A generalized preconditioned HSS method for non-Hermitian positive definite linear systems

We study the role of preconditioning strategies recently developed for coercive problems in connection with a two-step iterative method, based on the Hermitian skew-Hermitian splitting (HSS) of the coefficient matrix, proposed by Bai, Golub and Ng for the solution of nonsymmetric linear systems whose real part is coercive. As a model problem we consider Finite Differences (FD) matrix sequences {A n (a,p)} n discretizing the elliptic (convection-diffusion) problem with Ω being a plurirectangle of R d with a(x) being a uniformly positive function and p(x) denoting the Reynolds function: here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes. More precisely, in connection with preconditioned HSS/GMRES like methods, we consider the preconditioning sequence {P n (a)} n , P n (a):= D n 1/2(a)A n (1,0) D n 1/2(a) where D n (a) is the suitably scaled main diagonal of A n (a,0). If a(x) is positive and regular enough, then the preconditioned sequence shows a strong clustering at unity so that the sequence {P n (a)} n turns out to be a superlinear preconditioning sequence for {A n (a,0)} n where A n (a,0) represents a good approximation of Re(A n (a,p)) namely the real part of A n (a,p). The computational interest is due to the fact that the preconditioned HSS method has a convergence behavior depending on the spectral properties of {P n -1(a)Re(A n (a,p))} n ≈ {P n -1(a)A n (a,0)} n : therefore the solution of a linear system with coefficient matrix A n (a,p) is reduced to computations involving diagonals and to the use of fast Poisson solvers for {A n (1,0)} n . Some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques.

Some convergence conditions on successive over-relaxed (SOR) iterative method and symmetric SOR (SSOR) iterative method are proposed for non-Hermitian positive definite linear systems. Some examples are given to demonstrate the results obtained.

A practical formula for computing optimal parameters in the HSS iteration methods

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 © 2013 John Wiley & Sons, Ltd.

Hermitian and skew-Hermitian splitting (HSS) method converges unconditionally, which is efficient and robust for solving non-Hermitian positive-definite systems of linear equations. For solving systems of nonlinear equations with non-Hermitian positive-definite Jacobian matrices, Bai and Guo proposed the Newton-HSS method and gave numerical comparisons to show that the Newton-HSS method is superior to the Newton-USOR, the Newton-GMRES and the Newton-GCG methods. Recently, Wu and Chen proposed the modified Newton-HSS (MN-HSS) method which outperformed the Newton-HSS method. In this paper, we will establish a new accelerated modified Newton-HSS (AMN-HSS) method and give the local convergence theorem. Moreover, numerical results show that the AMN-HSS method outperforms the MN-HSS method.

Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations

Large sparse linear algebraic systems can be found in a variety of scientific and engineering fields and many scientists strive to solve them in an efficient and robust manner. In this paper, we propose an interpretable neural solver, the Fourier neural solver (FNS), to address them. FNS is based on deep learning and a fast Fourier transform. Because the error between the iterative solution and the ground truth involves a wide range of frequency modes, the FNS combines a stationary iterative method and frequency space correction to eliminate different components of the error. Local Fourier analysis shows that the FNS can pick up on the error components in frequency space that are challenging to eliminate with stationary methods. Numerical experiments on the anisotropic diffusion equation, convection–diffusion equation, and Helmholtz equation show that the FNS is more efficient and more robust than the state-of-the-art neural solver.

The simulation of several physical phenomena, arising from a wide class of scientific fields, requires solving large sparse linear systems effectively. In order to exploit appropriately the available supercomputing infrastructures, designing efficient and scalable solvers for large sparse linear systems is necessary. A hybrid parallel algebraic iterative method based on the Schur complement method and multiprojection techniques that utilizes semiaggregated subdomains is proposed. The graph corresponding to the coefficient matrix of the sparse linear system is partitioned by a graph partitioning algorithm and the Schur complement problem is solved by a preconditioned Krylov subspace method. The preconditioning scheme is based on domain decomposition and the resulting subdomains consist of fine and aggregated components from the inner boundaries. Moreover, two variants for approximating the local Schur complements, derived from the semiaggregation procedure, are proposed in order to improve memory requirements and preprocessing time. The proposed method is suitable for distributed memory systems with multicore nodes and has improved convergence behavior for a large number of workstations. Numerical and comparative results concerning the applicability, the convergence behavior, and the parallel performance of the proposed scheme are given.

