Unformatted text preview: LU(0) preconditioning
may not provide enough acceleration to the basic conjugate gradient algorithm. Thus,
we consider continuing the factorization to additional levels by allowing some ll in. We
will call a preconditioner ILU(p) if there are p levels of ll in. The next preconditioner
beyond ILU(0) allows one level of ll in. The structure of L1 for this ILU(1) algorithm
is shown on the right of Figure 9.3.8. One row nearest the bandwidth has been lled in.
It's simplest to imagine A as having the two bands corresponding to the lled band of L1
as also being fulled in. This situation is shown on the upper right of Figure 9.3.8. The
product L1 DLT lls in yet another band (cf. the lower portion of Figure 9.3.8), which is
ignored in the elimination.
An algorithm for performing incomplete factorization to level p for elliptic problems
on rectangular meshes follows the procedure described in Example 9.3.5. Let's, instead,
illustrate the method for a more general sparse matrix. This is typically done by de ning
a level of ll for each element created by Gaussian elimination. The assumption is that
elements get smaller as the ll in progresses. A size k , 0 < < 1, is assigned to an
element at ll level k. Initially, a nonzero element has a unit level of ll and a zero
element has an in nite level of ll. Rather than work with the LDLT factorization, let's
simplify our task and work with a more general LU factorization such as (9.1.3). The 9.3. Conjugate Gradient Methods J 51 J A= L 1= J-1
J L D LT =
1 Figure 9.3.8: Imagined structure of A (top left) and structure of L1 (top right) for an
ILU(1) preconditioner. The ll in when calculating L1 DLT is shown at the bottom.
1 computation is generated within a loop and, typically, the elements of L replace the
elements in the lower triangular portion of A and the elements of U replace the elements
in the upper triangular portion of A. A sample loop structure for the elimination is
shown in Figure 9.3.9. This is very basic Gaussian elimination. No piv...
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