24 research outputs found
A domain decomposition strategy to efficiently solve structures containing repeated patterns
This paper presents a strategy for the computation of structures with
repeated patterns based on domain decomposition and block Krylov solvers. It
can be seen as a special variant of the FETI method. We propose using the
presence of repeated domains in the problem to compute the solution by
minimizing the interface error on several directions simultaneously. The method
not only drastically decreases the size of the problems to solve but also
accelerates the convergence of interface problem for nearly no additional
computational cost and minimizes expensive memory accesses. The numerical
performances are illustrated on some thermal and elastic academic problems
Krylov subspaces associated with higher-order linear dynamical systems
A standard approach to model reduction of large-scale higher-order linear
dynamical systems is to rewrite the system as an equivalent first-order system
and then employ Krylov-subspace techniques for model reduction of first-order
systems. This paper presents some results about the structure of the
block-Krylov subspaces induced by the matrices of such equivalent first-order
formulations of higher-order systems. Two general classes of matrices, which
exhibit the key structures of the matrices of first-order formulations of
higher-order systems, are introduced. It is proved that for both classes, the
block-Krylov subspaces induced by the matrices in these classes can be viewed
as multiple copies of certain subspaces of the state space of the original
higher-order system
Total and selective reuse of Krylov subspaces for the resolution of sequences of nonlinear structural problems
This paper deals with the definition and optimization of augmentation spaces
for faster convergence of the conjugate gradient method in the resolution of
sequences of linear systems. Using advanced convergence results from the
literature, we present a procedure based on a selection of relevant
approximations of the eigenspaces for extracting, selecting and reusing
information from the Krylov subspaces generated by previous solutions in order
to accelerate the current iteration. Assessments of the method are proposed in
the cases of both linear and nonlinear structural problems.Comment: International Journal for Numerical Methods in Engineering (2013) 24
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Pade-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems
A standard approach to reduced-order modeling of higher-order linear
dynamical systems is to rewrite the system as an equivalent first-order system
and then employ Krylov-subspace techniques for reduced-order modeling of
first-order systems. While this approach results in reduced-order models that
are characterized as Pade-type or even true Pade approximants of the system's
transfer function, in general, these models do not preserve the form of the
original higher-order system. In this paper, we present a new approach to
reduced-order modeling of higher-order systems based on projections onto
suitably partitioned Krylov basis matrices that are obtained by applying
Krylov-subspace techniques to an equivalent first-order system. We show that
the resulting reduced-order models preserve the form of the original
higher-order system. While the resulting reduced-order models are no longer
optimal in the Pade sense, we show that they still satisfy a Pade-type
approximation property. We also introduce the notion of Hermitian higher-order
linear dynamical systems, and we establish an enhanced Pade-type approximation
property in the Hermitian case
Updating the QR decomposition of block tridiagonal and block Hessenberg matrices
Abstract We present an efficient block-wise update scheme for the QR decomposition of block tridiagonal and block Hessenberg matrices. For example, such matrices come up in generalizations of the Krylov space solvers MinRes, SymmLQ, GMRes, and QMR to block methods for linear systems of equations with multiple right-hand sides. In the non-block case it is very efficient (and, in fact, standard) to use Givens rotations for these QR decompositions. Normally, the same approach is also used with column-wise updates in the block case. However, we show that, even for small block sizes, block-wise updates using (in general, complex) Householder reflections instead of Givens rotations are far more efficient in this case, in particular if the unitary transformations that incorporate the reflections determined by a whole block are computed explicitly. Naturally, the bigger the block size the bigger the savings. We discuss the somewhat complicated algorithmic details of this block-wise update, and present numerical experiments on accuracy and timing for the various options (Givens vs. Householder, block-wise vs. column-wise update, explicit vs. implicit computation of unitary transformations). Our treatment allows variable block sizes and can be adapted to block Hessenberg matrices that do not have the special structure encountered in the above mentioned block Krylov space solvers