MA471 Fall 2003

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Lecture 20

• MA471 Fall 2003

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Relaxed Jacobi Iteration

• Instead of the usual Jacobi iteration
• We will instead make a partial step in the
  direction of the residual
• Where w is a relaxation parameter to be chosen.

i.e. a linear combination of the previous x and
the residual
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Stability Analysis

• Does the relaxed Jacobi iteration converge?
• The new error iteration matrix is
• Compared with the unrelaxed Jacobi iterator

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Eigenvalues of The Relaxed Jacobi Iterator Matrix

• Given the two matrices
• We instantly realize that their eigenvalues are
  related
• Since the new iterator only involves a scaling
  and terms added to the diagonal..

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cont

• Because the eigenvalues of the relaxed scheme are
  a linear combination of (1-w) and the
  weigenvalues of the Jacobi iterator we are
  guaranteed convergence as long as 0ltwlt1

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Example System

• We will now use a 1D heat equation to demonstrate
  how fast the relaxed Jacobi scheme converges.

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Revisit 1-D Diffusion With Finite Difference and
Jacobi Iteration

• Recall the 1-D finite difference system
• From which we obtain a system to be solved at
  every time step (i.e. each n)

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Linear System

• So consider
• A relaxed Jacobi scheme for solving this is
• i.e.

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Error Equation For Relaxed Jacobi Iterate

• First define the error at the ith node at the
  mth Jacobi iterate as
• Then the error satisfies

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Error Analysis cont

• Now we wish to investigate how fast the error
  decays with iteration.
• Suppose the error is of the form
• Then the error equation
• Becomes

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Error Analysis cont

• So we can simplify
• To
• From which we deduce

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Restrictions on Theta

• Assume now that a finite interval is chosen and
  divided into N subintervals.
• In addition assume that the intervals is set to
  be periodic (i.e. uN u0)
• Since u is periodic, then the error will be
  periodic so

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Rate of Convergence

• We now see that the error decreases as
• Where
• What does this mean?

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Rates of Decay Depending on Alpha

• We now see that the the rate of decay of the
  Fourier components of the error depends on the
  alpha parameter.
• The error was assumed to take the form
• And we found that the g multiplier had the form

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Special Cases

• The alpha0 mode decays by a factor of
  every iteration.
• The alpha1 mode decays by a factor of
• i.e. the decay rate depends of an error mode
  dependson its mode number alpha.

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Matlab Script To Display g(alpha)
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Decay Rate As a Function of alpha (w1)
Slowdecaying
Fastdecaying
Fastoscillating
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Decay Rate For Range of w
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Interpretation

• If we think of the error as a function of
  iteration and space, and expand this in terms of
  a discrete Fourier expansion then it appears that
  the longer wavelength components of the error
  decay the slowest.
• We will use this information as the basis of a
  new iterative scheme, namely the multigrid method.