CSE 245: Computer Aided Circuit Simulation and Verification

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CSE 245 Computer Aided Circuit Simulation and
Verification
Lecture 5 Matrix Solver II -Iterative
Method
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Outline

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

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Iterative Methods

• Stationary
• x(k1) Gx(k)c
• where G and c do not depend on iteration count
  (k)
• Non Stationary
• x(k1) x(k)akp(k)
• where computation involves information that
  change at each iteration

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Stationary Jacobi Method

• In the i-th equation solve for the value of xi
  while assuming the other entries of x remain
  fixed
• In matrix terms the method becomes
• where D, -L and -U represent the diagonal, the
  strictly lower-trg and strictly upper-trg parts
  of M
• MD-L-U

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Stationary-Gause-Seidel

• Like Jacobi, but now assume that previously
  computed results are used as soon as they are
  available
• In matrix terms the method becomes
• where D, -L and -U represent the diagonal, the
  strictly lower-trg and strictly upper-trg parts
  of M
• MD-L-U

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Stationary Successive Overrelaxation (SOR)

• Devised by extrapolation applied to Gauss-Seidel
  in the form of weighted average
• In matrix terms the method becomes
• where D, -L and -U represent the diagonal, the
  strictly lower-trg and strictly upper-trg parts
  of M
• MD-L-U

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SOR

• Choose w to accelerate the convergence
• W 1 Jacobi / Gauss-Seidel
• 2gtWgt1 Over-Relaxation
• W lt 1 Under-Relaxation

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Convergence of Stationary Method

• Linear Equation MXb
• A sufficient condition for convergence of the
  solution(GS,Jacob) is that the matrix M is
  diagonally dominant.
• If M is symmetric positive definite, SOR
  converges for any w (0ltwlt2)
• A necessary and sufficient condition for the
  convergence is the magnitude of the largest
  eigenvalue of the matrix G is smaller than 1
• Jacobi
• Gauss-Seidel
• SOR

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Outline

• Iterative Method
• Stationary Iterative Method (SOR, GS,Jacob)
• Krylov Method (CG, GMRES)
• Steepest Descent
• Conjugate Gradient
• Preconditioning
• Multigrid Method

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Linear Equation an optimization problem

• Quadratic function of vector x
• Matrix A is positive-definite, if
  for any nonzero vector x
• If A is symmetric, positive-definite, f(x) is
  minimized by the solution

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Linear Equation an optimization problem

• Quadratic function
• Derivative
• If A is symmetric
• If A is positive-definite
• is minimized by setting to 0

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For symmetric positive definite matrix A
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Gradient of quadratic form
The points in the direction of steepest increase
of f(x)
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Symmetric Positive-Definite Matrix A

• If A is symmetric positive definite
• P is the arbitrary point
• X is the solution point

since
We have,
If p ! x
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If A is not positive definite

• Positive definite matrix b) negative-definite
  matrix
• c) Singular matrix d) positive
  indefinite matrix

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Non-stationary Iterative Method

• State from initial guess x0, adjust it until
  close enough to the exact solution
• How to choose direction and step size?

i0,1,2,3,
Adjustment Direction
Step Size
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Steepest Descent Method (1)

• Choose the direction in which f decrease most
  quickly the direction opposite of
• Which is also the direction of residue

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Steepest Descent Method (2)

• How to choose step size ?
• Line Search
• should minimize f, along the direction of
  , which means

Orthogonal
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Steepest Descent Algorithm
Given x0, iterate until residue is smaller than
error tolerance
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Steepest Descent Method example

• Starting at (-2,-2) take the
• direction of steepest descent of f
• b) Find the point on the intersec-
• tion of these two surfaces that
• minimize f
• c) Intersection of surfaces.
• d) The gradient at the bottommost
• point is orthogonal to the gradient
• of the previous step

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Iterations of Steepest Descent Method
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Convergence of Steepest Descent-1
let
Eigenvector
EigenValue
j1,2,,n
Energy norm
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Convergence of Steepest Descent-2
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Convergence Study (n2)
assume
let
Spectral condition number
let
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Plot of w
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Case Study
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Bound of Convergence
It can be proved that it is also valid for ngt2,
where
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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
Pick orthogonal search direction

• We dont 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 gt 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 ?(A) A2 A-12
  ?max(A) / ?min(A)
• Residual is reduced by a constant factor by
  O(?1/2(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 gt save old vectors
  gt 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 gt 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 ?(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, . . . ak
(yTk-1rk-1) / (dTk-1Adk-1) step length xk
xk-1 ak dk-1 approx
solution rk rk-1 ak Adk-1
residual yk B-1 rk
preconditioning
solve ßk (yTk rk) / (yTk-1rk-1)
improvement dk yk ßk dk-1
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
• Axb
• 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
Ax b
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Simple iterative method

• x(0) -gt x(1) -gt -gt 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

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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
A2hx2hb2h

Ahxhbh
?h
Axb
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Idea 1 the V-cycle iteration

• Also called the nested iteration

Start with
?2h
A2hx2h b2h
A2hx2h b2h
Iterate gt
Prolongation ?
Restriction ?
?h
Ahxh bh
Iterate to get
Question 1 Why we need the coarse grid ?
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Prolongation

• Prolongation (interpolation) operator
• xh x2h

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Restriction

• Restriction operator
• xh x2h

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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 Axb 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 4h
h 2h 4h 8h
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Performance of Multigrid

• Complexity comparison

Gaussian elimination O(N2)
Jacobi iteration O(N2log?)
Gauss-Seidel O(N2log?)
SOR O(N3/2log?)
Conjugate gradient O(N3/2log?)
Multigrid ( iterative ) O(Nlog?)
Multigrid ( FMG ) O(N)
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Summary of MG ideas

• Three important ideas of MG
• Nested iteration
• Residual correction
• Elimination of error
• high frequency fine grid
• low frequency coarse grid

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AMG for unstructured grids

• Axb, no regular grid structure
• Fine grid defined from A

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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 ltlt 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
More Robust
Less Storage (if sparse)