Publication: Krylov methods for large-scale modern problems in numerical linear algebra
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2018-07
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2018-09-28
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Abstract
Large-scale problems have attracted much attention in the last decades since
they arise from different applications in several fields. Moreover, the matrices that
are involved in those problems are often sparse, this is, the majority of their entries
are zero. Around 40 years ago, the most common problems related to large-scale and
sparse matrices consisted in solving linear systems, finding eigenvalues and/or eigenvectors,
solving least square problems or computing singular value decompositions.
However, in the last years, large-scale and sparse problems of different natures have
appeared, motivating and challenging numerical linear algebra to develop effective
and efficient algorithms to solve them.
Common difficulties that appear during the development of algorithms for solving
modern large-scale problems are related to computational costs, storage issues and
CPU time, given the large size of the matrices, which indicate that direct methods
can not be used. This suggests that projection methods based on Krylov subspaces
are a good option to develop procedures for solving large-scale and sparse modern
problems.
In this PhD Thesis we develop novel and original algorithms for solving two
large-scale modern problems in numerical linear algebra: first, we introduce the
R-CORK method for solving rational eigenvalue problems and, second, we present
projection methods to compute the solution of T-Sylvester matrix equations, both
based on Krylov subspaces.
The R-CORK method is an extension of the compact rational Krylov method
(CORK) [104] introduced to solve a family of nonlinear eigenvalue problems that can
be expressed and linearized in certain particular ways and which include arbitrary
polynomial eigenvalue problems, but not arbitrary rational eigenvalue problems.
The R-CORK method exploits the structure of the linearized problem by representing
the Krylov vectors in a compact form in order to reduce the cost of storage,
resulting in a method with two levels of orthogonalization. The first level of orthogonalization
works with vectors of the same size as the original problem, and the
second level works with vectors of size much smaller than the original problem. Since
vectors of the size of the linearization are never stored or orthogonalized, R-CORK
is more efficient from the point of view of memory and orthogonalization costs than
the classical rational Krylov method applied to the linearization. Moreover, since
the R-CORK method is based on a classical rational Krylov method, the implementation
of implicit restarting is possible and we present an efficient way to do it, that
preserves the compact representation of the Krylov vectors.
We also introduce in this dissertation projection methods for solving the TSylvester
equation, which has recently attracted considerable attention as a consequence
of its close relation to palindromic eigenvalue problems and other applications.
The theory concerning T-Sylvester equations is rather well understood, and before the work in this thesis, there were stable and efficient numerical algorithms
to solve these matrix equations for small- to medium- sized matrices. However,
developing numerical algorithms for solving large-scale T-Sylvester equations was a
completely open problem. In this thesis, we introduce several projection methods
based on block Krylov subspaces and extended block Krylov subspaces for solving
the T-Sylvester equation when the right-hand side is a low-rank matrix. We also offer
an intuition on the expected convergence of the algorithm based on block Krylov
subspaces and a clear guidance on which algorithm is the most convenient to use in
each situation.
All the algorithms presented in this thesis have been extensively tested, and the
reported numerical results show that they perform satisfactorily in practice.
Description
Keywords
Numerical Linear Algebra, Krylov subspace, R-CORK method, Matrix equations, T-Sylvester equations