Bruno Carpentieri

Xian-Ming Gu, Bruno Carpentieri, Ting-Zhu Huang et al.

In the present study, we establish two new block variants of the Conjugate Orthogonal Conjugate Gradient (COCG) and the Conjugate A-Orthogonal Conjugate Residual (COCR) Krylov subspace methods for solving complex symmetric linear systems with multiple right hand sides. The proposed Block iterative solvers can fully exploit the complex symmetry property of coefficient matrix of the linear system. We report on extensive numerical experiments to show the favourable convergence properties of our newly developed Block algorithms for solving realistic electromagnetic simulations.

B. Carpentieri, Y. -F. Jing, T. -Z. Huang et al.

We present economical iterative algorithms built on the Biconjugate $A$-Orthonormalization Procedure for real unsymmetric and complex non-Hermitian systems. The principal characteristics of the developed solvers is that they are fast convergent and cheap in memory. We report on a large combination of numerical experiments to demonstrate that the proposed family of methods is highly competitive and often superior to other popular algorithms built upon the Arnoldi method and the biconjugate Lanczos procedures for unsymmetric linear sytems.

Xian-Ming Gu, Ting-Zhu Huang, Guojian Yin et al.

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, Méthode de Hessenberg Généralisée et Applications, Ph.D. Thesis, Université 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.

Yiming Bu, Bruno Carpentieri, Zhaoli Shen et al.

In this paper we introduce an algebraic recursive multilevel incomplete factorization preconditioner, based on a distributed Schur complement formulation, for solving general linear systems. The novelty of the proposed method is to combine factorization techniques of both implicit and explicit type, recursive combinatorial algorithms, multilevel mechanisms and overlapping strategies to maximize sparsity in the inverse factors and consequently reduce the factorization costs. Numerical experiments demonstrate the good potential of the proposed solver to precondition effectively general linear systems, also against other state-of-the-art iterative solvers of both implicit and explicit form.

Bruno Carpentieri, Jia Liao, Masha Sosonkina et al.

The paper describes an improved parallel MPI-based implementation of VBARMS, a variable block variant of the pARMS preconditioner proposed by Li,~Saad and Sosonkina [NLAA, 2003] for solving general nonsymmetric linear systems. The parallel VBARMS solver can detect automatically exact or approximate dense structures in the linear system, and exploits this information to achieve improved reliability and increased throughput during the factorization. A novel graph compression algorithm is discussed that finds these approximate dense blocks structures and requires only one simple to use parameter. A complete study of the numerical and parallel performance of parallel VBARMS is presented for the analysis of large turbulent Navier-Stokes equations on a suite of three-dimensional test cases.