17 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
Off-diagonal low-rank preconditioner for difficult PageRank problems
PageRank problem is the cornerstone of Google search engine and is usually stated as solving a huge linear system. Moreover, when the damping factor approaches 1, the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve. In this paper, we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets. In particular, the off-diagonal part of the block coefficient matrix can be compressed by a simple low-rank factorization, which can be beneficial for solving the PageRank problem. Hence, a matrix partition method is proposed to discover the special sets of rows for supporting the low rank factorization. Then a preconditioner based on the low-rank factorization is proposed for solving difficult PageRank problems. Numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods. (C) 2018 Elsevier B.V. All rights reserved
A Hessenberg-type algorithm for computing PageRank Problems
PageRank is a widespread model for analysing the relative relevance of nodes within large graphs arising in several applications. In the current paper, we present a cost-effective Hessenberg-type method built upon the Hessenberg process for the solution of difficult PageRank problems. The new method is very competitive with other popular algorithms in this field, such as Arnoldi-type methods, especially when the damping factor is close to 1 and the dimension of the search subspace is large. The convergence and the complexity of the proposed algorithm are investigated. Numerical experiments are reported to show the efficiency of the new solver for practical PageRank computations
Efficient variants of the CMRH method for solving a sequence of multi-shifted non-Hermitian linear systems simultaneously
Multi-shifted linear systems with non-Hermitian coefficient matrices arise in
numerical solutions of time-dependent partial/fractional differential equations
(PDEs/FDEs), in control theory, PageRank problems, and other research fields.
We derive efficient variants of the restarted Changing Minimal Residual method
based on the cost-effective Hessenberg procedure (CMRH) for this problem class.
Then, we introduce a flexible variant of the algorithm that allows to use
variable preconditioning at each iteration to further accelerate the
convergence of shifted CMRH. We analyse the performance of the new class of
methods in the numerical solution of PDEs and FDEs, also against other
multi-shifted Krylov subspace methods.Comment: Techn. Rep., Univ. of Groningen, 34 pages. 11 Tables, 2 Figs. This
manuscript was submitted to a journal at 20 Jun. 2016. Updated version-1: 31
pages, 10 tables, 2 figs. The manuscript was resubmitted to the journal at 9
Jun. 2018. Updated version-2: 29 pages, 10 tables, 2 figs. Make it concise.
Updated version-3: 27 pages, 10 tables, 2 figs. Updated version-4: 28 pages,
10 tables, 2 fig
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