Lev A. Krukier

Generalized skew‐Hermitian triangular splitting iteration methods for saddle‐point linear systems

SUMMARY A generalized skew-Hermitian triangular splitting iteration method is presented for solving non-Hermitian linear systems with strong skew-Hermitian parts. We study the convergence of the generalized skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew-Hermitian triangular splitting. Then the generalized skew-Hermitian triangular splitting iteration method is applied to non-Hermitian positive semidefinite saddle-point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew-Hermitian triangular splitting iteration methods are effective for solving non-Hermitian saddle-point linear systems with strong skew-Hermitian parts. Copyright

Product-type skew-Hermitian triangular splitting iteration methods for strongly non-Hermitian positive definite linear systems

By further generalizing the modified skew-Hermitian triangular splitting iteration methods studied in [L. Wang, Z.-Z. Bai, Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts, BIT Numer. Math. 44 (2004) 363-386], in this paper, we present a new iteration scheme, called the product-type skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this method. Moreover, when it is applied to precondition the Krylov subspace methods, the preconditioning property of the product-type skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that the product-type skew-Hermitian triangular splitting iteration method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.

Triangular skew-symmetric iterative solvers for strongly nonsymmetric positive real linear system of equations

A new class of triangular iterative methods for solving nonsymmetric linear systems of equations with strongly nonsymmetric positive real matrices is proposed. The triangular operator of these iterative methods uses the skew-symmetric part of the initial matrix. For the new methods, a convergence analysis, a technique for choosing the optimal parameter, and an accelerating procedure are presented. Several numerical experiments include the solution of the strongly nonsymmetric linear systems arising from a central finite-difference approximation of the steady convection-diffusion equation with the Peclet numbers Pe = 103, 104, and 105. The relative performance of these methods is compared to the popular SOR procedure.

Convergence acceleration of triangular iterative methods based on the skew-symmetric part of the matrix

Abstract The numerical investigation was done on a 2-D convection–diffusion model problem in a unit square with a small parameter (Peclet number) at the higher derivative. Various ways of assigning coefficients for convective terms were considered. Central difference approximation of this equation produces a system of linear algebraic equations with an essential non-symmetric matrix. The calculations were performed with Peclet number equal to 103–105. Triangular and product triangular iterative methods, which have been built up by special way using only the triangular parts of the skew-symmetric component of the matrix, are used to solve such systems. A way to accelerate convergence of these methods for the case when coefficients of convective terms change very quickly is presented. Comparison with GMRES method was produced.

Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid

A stationary convection-diffusion problem with a small parameter multiplying the highest derivative is considered. The problem is discretized on a uniform rectangular grid by the central-difference scheme. A new class of two-step iterative methods for solving this problem is proposed and investigated. The convergence of the methods is proved, optimal iterative methods are chosen, and the rate of convergence is estimated. Numerical results are presented that show the high efficiency of the methods.

General resolution of a convergence question of L. Krukier

It was shown that there exists a matrix norm N such that N(A−1B)<1 if A, B∈Mn(), 0∉F(A), and g(A, B)<1 where g(A, B)=max|z|, z∈G(A, B), Copyright

Numerical Solution of the Steady Convection-diffusion Equation with Dominant Convection

Abstract Steady convection-diffusion equation in 2-D domain is considered. Central finite-difference approximation has been taken to obtain a large sparse nonsymmetric linear system with positive real matrix. New class of product triangular skew-symmetric iterative methods for solution of such system is presented and considered. Using this method as preconditioner for GMRES and BiCG has been made. Results of numerical experiments for two-dimensional convection-diffusion equation for different big Peclet numbers and velocity coefficients have been presented.

Probing the last scattering surface through recent and future CMB observations

We have constrained the extended (delayed and accelerated) models of hydrogen recombination, by investigating associated changes of the position and the width of the last scattering surface. Using the recently obtained CMB and SDSS data, we find that the recently derived data constraints favor the accelerated recombination model, although the other models (standard, delayed recombination) are not ruled out at 1σ confidence level. If the accelerated recombination had actually occurred in our early Universe, it is likely that baryonic clustering on small scales would have been the cause of it. By comparing the ionization history of baryonic cloud models with that of the best-fit accelerated recombination model, we find that some portion of our early Universe has baryonic underdensity. We have made a forecast for the PLANCK data constraint, which shows that we will be able to rule out the standard or delayed recombination models if the recombination in our early Universe had proceeded with α~−0.01 or lower, and residual foregrounds and systematic effects are negligible.

Special iterative methods for solution of the steady Convection-Diffusion-Reaction equation with dominant convection

Iterative methods based on skew-symmetric splitting of initial matrix, arising from central finite-difference approximation of steady convection-diffusion-reaction (CDR) equation in 2-D domain are considered. The property of obtained large sparse nonsymmetric linear system Au = f is investigated. A new class of triangular and product triangular skew-symmetric iterative methods is presented. Sufficient conditions of convergence for iterative methods of solution CDR with variable coefficient of reaction and dominant convection are obtained. The results of numerical experiments for the solution of a two-dimensional CDR equation are presented. The uniform grid, central differences for the first derivatives, natural ordering of points, Peclet numbers Pe = 103, 104, 105, variable coefficient of reaction and different velocity coefficients have been used.

Efficient iteration method for saddle point problems

An algorithm that is a modification of Hermitian and skew-Hermitian splitting is considered to be used to solve a large system of linear algebraic equations with a saddle point. The method is applied to solve constrained optimization problems. The numerical experiments show the high efficiency of the method when it is used to solve problems of the given class.