THESE de DOCTORAT - cerfacs

56 Chapter 3: Development of a numerical tool for combustion noise analysis, AVSP-f
and the Incomplete Cholesky or LU factorization. These last are known as ‘incomplete’ since M −1
is allowed only to have nonzeros in the positions where A has nonzeros. This is done in order
to keep the sparsity of the original matrix A. A Complete Cholesky or LU factorization
would destroy zeros leading to a dense M matrix. These preconditioners are appropiate for illconditioned
problems of small size. But for big and ill-conditioned A matrices, these methods
are not the best suited.
3.4.2 Dynamic preconditioning: the embedded GMRES
The right preconditioning of the Ax = b problem reads
Ax = AMc = b (3.57)
The traditional way to solve this system would be first to find a matrix M such that AM is well
conditioned; second, to solve the system for the vector c; and finally to find x from x = Mc.
In this procedure it is imperative to explicitly construct and store the matrix M. A better way
to apply preconditioning would be to find a preconditioning matrix of size n × m by solving
an second linear system. In other words, applying GMRES to find a good preconditioner.
Recalling Eq. (3.18), we add the preconditioning matrix M into the computation
( )
(
)
Ax = A x 0 + ̂Q m y = b − r m → Ax = A x 0 + M ̂Q m c = b − r m (3.58)
where x = x 0 + M ̂Q m c. The matrix Ẑ m , which is the projection of M in the Krylov space, is
defined as Ẑ m = M ̂Q m . Eq. (3.58) becomes
( )
A x 0 + Ẑ m c = b − r m (3.59)
The preconditioning matrix M is let to be A −1 , so that AM −1 = I. The vector z m of the matrix
Ẑ m can now be found by solving the linear system
z m = Mq m → Az m = q m (3.60)
which is clearly a problem of the type Ax = b. Once the vector z m is found, the orthogonalization
(finding q m ) is carried out applying the Arnoldi procedure to AẐ m instead of A ̂Q m ,
resulting in a faster convergence. Figure 3.8 represents the procedure just described.
As it can be observed from Fig. 3.8 the system Ax = b is solved by one GMRES algorithm
embedded in another GMRES. As a consequence, the present strategy is called the embedded