Steepest decent and conjugate gradients cg
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Steepest Decent and Conjugate Gradients (CG). Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system. Steepest Decent and Conjugate Gradients (CG). Solving of the linear equation system Problem : dimension n too big, or not enough time for gauss elimination

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Steepest Decent and Conjugate Gradients (CG)

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Steepest Decent and Conjugate Gradients (CG)


Steepest Decent and Conjugate Gradients (CG)

  • Solving of the linear equation system


Steepest Decent and Conjugate Gradients (CG)

  • Solving of the linear equation system

  • Problem: dimension n too big, or not enough time for gauss elimination

    Iterative methods are used to get an approximate solution.


Steepest Decent and Conjugate Gradients (CG)

  • Solving of the linear equation system

  • Problem: dimension n too big, or not enough time for gauss elimination

    Iterative methods are used to get an approximate solution.

  • Definition Iterative method: given starting point , do steps

    hopefully converge to the right solution


starting issues


starting issues

  • Solving is equivalent to minimizing


starting issues

  • Solving is equivalent to minimizing

  • A has to be symmetric positive definite:


starting issues


starting issues

  • If A is also positive definite the solution of is the minimum


starting issues

  • If A is also positive definite the solution of is the minimum


starting issues

  • error:

    The norm of the error shows how far we are away from the exact solution, but can’t be computed without knowing of the exact solution .


starting issues

  • error:

    The norm of the error shows how far we are away from the exact solution, but can’t be computed without knowing of the exact solution .

  • residual:

    can be calculated


Steepest Decent


Steepest Decent

  • We are at the point . How do we reach ?


Steepest Decent

  • We are at the point . How do we reach ?

  • Idea: go into the direction in which decreases most quickly ( )


Steepest Decent

  • We are at the point . How do we reach ?

  • Idea: go into the direction in which decreases most quickly ( )

  • how far should we go?


Steepest Decent

  • We are at the point . How do we reach ?

  • Idea: go into the direction in which decreases most quickly ( )

  • how far should we go?

    Choose so that is minimized:


Steepest Decent

  • We are at the point . How do we reach ?

  • Idea: go into the direction in which decreases most quickly ( )

  • how far should we go?

    Choose so that is minimized:


Steepest Decent

  • We are at the point . How do we reach ?

  • Idea: go into the direction in which decreases most quickly ( )

  • how far should we go?

    Choose so that is minimized:


Steepest Decent

  • We are at the point . How do we reach ?

  • Idea: go into the direction in which decreases most quickly ( )

  • how far should we go?

    Choose so that is minimized:


Steepest Decent

  • We are at the point . How do we reach ?

  • Idea: go into the direction in which decreases most quickly ( )

  • how far should we go?

    Choose so that is minimized:


Steepest Decent

  • We are at the point . How do we reach ?

  • Idea: go into the direction in which decreases most quickly ( )

  • how far should we go?

    Choose so that is minimized:


Steepest Decent

one step of steepest decent can be calculated as follows:


Steepest Decent

one step of steepest decent can be calculated as follows:

  • stopping criterion: or with an given small

    It would be better to use the error instead of the residual, but you can’t calculate the error.


Steepest Decent

Method of steepest decent:


Steepest Decent

  • As you can see the starting point is important!


Steepest Decent

  • As you can see the starting point is important!

    When you know anything about the solution use it to guess a good starting point. Otherwise you can choose a starting point you want e.g. .


Steepest Decent - Convergence


Steepest Decent - Convergence

  • Definition energy norm:


Steepest Decent - Convergence

  • Definition energy norm:

  • Definition condition:

    ( is the largest and the smallest eigenvalue of A)


Steepest Decent - Convergence

  • Definition energy norm:

  • Definition condition:

    ( is the largest and the smallest eigenvalue of A)

    convergence gets worse when the condition gets larger


Conjugate Gradients


Conjugate Gradients

  • is there a better direction?


Conjugate Gradients

  • is there a better direction?

  • Idea: orthogonal search directions


Conjugate Gradients

  • is there a better direction?

  • Idea: orthogonal search directions


Conjugate Gradients

  • is there a better direction?

  • Idea: orthogonal search directions

  • only walk once in each direction and minimize


Conjugate Gradients

  • is there a better direction?

  • Idea: orthogonal search directions

  • only walk once in each direction and minimize

    maximal n steps are needed to reach the exact solution


Conjugate Gradients

  • is there a better direction?

  • Idea: orthogonal search directions

  • only walk once in each direction and minimize

    maximal n steps are needed to reach the exact solution

    has to be orthogonal to


Conjugate Gradients

  • example with the coordinate axes as orthogonal search directions:


Conjugate Gradients

  • example with the coordinate axes as orthogonal search directions:

    Problem: can’t be computed because (you don’t know !)


Conjugate Gradients

  • new idea: A-orthogonal


Conjugate Gradients

  • new idea: A-orthogonal

  • Definition A-orthogonal: A-orthogonal

    (reminder: orthogonal: )


Conjugate Gradients

  • new idea: A-orthogonal

  • Definition A-orthogonal: A-orthogonal

    (reminder: orthogonal: )

  • now has to be A-orthogonal to


Conjugate Gradients

  • new idea: A-orthogonal

  • Definition A-orthogonal: A-orthogonal

    (reminder: orthogonal: )

  • now has to be A-orthogonal to


Conjugate Gradients

  • new idea: A-orthogonal

  • Definition A-orthogonal: A-orthogonal

    (reminder: orthogonal: )

  • now has to be A-orthogonal to

    can be computed!


Conjugate Gradients

  • A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram-Schmidt (same idea as Gram-Schmidt).


Conjugate Gradients

  • Gram-Schmidt:

    linearly independent vectors


Conjugate Gradients

  • Gram-Schmidt:

    linearly independent vectors


Conjugate Gradients

  • Gram-Schmidt:

    linearly independent vectors

  • conjugate Gram-Schmidt:


Conjugate Gradients

  • A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram-Schmidt (same idea as Gram-Schmidt).

  • CG works by setting (makes conjugate Gram-Schmidt easy)


Conjugate Gradients

  • A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram-Schmidt (same idea as Gram-Schmidt).

  • CG works by setting (makes conjugate Gram-Schmidt easy)

    with


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Conjugate Gradients


Method of Conjugate Gradients:


Conjugate Gradients - Convergence


Conjugate Gradients - Convergence


Conjugate Gradients - Convergence

  • for steepest decent for CG

    Convergence of CG is much better!


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