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1. In each processor Do:2 diii¢’G"$t C
yt h h h t e D w
2. Forward solve:
3. Perform the forward solve for the interior nodes.
4. Receive the updated values from the adjacent processors H
5. with . 4 ¢ ) )z 4 ¢ Cz
6. Perform the forward solve for the interface nodes.
7. Send the updated values of boundary nodes to the adjacent
8. processors with . )z 4 ¢ 4¢
q Cz
H
9. Backward solve:
10. Receive the updated values from the adjacent processors H
11. with . )z 4 ¢ 4¢
q Cz
12. Perform the backward solve for the boundary nodes.
13. Send the updated values of boundary nodes to the adjacent
14. processors, with . 4 ¢ ) )z 4 ¢ Cz
H
15. Perform the backward solve for the interior nodes.
16. EndDo
As in the ILU(0) factorization, the interior nodes do not depend on the nodes from the
external processors and can be computed in parallel in lines 3 and 15. In the forward solve,
the solution of the interior nodes is followed by an exchange of data and the solution on
the interface. The backward solve works in reverse in that the boundary nodes are ¬rst
computed, then they are sent to adjacent processors. Finally, interior nodes are updated.
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This section gives a brief account of other parallel preconditioning techniques which are
sometimes used. The next chapter also examines another important class of methods, which
were brie¬‚y mentioned before, namely, the class of Domain Decomposition methods.
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7 —— p w7 z p — w z
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Another class of preconditioners that require only matrixbyvector products, is the class
of approximate inverse preconditioners. Discussed in Chapter 10, these can be used in
many different ways. Besides being simple to implement, both their preprocessing phase
and iteration phase allow a large degree of parallelism. Their disadvantage is similar to
polynomial preconditioners, namely, the number of steps required for convergence may be
large, possibly substantially larger than with the standard techniques. On the positive side,
they are fairly robust techniques which can work well where standard methods may fail.
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A somewhat specialized set of techniques is the class of ElementByElement (EBE) pre
conditioners which are geared toward ¬nite element problems and are motivated by the
desire to avoid assembling ¬nite element matrices. Many ¬nite element codes keep the
data related to the linear system in unassembled form. The element matrices associated
with each element are stored and never added together. This is convenient when using di
rect methods since there are techniques, known as frontal methods, that allow Gaussian
elimination to be performed by using a few elements at a time. W ¢R
˜ ˜
It was seen in Chapter 2 that the global stiffness matrix is the sum of matrices
associated with each element, i.e.,
™¢ G ‚
W ¢R
˜ ˜
D h
c ¢
W ¢R
˜
Here, the matrix is an matrix de¬ned as
¢
i i
W ¢R
˜ P˜¢ 2 w'2
¢
D §¥
¦¤
˜
in which is the element matrix and is a Boolean connectivity matrix which maps ¢2
©¨
¦¤
˜ ˜
the coordinates of the small matrix into those of the full matrix . Chapter 2 showed
¤ ¦
how matrixbyvector products can be performed in unassembled form. To perform this
product in parallel, note that the only potential obstacle to performing the matrixbyvector
product in parallel, i.e., across all elements, is in the last phase, i.e., when the contributions W ¢R
˜
are summed to the resulting vector . In order to add the contributions in paral

lel, group elements that do not have nodes in common. Referring to Equation (2.35), the
contributions
˜ }
¢ 2x  w¢
D ¤ ¦