Krylov Subspace Iterative Methods Research Articles

The generalized double shift-splitting preconditioner for nonsymmetric generalized saddle point problems from the steady Navier–Stokes equations

In this paper, a generalized double shift-splitting (GDSS) preconditioner induced by a new matrix splitting method is proposed and implemented for nonsymmetric generalized saddle point problems having a nonsymmetric positive definite (1,1)-block and a positive definite (2,2)-block. Detailed theoretical analysis of the iteration matrix is provided to show the GDSS method, which corresponds to the GDSS preconditioner, is unconditionally convergent. Additionally, a deteriorated GDSS (DGDSS) method is proposed. It is shown that, with suitable choice of parameter matrix, the DGDSS preconditioned matrix has an eigenvalue at 1 with multiplicity n, and the other m eigenvalues are of the form $$1-\lambda $$ with $$|\lambda |<1$$ , independently of the Schur complement matrix related. Finally, numerical experiments arising from a model Navier–Stokes problem are provided to validate and illustrate the effectiveness of the proposed preconditioner, with which a faster convergence for Krylov subspace iteration methods can be achieved.

An iterative solver for the 3D Helmholtz equation

We develop a frequency–domain iterative solver for numerical simulation of acoustic waves in 3D heterogeneous media. It is based on the application of a unique preconditioner to the Helmholtz equation that ensures convergence for Krylov subspace iteration methods. Effective inversion of the preconditioner involves the Fast Fourier Transform (FFT) and numerical solution of a series of boundary value problems for ordinary differential equations. Matrix-by-vector multiplication for iterative inversion of the preconditioned matrix involves inversion of the preconditioner and pointwise multiplication of grid functions. Our solver has been verified by benchmarking against exact solutions and a time-domain solver.

Electromagnetic scattering from inhomogeneous objects embedded in spherically multilayered media solved by the method of moments

ABSTRACTThe method of moments (MOM) is applied to solve electromagnetic (EM) scattering problems in spherically multilayered media. A spectral‐domain dyadic Green's function (DGF) in a spherically multilayered medium is constructed in terms of the spherical vector wave functions by using the method of scattering superposition in the spherical coordinate system. Its expression is later transformed to that in the Cartesian coordinate. We discretize the computational domain which contains the scatterer into N size‐independent cells in the Cartesian coordinate. Based on the transformed DGFs, the volume integral equations can be solved by MOM combined with Krylov subspace iterative methods. In this letter, we choose the bi‐conjugate gradient stabilized (BCGS) iteration method due to its fast convergence. Numerical results compared with FDTD solutions from a commercial software are presented to validate the accuracy and efficiency of our method. © 2017 Wiley Periodicals, Inc. Microwave Opt Technol Lett 59:526–530, 2017

Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions

A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynomials with unknown coefficients. By collocating the main equations together with the initial and boundary conditions at some special points (i.e., CGL collocation points), equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. Operational matrices of differentiation are implemented for speeding up the operations. In both of the one-dimensional and two-dimensional diffusion and wave equations, the geometrical distributions of the collocation points are depicted for clarity of presentation. Several numerical examples are provided to show the efficiency and spectral (exponential) accuracy of the proposed method.

A Robust Preconditioned Iterative Method for the Navier-Stokes Equations with High Reynolds Numbers

In this paper, we proposed a new solver for the Navier-Stokes equations coming from the channel flow with high Reynolds number. We use the preconditioned Krylov subspace iterative methods such as Generalized Minimum Residual Methods (GMRES). We consider the variation of the Hermitian and Skew-Hermitian splitting to construct the preconditioner. Convergence of the preconditioned iteration is analyzed. We can show that the proposed preconditioner has a robust behavior for the Navier-Stokes problems in variety of models. Numerical experiments show the robustness and efficiency of the preconditioned GMRES for the Navier-Stokes problems with Reynolds numbers up to ten thousands.

A shift-splitting preconditioner for a class of block two-by-two linear systems

In this paper, we construct a shift-splitting preconditioner for a class of block two-by-two linear systems. The proposed preconditioner is extracted from a stationary iterative method which is unconditionally convergent. Moreover, the eigenvalue distribution of the corresponding preconditioned matrix is studied. Numerical experiments are presented to show that our new preconditioner can be quite competitive when used to precondition Krylov subspace iterative methods such as GMRES.

Some results on the regularization of LSQR for large-scale discrete ill-posed problems

LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and its mathematically equivalent CGLS applied to normal equations system, are commonly used for large-scale discrete ill-posed problems. It is well known that LSQR and CGLS have regularizing effects, where the number of iterations plays the role of the regularization parameter. However, it has long been unknown whether the regularizing effects are good enough to find best possible regularized solutions. Here a best possible regularized solution means that it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition (TSVD) method. In this paper, we establish bounds for the distance between the $k$-dimensional Krylov subspace and the $k$-dimensional dominant right singular space. They show that the Krylov subspace captures the dominant right singular space better for severely and moderately ill-posed problems than for mildly ill-posed problems. Our general conclusions are that LSQR has better regularizing effects for the first two kinds of problems than for the third kind, and a hybrid LSQR with additional regularization is generally needed for mildly ill-posed problems. Exploiting the established bounds, we derive an estimate for the accuracy of the rank $k$ approximation generated by Lanczos bidiagonalization. Numerical experiments illustrate that the regularizing effects of LSQR are good enough to compute best possible regularized solutions for severely and moderately ill-posed problems, stronger than our theory predicts, but they are not for mildly ill-posed problems and additional regularization is needed.

