Iterative Methods Mostly for SPD systems Iterative Linear - - PDF document

Iterative Linear System Solvers for Large, Sparse

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Matrices Iterative Methods

Mostly for SPD systems

 conjugate gradient and its variants

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 minimize through successive line

searches

Steepest Descent

Decrease function

 negative of gradient

h h

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 how much?

Steepest Descent

How much?

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Steepest Descent

Algorithm

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Let’s do Better

Conjugate directions

 don’t undo earlier gains

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Conjugate Gradient

Directions A-orthogonal to error

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Conjugate Gradient

Next direction?

 orthogonal to all others  easy update:

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Algorithm

Sweeps matrix multiply dot product acc.

What happened to d? (see section 7.3 of Shewchuk)

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Iterative Methods

What you’ll need

 you supply  given topological datastructure

i l i l i li

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 implement matrix multiplies

 stopping criterion: relative residual

Typical Implementation

What lives where?

 degrees of freedom at vertices

 need additional variables for solver

t i t i d

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 matrix entries on edges

Matrix multiplies

 iterate over mesh!  mesh is sparse matrix data structure

Iterative Methods

Other issues

 number of iterations proportional

to  (steepest descent),  (CG)

 preconditioning

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 preconditioning

 use approximate inverse

 diagonal (Jacobi)  incomplete Cholesky,

hierarchical,…

Other System Types

SPD

 conjugate gradients  Shewchuk, “CG w/o the Agony”

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non-S PD & non-S non-PD

 bi-conjugate gradients  see Numerical Recipes

 http://www.nr.com/ (chapter 2.7)