioners Suppose you had a matrix B such that:**

condition number κ(B-1A) is small By = z is easy to solve Then you could solve (B-1A)x = B-1b 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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**Preconditioned conjugate gradient iteration**

x0 = 0, r0 = b, d0 = B-1 r0, y0 = B-1 r0 for k = 1, 2, 3, . . . αk = (yTk-1rk-1) / (dTk-1Adk-1) step length xk = xk-1 + αk dk approx solution rk = rk-1 – αk Adk residual yk = B-1 rk preconditioning solve βk = (yTk rk) / (yTk-1rk-1) improvement dk = yk + βk dk search direction One matrix-vector multiplication per iteration One solve with preconditioner per iteration

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**Outline Iterative Method Stationary Iterative Method (SOR, GS,Jacob)**

Krylov Method (CG, GMRES) Multigrid Method

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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.

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The model problem + v1 v2 v3 v4 v5 v6 v7 v8 vs Ax = b

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**Simple iterative method**

x(0) -> x(1) -> … -> x(k) Jacobi iteration Matrix form : x(k) = Rjx(k-1) + Cj General form: x(k) = Rx(k-1) + C (1) Stationary: x* = Rx* + C (2)

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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) = Rke(0) (6) ((1)+(2)+(3)) Convergence:

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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

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**Iteration reduce low frequency error efficiently**

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

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**Multigrid – a first glance**

Two levels : coarse and fine grid 2h A2hx2h=b2h 1 2 3 4 h Ahxh=bh 1 2 3 4 5 6 7 8 Ax=b

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**Idea 1: the V-cycle iteration**

Also called the nested iteration Start with 2h A2hx2h = b2h A2hx2h = b2h Iterate => Prolongation: Restriction: h Iterate to get Ahxh = bh Question 1: Why we need the coarse grid ?

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**Prolongation Prolongation (interpolation) operator xh = x2h 1 2 3 4 5**

6 7 8

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Restriction Restriction operator xh = x2h 1 2 3 4 5 6 7 8

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**Smoothing The basic iterations in each level In ph: xphold xphnew**

Iteration reduces the error, makes the error smooth geometrically. So the iteration is called smoothing.

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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 )

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

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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 Calculate rh b2h= Ih2h rh ( Restriction ) eh = Ih2h x2h ( Prolongation ) xh = xh + eh

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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.

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**Revised V-cycle with idea 2**

2h h Smoothing on xh Calculate rh b2h= Ih2h rh Smoothing on x2h eh = Ih2h x2h Correct: xh = xh + eh ` Restriction Prolongation

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What is A2h Galerkin condition

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**Going to multilevels V-cycle and W-cycle Full Multigrid V-cycle h 2h**

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**Performance of Multigrid**

Complexity comparison Gaussian elimination O(N2) Jacobi iteration O(N2log) Gauss-Seidel SOR O(N3/2log) Conjugate gradient Multigrid ( iterative ) O(Nlog) Multigrid ( FMG ) O(N)

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**Summary of MG ideas Important ideas of MG Hierarchical iteration**

Residual correction Galerkin condition Smoothing the error: high frequency : fine grid low frequency : coarse grid

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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

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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

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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 4 3 5 6

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**How to define the smoothness of error**

AMG fundamental concept: Smooth error = small residuals ||r|| << ||e||

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**How are Prolongation and Restriction done**

Prolongation is based on smooth error and strong connections Common practice: I

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AMG Prolongation (2)

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AMG Prolongation (3) Restriction :

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**Summary Multigrid is a multilevel iterative method.**

Advantage: scalable If no geometrical grid is available, try Algebraic multigrid method

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**The landscape of Solvers**

Direct A = LU Iterative y’ = Ay More Robust More General Non- symmetric Symmetric positive definite More Robust Less Storage (if sparse)

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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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