Extended shift-splitting preconditioners for saddle point problems

In this paper we consider to solve the linear systems of the saddle point problems by preconditioned Krylov subspace methods. The preconditioners are based on a special splitting of the saddle point matrix. The convergence theory of this class of the extended shift-splitting preconditioned iteration methods is established. The spectral properties of the preconditioned matrices are analyzed. Numerical implementations show that the resulting preconditioners lead to fast convergence when they are used to precondition Krylov subspace iteration methods such as GMRES.

Shifted SSOR-like preconditioner for non-Hermitian positive definite matrices

Based on the SSOR-like iteration method of Bai (Numer. Linear Algebra Appl. 23, 37-60, 2016), we give a shifted SSOR-like preconditioner which is positive definite for solving the non-Hermitian positive definite linear system with a dominant Hermitian part. Estimated eigenvalue bounds for the preconditioned matrix and convergence conditions for the iteration matrix are derived. Numerical tests illustrate the efficiency of the new preconditioner when applied to precondition the Krylov subspace iteration methods such as GMRES.

Two improvements of the deteriorated PSS preconditioner for generalized saddle point problems

For generalized saddle point problems, we present two improved variants of the deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner to accelerate the convergence rate of the Krylov subspace iteration method. The new preconditioners are not only better approximations to the generalized saddle point matrix than the DPSS preconditioner but also easier to implement than the DPSS preconditioner. Theoretical analyses show that the corresponding splitting iteration methods are also convergent unconditionally. The quasi-optimal choices and practical estimations of the iteration parameters are discussed. Moreover, eigenproperties of the preconditioned matrices are described and upper bounds of the degree of the minimal polynomial of the preconditioned matrices are obtained. Finally, numerical experiments arising from the discretization of a model Navier-Stokes equation are presented to show the efficiency of the proposed preconditioners.

A Block Diagonal Preconditioner for Generalised Saddle Point Problems

AbstractA lopsided alternating direction iteration (LADI) method and an induced block diagonal preconditioner for solving block two-by-two generalised saddle point problems are presented. The convergence of the LADI method is analysed, and the block diagonal preconditioner can accelerate the convergence rates of Krylov subspace iteration methods such as GMRES. Our new preconditioned method only requires a solver for two linear equation sub-systems with symmetric and positive definite coefficient matrices. Numerical experiments show that the GMRES with the new preconditioner is quite effective.

One-way domain decomposition method with exact radiation condition and fast GMRES solver for the solution of Maxwell's equations

For the solution of the time-harmonic electromagnetic scattering problem by inhomogeneous 3-D objects, a one-way domain decomposition method (DDM) is considered: the computational domain is partitioned into concentric subdomains on the interfaces of which Robin-type transmission conditions (TCs) are prescribed; an integral representation of the electromagnetic fields on the outer boundary constitutes an exact radiation condition. The global system obtained after discretization of the finite element (FE) formulations is solved via a Krylov subspace iterative method (GMRES). It is preconditioned in such a way that, essentially, only the solution of the FE subsystems in each subdomain is required. This is made possible by a computationally cheap H(curl)–H(div) transformation performed on the interfaces that separate the two outermost subdomains. The eigenvalues of the preconditioned matrix of the system are bounded by two, and optimized values of the coefficients involved in the local TCs on the interfaces are determined so as to maximize the minimum eigenvalue. Numerical experiments are presented that illustrate the numerical accuracy of this technique, its fast convergence, and legitimate the choices made for the optimized coefficients.

A new matrix method for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions

This paper is devoted to develop a new matrix scheme for solving two-dimensional time-dependent diffusion equations with Dirichlet boundary conditions. We first transform these equations into equivalent integro partial differential equations (PDEs). Such these integro-PDEs contain both of the initial and boundary conditions and can be solved numerically in a more appropriate manner. Subsequently, all the existing known and unknown functions in the latter equations are approximated by Bernoulli polynomials and operational matrices of differentiation and integration together with the completeness of these polynomials can be used to reduce the integro-PDEs into the associated algebraic generalized Sylvester equations. For solving these algebraic equations, an efficient Krylov subspace iterative method (i.e., BICGSTAB) is implemented. Two numerical examples are given to demonstrate the efficiency, accuracy, and versatility of the proposed method.

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

The governing dynamics of fluid flow is stated as a system of partial differential equations referred to as the Navier-Stokes system. In industrial and scientific applications, fluid flow control becomes an optimization problem where the governing partial differential equations of the fluid flow are stated as constraints. When discretized, the optimal control of the Navier-Stokes equations leads to large sparse saddle point systems in two levels. In this paper, we consider distributed optimal control for the Stokes system and test the particular case when the arising linear system can be compressed after eliminating the control function. In that case, a system arises in a form which enables the application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain conditions, the condition number of the so preconditioned matrix is bounded by 2. The numerical and computational efficiency of the method in terms of number of iterations and execution time is favorably compared with other published methods.

