47 research outputs found
A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems
In this paper, two efficient iterative algorithms based on the simpler GMRES
method are proposed for solving shifted linear systems. To make full use of the
shifted structure, the proposed algorithms utilizing the deflated restarting
strategy and flexible preconditioning can significantly reduce the number of
matrix-vector products and the elapsed CPU time. Numerical experiments are
reported to illustrate the performance and effectiveness of the proposed
algorithms.Comment: 17 pages. 9 Tables, 1 figure; Newly update: add some new numerical
results and correct some typos and syntax error
Recycling Krylov Subspaces for Efficient Partitioned Solution of Aerostructural Adjoint Systems
Robust and efficient solvers for coupled-adjoint linear systems are crucial
to successful aerostructural optimization. Monolithic and partitioned
strategies can be applied. The monolithic approach is expected to offer better
robustness and efficiency for strong fluid-structure interactions. However, it
requires a high implementation cost and convergence may depend on appropriate
scaling and initialization strategies. On the other hand, the modularity of the
partitioned method enables a straightforward implementation while its
convergence may require relaxation. In addition, a partitioned solver leads to
a higher number of iterations to get the same level of convergence as the
monolithic one.
The objective of this paper is to accelerate the fluid-structure
coupled-adjoint partitioned solver by considering techniques borrowed from
approximate invariant subspace recycling strategies adapted to sequences of
linear systems with varying right-hand sides. Indeed, in a partitioned
framework, the structural source term attached to the fluid block of equations
affects the right-hand side with the nice property of quickly converging to a
constant value. We also consider deflation of approximate eigenvectors in
conjunction with advanced inner-outer Krylov solvers for the fluid block
equations. We demonstrate the benefit of these techniques by computing the
coupled derivatives of an aeroelastic configuration of the ONERA-M6 fixed wing
in transonic flow. For this exercise the fluid grid was coupled to a structural
model specifically designed to exhibit a high flexibility. All computations are
performed using RANS flow modeling and a fully linearized one-equation
Spalart-Allmaras turbulence model. Numerical simulations show up to 39%
reduction in matrix-vector products for GCRO-DR and up to 19% for the nested
FGCRO-DR solver.Comment: 42 pages, 21 figure
A New Implementation of GMRES Using Generalized Purcell Method
In this paper, a new method based on the generalized Purcell method is proposed to solve the usual least-squares problem arising in the GMRES method. The theoretical aspects and computational results of the method are provided. For the popular iterative method GMRES, the decomposition matrices of the Hessenberg matrix is obtained by using a simple recursive relation instead of Givens rotations. The other advantages of the proposed method are low computational cost and no need for orthogonal decomposition of the Hessenberg matrix or pivoting. The comparisons for ill-conditioned sparse standard matrices are made. They show a good agreement with available literature
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
Block GMRES method with inexact breakdowns and deflated restarting
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