CS475 / CM375 Lecture 23: Nov 29, 2011 Convergence of Iterative - - PDF document

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CS475 / CM375 Lecture 23: Nov 29, 2011 Convergence of Iterative - - PDF document

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29/11/2011 CS475 / CM375 Lecture 23: Nov 29, 2011 Convergence of Iterative Methods CS475/CM375 (c) 2011 P. Poupart & J. Wan 1 Richardson Convergence The iteration matrix is given by:

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29/11/2011 1

CS475 / CM375 Lecture 23: Nov 29, 2011

Convergence of Iterative Methods

CS475/CM375 (c) 2011 P. Poupart & J. Wan 1

Richardson Convergence

  • The iteration matrix is given by:

≡

  • Suppose , is an eigenpair of . Then

1

  • Hence ≡ 1 is an eigenvalue of

CS475/CM375 (c) 2011 P. Poupart & J. Wan 2

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Richardson Convergence

  • Lemma: Let and be the smallest and

largest eigenvalue of . Then max 1 , 1

  • Proof:

CS475/CM375 (c) 2011 P. Poupart & J. Wan 3

Richardson Convergence

  • Notes:

– If 0 and 0, then either 1 1 0

  • r

1 1 0 ⟹ 1 ⟹ Richardson method diverges

CS475/CM375 (c) 2011 P. Poupart & J. Wan 4

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Richardson Convergence

  • Theorem: Assume all eigenvalues of are positive.

Then Richardson converges if and only if 0 2/

  • Proof: if 0 2/, then

CS475/CM375 (c) 2011 P. Poupart & J. Wan 5

Richardson Convergence

  • Proof continued: Assume 1. Then

CS475/CM375 (c) 2011 P. Poupart & J. Wan 6

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Richardson Convergence

  • : 1 1

2/

  • 1
  • /

/

  • Picture:

CS475/CM375 (c) 2011 P. Poupart & J. Wan 7

Jacobi Convergence

  • Theorem: If and 2 are SPD, then Jacobi

converges

  • Proof: Let be an eigenval. of

CS475/CM375 (c) 2011 P. Poupart & J. Wan 8

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Jacobi Convergence

  • Proof continued: Since 2 is SPD,

CS475/CM375 (c) 2011 P. Poupart & J. Wan 9

Gauss‐Seidel & SOR Convergence

  • Theorem: If is SPD, then GS & SOR 0 2

converge.

  • Definition: is an M‐matrix if

i. ii. iii. exists and 0 ∀

  • Theorem: If is an M‐matrix, Jacobi and GS
  • converge. Moreover
  • 1

i.e. the convergence rate of GS is better than that of Jacobi

CS475/CM375 (c) 2011 P. Poupart & J. Wan 10

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Example: Poisson Equation

  • Recall:
  • Theorem: Let be the 2D Laplacian matrix. The

eigenvalues of are given by:

  • sin
  • sin
  • 1 ,

CS475/CM375 (c) 2011 P. Poupart & J. Wan 11

Example: Poisson Equation

  • The smallest eigenvalue is attained for 1
  • sin
  • The largest eigenvalue is attained for
  • sin
  • sin
  • 1
  • , 1
  • cos
  • is SPD and an M‐matrix

CS475/CM375 (c) 2011 P. Poupart & J. Wan 12

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Richardson

max 1

  • sin
  • , 1
  • cos
  • Convergence holds for 0
  • and
  • 1 2 sin
  • CS475/CM375 (c) 2011 P. Poupart & J. Wan

13

Jacobi

  • ptimal Richardson

∴ 1 2 sin 2 cos

  • By Taylor expansion: cos 1
  • ! ⋯

∴ cos 1 2

  • For small mesh size , 1 → slow convergence

CS475/CM375 (c) 2011 P. Poupart & J. Wan 14

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Gauss Seidel (GS)

  • cos

1 sin 1

  • For small , slow convergence for GS
  • Convergence rate 2 convergence rate of Jacobi

CS475/CM375 (c) 2011 P. Poupart & J. Wan 15

Successive Over Relaxation (SOR)

  • For SOR:
  • 1
  • 1 2
  • Optimal SOR is an order of magnitude better than GS

and Jacobi

CS475/CM375 (c) 2011 P. Poupart & J. Wan 16