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f f

§

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This establishes that . As a result, this strategy guarantees that the row sums of

§

¦

are equal to those of . For PDEs, the vector of all ones represents the discretization

of a constant function. This additional constraint forces the ILU factorization to be exact

for constant functions in some sense. Therefore, it is not surprising that often the algorithm

does well for such problems. For other problems or problems with discontinuous coef¬-

cients, MILU algorithms usually are not better than their ILU counterparts, in general.

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For regularly structured matrices there are two elements dropped at the

h S ¦S h S ¨S

-th step of ILU(0). These are and located on the north-west and south-

C

d

f d

f

east corners of the stencil, respectively. Thus, the row sum associated with step is 5S

C

h ¨ c w h S ¡S S

S

ªS ¢

f'd

d f

h S¢ S¢

f'd

d

and the MILU variant of the recurrence (10.19) is

h ¨S w h S S S

iS ¢

f'd

d f

h S¢ S

f¢

¨ 'd

d

S S S S y

ªS0¢

¨ ¨ ¬S ¢ ¨

S

h S¢ S

d ¢

f

d

§

¦

¨

The new ILU factorization is now such that in which according to (10.21)

¤

the -th row of the new remainder matrix is given by ¤

vt s C

u

¨ ˜S a 5 S C `

E C

5 8S 5S

C

whose row sum is zero.

This generic idea of lumping together all the elements dropped in the elimination pro-

¦

cess and adding them to the diagonal of can be used for any form of ILU factorization.

In addition, there are variants of diagonal compensation in which only a fraction of the

dropped elements are added to the diagonal. Thus, the term in the above example would ¢ S

´

be replaced by before being added to , where is typically between 0 and 1. Other

¢¡ ¡

S SS ¦

strategies distribute the sum among nonzero elements of and , other than the diago-

¢ S

nal.

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Incomplete factorizations which rely on the levels of ¬ll are blind to numerical values be-

cause elements that are dropped depend only on the structure of . This can cause some

dif¬culties for realistic problems that arise in many applications. A few alternative methods

are available which are based on dropping elements in the Gaussian elimination process

according to their magnitude rather than their locations. With these techniques, the zero

pattern is determined dynamically. The simplest way to obtain an incomplete factor-

ization of this type is to take a sparse direct solver and modify it by adding lines of code

which will ignore “small” elements. However, most direct solvers have a complex imple-

mentation which involves several layers of data structures that may make this approach

ineffective. It is desirable to develop a strategy which is more akin to the ILU(0) approach.

This section describes one such technique.

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A generic ILU algorithm with threshold can be derived from the IKJversion of Gaussian

elimination, Algorithm 10.2, by including a set of rules for dropping small elements. In

what follows, applying a dropping rule to an element will only mean replacing the element

by zero if it satis¬es a set of criteria. A dropping rule can be applied to a whole row by

applying the same rule to all the elements of the row. In the following algorithm, is a

full-length working row which is used to accumulate linear combinations of sparse rows in

the elimination and is the -th entry of this row. As usual, denotes the -th row of £

5S

.

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