Einladung zu einem Vortrag von:
Gergana Bencheva
Institute for Parallel Processing, Bulgarian Academy of Sciences
Montag, 20. Dezember 2004
15:30 Uhr, HF 136
Parallel Algorithms for Separation of Variables
and Sparse Matrices Factorization
The appearance of parallel architectures and the recent progress
in computational technologies has inspired quite a lot of interest in
development of efficient parallel algorithms for solution of problems
in almost all nowadays scientific areas. Presented dissertation is
devoted to construction and analysis of parallel methods.
Subject of investigations in the thesis are numerical methods
for solution of large systems of linear algebraic equations obtained
after discretization of a second order elliptic boundary value problems.
Two approximation approaches are considered: a) by finite differences
with five-point stencil for a separable problem; b) by rotated
bilinear nonconforming finite elements (NFE) for a problem in a
general form.
The main results, obtained in the thesis, are constructive.
They are presented in three chapters.
A theoretical and experimental comparative analysis of five
separable solvers is made in the first part. Namely, the attention is
focused on the standard (SV) and the fast (FASV) algorithm for separation
of variables as well as on the standard (SM) and two variants (GMF and GMS)
of the generalized marching algorithm. The proposed original modification
GMS of GMF simplifies the structure of the generalized marching algorithm
and has advantages with respect to parallel implementation.
An almost optimal separable preconditioner for anisotropic
elliptic boundary value problems discretized by rotated bilinear NFE
is proposed and investigated theoretically and experimentally.
A characterization of the anisotropy coefficient influence on the
efficiency of the preconditioner is obtained.
The second part deals with parallel direct solvers. New parallel
implementations of the algorithms FASV, GMF and GMS are proposed and a
theoretical comparative analysis of their properties is derived.
The third part is devoted to parallel iterative methods. A new
parallel preconditioner for anisotropic elliptic boundary value problems,
discretized by rotated bilinear NFE is proposed.
The modified incomplete Cholesky factorization is applied to a locally
constructed approximation of the stiffness matrix. Uniform estimates for
the condition number are obtained with respect to the problem size and
coefficient jumps.
A characterization for the parallel efficiency of the constructed
direct and iterative solvers is obtained for classes of distributed
memory parallel computing systems.
An experimental comparative analysis of the behaviour of each of
the proposed parallel solvers on distributed systems with different
characteristics is made.
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