Parallel Sparse Linear Algebra for Homotopy Methods
Globally convergent homotopy methods are used to solve difficult nonlinear systems of equations by tracking the zero curve of a homotopy map. Homotopy curve tracking involves solving a sequence of linear systems, which often vary greatly in difficulty. In this research, a popular iterative solution...
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Virginia Tech
2014
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| Online Access: | http://hdl.handle.net/10919/30718 http://scholar.lib.vt.edu/theses/available/etd-81897-131850/ |
| Summary: | Globally convergent homotopy methods are used to solve difficult
nonlinear systems of equations by tracking the zero curve of a homotopy
map. Homotopy curve tracking involves solving a sequence of linear
systems, which often vary greatly in difficulty. In this research, a
popular iterative solution tool, GMRES(k), is adapted to deal with the
sequence of such systems. The proposed adaptive strategy of GMRES(k)
allows tuning of the restart parameter k based on the GMRES
convergence rate for the given problem. Adaptive GMRES(k) is shown
to be superior to several other iterative techniques on analog
circuit simulation problems and on postbuckling structural analysis
problems.
Developing parallel techniques for robust but expensive sequential
computations, such as globally convergent homotopy methods, is
important. The design of these techniques encompasses the functionality
of the iterative method (adaptive GMRES(k)) implemented sequentially
and is based on the results of a parallel
performance analysis of several implementations. An implementation of
adaptive GMRES(k) with Householder reflections in its
orthogonalization phase is developed. It is shown that
the efficiency of linear system solution by the adaptive GMRES(k)
algorithm depends on the change in problem difficulty when the problem
is scaled.
In contrast, a standard GMRES(k) implementation using Householder
reflections maintains a
constant efficiency with increase in problem size and number of
processors, as concluded analytically and experimentally. The supporting
numerical results are obtained on three distributed memory homogeneous
parallel architectures: CRAY T3E, Intel Paragon, and IBM SP2. === Ph. D. |
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