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Advanced Incomplete Factorization Algorithms for Stiltijes Matrices

V.P.Il'in
Computing Center, Siberian Division RAS, Novosibirsk, Russia
The modern numerical methods for solving the linear algebraic systems
Au = f with high order sparse matrices A, which arise in grid approximations
of multidimensional boundary value problems, are based mainly on accelerated
iterative processes with easily invertible preconditioning matrices presented
in the form of approximate (incomplete) factorization of the original
matrix A, see [1], [2] for example. We consider some recent algorithmic
approaches, theoretical foundations, experimental data and open questions
for incomplete factorization of Stiltijes matrices which are "the best" ones
in the sense that they have the most advanced results. Special attention is
given to solving the elliptic differential equations with strongly variable
coefficients, singular perturbated diffusion-convection and parabolic
equations.
In general, the block tridiagonal Stiltijes matrix A=D-L-U with
block diagonal or diagonal matrix D, low- and upper-triangular
matrices L, U = L' can be approximated by the factorized matrix
B=(G - L)G^(G - U), where G is block-diagonal matrix defined by
some approximations of Shur complement matrices of A and G^ is inverse of G.
We describe two sets of explicit and implicit methods with closure in the
sense that in limit B is the exact factorization of A.
There are presented several modifications of the algorithms with various
iterative parameters which provide the generalization of wellknown methods and
application of extended row sum criteria
B yk = A yk, k=1,2,...,
the symmetrized alternating direction methods with nonsymmetric preconditioners,
adapted "flow-directed" algorithms, some versions of algebraic domain
decomposition and hierarhical multigrid approaches. Both theoretical estimates
and computational results are included.
References
1. V.P.Il'in. Iterative incomplete factorization methods, Wold Sci. Pub. Co.,
1992, 192 pp.
2. V.P.Il'in. Incomplete factorization methods for solution of the algebraic
systems (in Russian), Moscow: Nauka, 1995, 288 pp.