A new iterative method for solving a class of complex symmetric system of linear equations

We present a new stationary iterative method, called Scale-Splitting (SCSP) method, and investigate its convergence properties. The SCSP method naturally results in a simple matrix splitting preconditioner, called SCSP-preconditioner, for the original linear system. Some numerical comparisons are presented between the SCSP-preconditioner and several available block preconditioners, such as PGSOR (Hezari et al. Numer. Linear Algebra Appl. 22, 761---776, 2015) and rotate block triangular preconditioners (Bai Sci. China Math. 56, 2523---2538, 2013), when they are applied to expedite the convergence rate of Krylov subspace iteration methods for solving the original complex system and its block real formulation, respectively. Numerical experiments show that the SCSP-preconditioner can compete with PGSOR-preconditioner and even more effective than the rotate block triangular preconditioners.

A class of generic factored and multi-level recursive approximate inverse techniques for solving general sparse systems

Purpose – The purpose of this paper is to propose novel factored approximate sparse inverse schemes and multi-level methods for the solution of large sparse linear systems. Design/methodology/approach – The main motive for the derivation of the various generic preconditioning schemes lies to the efficiency and effectiveness of factored preconditioning schemes in conjunction with Krylov subspace iterative methods as well as multi-level techniques for solving various model problems. Factored approximate inverses, namely, Generic Factored Approximate Sparse Inverse, require less fill-in and are computed faster due to the reduced number of nonzero elements. A modified column wise approach, namely, Modified Generic Factored Approximate Sparse Inverse, is also proposed to further enhance performance. The multi-level approximate inverse scheme, namely, Multi-level Algebraic Recursive Generic Approximate Inverse Solver, utilizes a multi-level hierarchy formed using Block Independent Set reordering scheme and an approximation of the Schur complement that results in the solution of reduced order linear systems thus enhancing performance and convergence behavior. Moreover, a theoretical estimate for the quality of the multi-level approximate inverse is also provided. Findings – Application of the proposed schemes to various model problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than results by other researchers for some of the model problems. Research limitations/implications – Further enhancements are investigated for the proposed factored approximate inverse schemes as well as the multi-level techniques to improve quality of the schemes. Furthermore, the proposed schemes rely on the definition of multiple parameters that for some problems require thorough testing, thus adaptive techniques to define the values of the various parameters are currently under research. Moreover, parallel schemes will be investigated. Originality/value – The proposed approximate inverse preconditioning schemes as well as multi-level schemes are efficient computational methods that are valuable for computer scientists and for scientists and engineers in engineering computations.

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems : Poisson and convection–diffusion control

Uzawa Algorithms for Fully Fuzzy Linear Systems

AbstractRecently, there have been many studies on solving different kinds of fuzzy equations. In this paper, the solution of a trapezoidal fully fuzzy linear system (FFLS) is studied. Uzawa approach, which is a popular iterative technique for saddle point problems, is considered for solving such FFLSs. In our Uzawa approach, it is possible to compute the solution of a fuzzy system using various relaxation iterative methods such as Richardson, Jacobi, Gauss-Seidel, SOR, SSOR as well as Krylov subspace methods such as GMRES, QMR and BiCGSTAB. Krylov subspace iterative methods are known to converge for a larger class of matrices than relaxation iterative methods and they exhibit higher convergence rates. Thus, they are more widely used in practical problems. Numerical experiments are to illustrate the performance of our suggested methods.

Performance of Neumann Expansion Preconditioners for Iterative Methods with Geotechnical Elastoplastic Applications

AbstractBecause most geotechnical analyses may involve elastoplastic geomaterials, a robust solution scheme is of critical importance to the efficiency of the entire finite-element (FE) computation. To accelerate the Krylov subspace iterative methods, some preconditioning techniques have been developed based on the factorization of elastic stiffness. However, the idea of constructing a preconditioner based on elastic stiffness is heuristic. In this study, a class of Neumann expansion preconditioners constructed from the constant (elastic) partition and varying (plastic) partition of the elastoplastic stiffness matrix was proposed. Based on two numerical examples, the performances of truncated Neumann expansion preconditioners were examined with associated and nonassociated soil plasticity considered, respectively. It is interesting to note that the convergence behaviors of truncated Neumann expansion preconditioners closely depended on the approximation to the elastic stiffness part as well as the truncat...

Parameterized rotated block preconditioning techniques for block two-by-two systems with application to complex linear systems

In this paper, based on the rotated block triangular preconditioning technique, we present a class of parameterized rotated block preconditioners to accelerate Krylov subspace iterative methods with coefficient matrices of nonsymmetric sub-blocks. The parameterized rotated block preconditioners can be seen as a generalization of the rotated block triangular preconditioners. The eigen-properties of the corresponding preconditioned matrices are analyzed. Numerical experiments are used to demonstrate the feasibility and effectiveness of the parameterized rotated block preconditioners.