66 research outputs found
Recycling BiCGSTAB with an Application to Parametric Model Order Reduction
Krylov subspace recycling is a process for accelerating the convergence of
sequences of linear systems. Based on this technique, the recycling BiCG
algorithm has been developed recently. Here, we now generalize and extend this
recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving
sequences of dual linear systems, while the focus here is on efficiently
solving sequences of single linear systems (assuming non-symmetric matrices for
both recycling BiCG and recycling BiCGSTAB).
As compared with other methods for solving sequences of single linear systems
with non-symmetric matrices (e.g., recycling variants of GMRES), BiCG based
recycling algorithms, like recycling BiCGSTAB, have the advantage that they
involve a short-term recurrence, and hence, do not suffer from storage issues
and are also cheaper with respect to the orthogonalizations.
We modify the BiCGSTAB algorithm to use a recycle space, which is built from
left and right approximate invariant subspaces. Using our algorithm for a
parametric model order reduction example gives good results. We show about 40%
savings in the number of matrix-vector products and about 35% savings in
runtime.Comment: 18 pages, 5 figures, Extended version of Max Planck Institute report
(MPIMD/13-21
Restarted Hessenberg method for solving shifted nonsymmetric linear systems
It is known that the restarted full orthogonalization method (FOM)
outperforms the restarted generalized minimum residual (GMRES) method in
several circumstances for solving shifted linear systems when the shifts are
handled simultaneously. Many variants of them have been proposed to enhance
their performance. We show that another restarted method, the restarted
Hessenberg method [M. Heyouni, M\'ethode de Hessenberg G\'en\'eralis\'ee et
Applications, Ph.D. Thesis, Universit\'e des Sciences et Technologies de Lille,
France, 1996] based on Hessenberg procedure, can effectively be employed, which
can provide accelerating convergence rate with respect to the number of
restarts. Theoretical analysis shows that the new residual of shifted restarted
Hessenberg method is still collinear with each other. In these cases where the
proposed algorithm needs less enough CPU time elapsed to converge than the
earlier established restarted shifted FOM, weighted restarted shifted FOM, and
some other popular shifted iterative solvers based on the short-term vector
recurrence, as shown via extensive numerical experiments involving the recent
popular applications of handling the time fractional differential equations.Comment: 19 pages, 7 tables. Some corrections for updating the reference
Sparse grid based Chebyshev HOPGD for parameterized linear systems
We consider approximating solutions to parameterized linear systems of the
form , where . Here the matrix is
nonsingular, large, and sparse and depends nonlinearly on the parameters
and . Specifically, the system arises from a discretization of a
partial differential equation and , . This work combines companion linearization with the Krylov
subspace method preconditioned bi-conjugate gradient (BiCG) and a decomposition
of a tensor matrix of precomputed solutions, called snapshots. As a result, a
reduced order model of is constructed, and this model can be
evaluated in a cheap way for many values of the parameters. The decomposition
is performed efficiently using the sparse grid based higher-order proper
generalized decomposition (HOPGD), and the snapshots are generated as one
variable functions of or of . Tensor decompositions performed on
a set of snapshots can fail to reach a certain level of accuracy, and it is not
possible to know a priori if the decomposition will be successful. This method
offers a way to generate a new set of solutions on the same parameter space at
little additional cost. An interpolation of the model is used to produce
approximations on the entire parameter space, and this method can be used to
solve a parameter estimation problem. Numerical examples of a parameterized
Helmholtz equation show the competitiveness of our approach. The simulations
are reproducible, and the software is available online
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