¦

˜

Observe that when is an exact preconditioner, i.e., when , then according to

' ' D

£

Proposition 13.1, is also an exact preconditioner, i.e., . This induced precon-

D

ditioner can be expressed as

An4 1 H¦ j

c

˜

c

¤¦# 0 v

2 w¤ # v '

D h

It may be argued that this uses a preconditioner related to the original problem to be solved £

in the ¬rst place. However, even though the preconditioning on may be de¬ned from a

˜

preconditioning of , the linear system is being solved for the interface variables. That is

typically much smaller than the original linear system. For example, GMRES can be used

with a much larger dimension of the Krylov subspace since the Arnoldi vectors to keep in

memory are much smaller. Also note that from a Partial Differential Equations viewpoint,

systems of the form (13.44) correspond to the Laplace equation, the solutions of which

are “Harmonic” functions. There are fast techniques which provide the solution of such

equations inexpensively.

˜'

In the case where is an ILU factorization of , can be expressed in an ex- '

plicit form in terms of the entries of the factors of . This de¬nes a preconditioner to

˜

£

that is induced canonically from an incomplete LU factorization of . Assume that the

preconditioner is in a factored form , where

' '¢ D ' '

¥ ¥ c

¡¢

¢

¢ 5 #¢

v

'¢ '

D D h

c

£ ¢ ¢

v ¦ ¦

Then, the inverse of will have the following structure:

'

c c c

¢ v'

v' D v'

¡¥

£ £ ¡¥

£

¢

£

D c c

¢ ¢

¦ ¦

v v

£¥ £

£

D

¢c v

c

v ¦