Many applications in science and engineering give rise to large sparse linear systems of equations that need to be solved as efficiently as possible. As the size of the problems of interest increases, it can become necessary to consider exploiting multiprocessors to solve these systems. We report on the design and development of parallel frontal solvers for the numerical solution of large sparse linear systems. Three codes have been developed for the mathematical software library HSL (www.cse.clrc.ac.uk/Activity/HSL). The first is for unsymmetric finite-element problems; the second is for symmetric positive definite finite-element problems; and the third is for highly unsymmetric linear systems such as those that arise in chemical process engineering. In each case, the problem is subdivided into a small number of loosely connected subproblems and a frontal method is then applied to each of the subproblems in parallel. We discuss how our software is designed to achieve the goals of portability, ease of use, efficiency, and flexibility, and illustrate the performance using problems arising from real applications.

In recent years a number of solvers for the direct solution of large sparse symmetric linear systems of equations have been developed. These include solvers that are designed for the solution of positive definite systems as well as those that are principally intended for solving indefinite problems. In this study, we use performance profiles as a tool for evaluating and comparing the performance of serial sparse direct solvers on an extensive set of symmetric test problems taken from a range of practical applications.

Traditional numerical methods often struggle to handle time-fractional derivatives effectively, so new numerical methods are required for time-fractional diffusion equations. To solve multi-time fractional diffusion equations efficiently, based on Hermitian and skew-Hermitian splitting (HSS) iterative methods (a unique solution converging to a system of linear equations), new parameters are introduced, and relaxation techniques of the iterative method are combined to accelerate the HSS iterative method. A new extrapolated HSS iterative method is proposed, and its Wilson-like element structure is analyzed. Using the special properties of the element and fractional derivative processing, the super-approximation results are derived, and the super-convergence estimates are derived by interpolation post-processing technology. The test results show that the minimum error of this method is 0.15 %, and the error is small.

Solution of a large sparse or dense linear systems often constitutes the most time consuming step when solving large multidimensional industrial problems on modern high performance supercomputers. In most practical cases these problems produce linear systems with ill-conditioned and off-diagonal dominant coefficient matrices. As the problem size grows, especially in the 3-dimensional case, direct methods for solving these systems (such as triangular decomposition) begin to lose their effectiveness due to the large arithmetic costs and memory requirements. These difficulties motivate resorting to iterative solution methods which are, at least theoretically, more efficient with respect to the arithmetic cost and memory requirements. On March 12, 1993 Qasim Sheikh from Cray Research presented numerical experiments with iterative solvers that are capable of solving large linear systems which arise in industrial applications. He then compared the effectiveness and some performance aspects of these iterative methods with other state of the art, highly optimized, iterative as well as direct solvers. The main results of his talk are based on a recent paper [1] written jointly with Kharchenko, Nikishin, Yeremin and Heroux.KeywordsIterative SolverPreconditioned MatrixLarge Linear SystemNonsymmetric Linear SystemTurbine PumpThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks

This work deals with the solution of large non-Hermitian linear systems on desktop workstations with multiple graphics processing units (GPUs). While our implementation is motivated by the need to accelerate volume conductor modeling for bioelectrical brain imaging, the problem itself is common in scientific computing. Whenever a complex partial differential equation is numerically solved, a typically non-Hermitian sparse complex linear system needs to be solved. For problem sizes in the millions, this can take a long time even with highly optimized CPU-based solvers. Our GPU-accelerated solver outperforms an optimized OpenMP-based reference running on two quad-core CPUs by a factor of up to 31× in single precision and up to 7× in double precision, at the cost of a very modest hardware upgrade of two dual-GPU GTX 295 graphics cards. A pair of stronger Fermi GPUs (GTX 480) achieves speedups of 30× in single precision and 15× in double precision.

During the last decades, research efforts have been focused on the derivation of effective preconditioned iterative methods. The preconditioned iterative methods are mainly categorized into implicit preconditioned methods and explicit preconditioned methods. In this manuscript we review implicit preconditioned methods, based on incomplete and approximate factorization, and explicit preconditioned methods, based on sparse approximate inverses and explicit approximate inverses. Modified Moore-Penrose conditions are presented and theoretical estimates for the sensitivity of the explicit approximate inverse matrix of the explicit preconditioned method are derived. Finally, the performance of the preconditioned iterative methods is illustrated by solving characteristic 2D elliptic problem and numerical results are given indicating a qualitative agreement with the theoretical estimates.