Iteration Methods

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Iteration Methods

• Jacobi, Gauss-Seidel, SOR
• By Kevin Jourdain

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Introduction

• PDEs Application
• Direct method vs. iteration methods
• Applications of iteration methods
• (a) Electric network consisting of resistors
• (b) Heat-conduction problems
• (c) Particule diffusion
• (d) Certain stress-strain problems
• (e) Fluid, magnetic, or electric potential

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PDEs Application

• Why are we using iteration methods?
• We use, for example, iteration methods in
• Elliptic PDE
• Parabolic PDE
• Hyperbolic PDE

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Elliptic PDE

• Once we discretise the PDEs, we obtain a linear
  system that a lot of time end up being a
  diagonally dominated matrix. This is when we will
  use the iteration methods. We will explain why
  later on. Example of linear system

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Direct methods vs. Iteration methods

• During the last Math Night, we talked about
  direct methods (Gauss-Jordan).
• Today, we will talk about iteration methods.

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Direct methods

• Theoretically, methods such as Gauss Jordan
  elimination will give us exact solutions.
• If the system is too big, we will have problems
  with
• the round-off errors
• The storage capacity of the computer.
• How to solve it ?

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Iteration Methods

• Iteration repeating a process over and over
  until an approximation of the solution is
  reached.

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Advantages

• It is useful to solve certain types of problems
• When the number of unknowns is very large but the
  coefficient matrix is sparse, Gauss Elimination
  becomes inefficient, and sometimes inapplicable
  if the methods are preferred. Additional
  advantages of iterative methods include (1)
  programming is simple, and (2) it is easily
  applicable when coefficients are nonlinear.
  Although there are many versions of iterative
  schemes, we introduce three iterative methods,
  Jacobi iterative, Gauss-Seidel, and
  successive-over-relaxation (SOR) methods.

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Advantages

• Iterative methods can be applied to system as
  many as 100,000 variables. Examples of these
  large systems arise in the solution of partial
  differential equations.

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Advantages

• Also the amount of storage, as stated earlier,
  is far less than directs methods. In our example,
  we have
• 100,000 100,000 variables for direct
• 3 100,000 2 variables for iteration

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Advantages

• Applications
• (a) Electric network consisting of resistors
• (b) Heat-conduction problems
• (c) Particule diffusion
• (d) Certain stress-strain problems
• (e) Fluid, magnetic, or electric potential

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Disadvantages

• Iteration cannot be applied to every system

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Iteration Methods

• Consider a linear system,
• A is a strictly diagonal dominated matrix, X
  unknown vector, b non-homogeneous vector.
• Therefore, a sufficient condition for iterative
  methods to converge is

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Iteration Method

• For all i

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Jacobi Method

• Jacobi iterative method can be derived as the
  following

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Jacobi

• Where

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Jacobi

• Exact formula obtained earlier
• Therefore, the iteration formula would be

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Jacobi

• Algebraic form
• Code

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Gauss-Seidel

• Gauss-Seidel iterative method can be derived as
  followed

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Gauss-Seidel

• Exact formula obtained earlier
• Therefore, the iteration formula would be

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Gauss-Seidel

• Algebraic form
• Code

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Gauss-Seidel

• Note
• Jacobi
• Gauss-Seidel
• Gauss-Seidel is a faster method because it uses
  the updated solution.

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SOR

• Successive-Over-Relaxation Method, is a modified
  version of the Gauss-Seidel Method
• Code

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Convergence

• The iterative process is terminated when a
  convergence criterion is satisfied. One commonly
  used stopping criterion, known as the relative
  change criteria, is to iterate until

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Convergence

• Is less than a prescribed tolerance ? gt 0. For
  most of cases, maximum iterations time is 30, by
  experience. Because of the completeness of Rn,
  the limit of X(k) is also in Rn.

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Conclusion

• Who is faster ??
• And the winner is

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Conclusion (final)

• SOR !!
• SOR should be faster than the two others,
  depending on how we peak geniously our ?.