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\title{Using multi-level ILU-factorizations within the
Jacobi-Davidson method}
\author{F.W. Wubs \\
       University of Groningen\\
       P.O. Box 800, 9700 AV Groningen, The Netherlands\\
       \and
       G.L.G. Sleijpen\\
       Utrecht University\\
       P.O. Box 80.010, 3508 TA Utrecht, The Netherlands}
\date{}
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\maketitle
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\subsection*{Abstract}
The solution of large sparse systems occurring in eigenvalue problems
often forms the bottle-neck in the computation.
A promising category of preconditioners for such systems
form the algebraic multi-level ILU
factorizations. The crucial aspects of these factorizations are ordering
and dropping. In the variant developed in Groningen, called MRILU,
the ordering is based on the magnitude of the elements and determined
during the factorization phase. For the dropping local and global criteria
are used. The thus developed method can handle matrices arising from
structured and unstructured problems including systems of PDEs.
Experiments reveal that the method is efficient and shows in
several cases near grid independent convergence.

MRILU has been applied to eigenvalue problems in combination with the
JDQZ (Jacobi-Davidson method with deflation and restart). JDQZ is a Newton-like method
with subspace acceleration. Due to orthogonality constraints, the
Jacobian of the Newton correction equation
has the form of an augmented matrix. The factorization
of the augmented matrix is used as an approximate Jacobian. Hence, only one solve and
matrix vector product are needed per iteration step in the resulting
JDQZ method.
The implementation is such that only the bordering part of
the matrix changes at every step. Thereby the factorization of a new
augmented matrix can be obtained by a small adjustment of the old
factorization.
The
combination of MRILU and JDQZ yields an effective eigenvalue solver. For
example, it was successfully used to find a range of critical
eigenvalues of a Navier-Stokes
Jacobian.
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\subsubsection*{References}
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E.F.F. Botta and F.W. Wubs. MRILU: An effective algebraic multi-level
ILU-preconditioner for sparse matrices. To appear in SIAM J. on Matrix Anal. Appl.\\[1mm]
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G.L.G. Sleijpen and H.A. van der Vorst. A Jacobi-Davidson iteration method
for linear eigenvalue problems, SIAM J. Matrix Anal. Appl, 17 (1996),
pp. 401-425\\[1mm]
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D.R. Fokkema, G.L.G. Sleijpen and H.A. van der Vorst. Jacobi-Davidson style
QR and QZ algorithms for the reduction of matrix pencils. To appear in
SIAM J. Sc. Comput. (1998)
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