Bai Zhongzhi

Rotated block triangular preconditioning based on PMHSS

Based on the PMHSS preconditioning matrix, we construct a class of rotated block triangular preconditioners for block two-by-two matrices of real square blocks, and analyze the eigen-properties of the corresponding preconditioned matrices. Numerical experiments show that these rotated block triangular preconditioners can be competitive to and even more efficient than the PMHSS pre-conditioner when they are used to accelerate Krylov subspace iteration methods for solving block two-by-two linear systems with coefficient matrices possibly of nonsymmetric sub-blocks.

Construction and analysis of structured preconditioners for block two-by-two matrices

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.

PARALLEL MULTILEVEL ITERATIVE METHODS

Abstract For large-scale system of linear equations with symmetric positive definite block coefficient matrix resulting from the discretization of a self-adjoint elliptic boundary-value problem, by making use of blocked multilevel iteration we construct preconditioning matrices for the coefficient matrix and set up a class of parallel multilevel iterative methods for solving such system. Theoretical analysis shows that besides lending themselves to strongly parallel computation these new methods have convergence rates independent of both the sizes and the level numbers of the grids, and their computational work loads are also bounded by linear functions about the step sizes of the finest grids.

A CLASS OF FACTORIZATION UPDATE ALGORITHM FOR SOLVING SYSTEMS OF SPARSE NONLINEAR EQUATIONS

In this paper, we establish a class of sparse update algorithm based on matrix triangular factorizations for solving a system of sparse equations. The localQ-superlinear convergence of the algorithm is proved without introducing anm-step refactorization. We compare the numerical results of the new algorithm with those of the known algorithms, The comparison implies that the new algorithm is satisfactory.

A unified framework for the construction of various algebraic multilevel preconditioning methods

A framework for algebraic multilevel preconditioning methods is presented for solving large sparse systems of linear equations with symmetric positive definite coefficient matrices, which arise in the discretization of second order elliptic boundary value problems by the finite element method. This framework covers not only all known algebraic multilevel preconditioning methods, but yields also new ones. It is shown that all preconditioners within this framework have optimal orders of complexities for problems in two-dimensional(2-D) and three-dimensional(3-D) problem domains, and their relatives condition numbers are bounded uniformly with respect to the numbers of both the levels and the nodes.

Parallel interval matrix multisplitting AOR methods and their convergence

This paper proposes a class of parallel interval matrix multisplitting AOR methods for solving systems of interval linear equations and discusses their convergence properties under the conditions that the coefficient matrices are interval H-matrices.

A framework of parallel algebraic multilevel preconditioning iterations

A framework for parallel algebraic multilevel preconditioning methods is presented for solving large sparse systems of linear equations with symmetric positive definite coefficient matrices, which arise in suitable finite element discretizations of many second-order self-adjoint elliptic boundary value problems. This framework not only covers all known parallel algebraic multilevel preconditioning methods, but also yields new ones. It is shown that all preconditioners within this framework have optimal orders of complexities for problems in two-dimensional (2-D) and three-dimensional (3-D) problem domains, and their relative condition numbers are bounded uniformly with respect to the numbers of both levels and nodes.