Symmetric Linear Systems Research Articles

A modified generalized shift-splitting iteration method for complex symmetric linear systems

In this work, we introduce a modified generalized shift-splitting (MGSS) iteration method for a class of complex symmetric linear systems. Meanwhile, we establish the convergence analysis and give the spectral properties of the corresponding preconditioned matrix. Finally, a numerical example illustrates the effectiveness of the MGSS.

Preconditioning for symmetric positive definite systems in balanced fractional diffusion equations

In this paper, we study the finite volume discretization method for balanced fractional diffusion equations where the fractional differential operators are comprised of both Riemann-Liouville and Caputo fractional derivatives. The main advantage of this approach is that new symmetric positive definite Toeplitz-like linear systems can be constructed for solving balanced fractional diffusion equations when diffusion functions are non-constant. It is different from non-symmetric Toeplitz-like linear systems usually obtained by existing numerical methods for fractional diffusion equations. The preconditioned conjugate gradient method with circulant and banded preconditioners can be applied to solve the proposed symmetric positive definite Toeplitz-like linear systems. Numerical examples, for both of one- and two- dimensional cases, are given to demonstrate the good accuracy of the finite volume discretization method and the fast convergence of the preconditioned conjugate gradient method.

A fast algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems

In this paper, we develop a new algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems. Numerical experiments are given in order to illustrate the validity and efficiency of our algorithm.

A fast forward algorithm for three-dimensional gravity gradient tensor using the compact difference schemes

We propose a fast and accurate 3D large-scale forward modeling method to compute the gravity gradient tensor, in terms of a fourth-order compact finite difference scheme and the extrapolation cascadic multigrid (EXCMG) method. Firstly, the 19-point fourth-order compact finite difference scheme with unequal mesh sizes in different coordinate directions is used to discretize the governing 3D Poisson equation. The resulted symmetric positive definite linear systems are solved by the EXCMG method. Then, the second-order derivatives (gravity gradient tensor) are calculated by solving a series of tridiagonal linear systems resulting from the fourth-order compact finite difference discretization. Finally, numerical result for a cubic model with positive density is used to verify the accuracy of our presented method. It shows that our method can give nearly fourth-order accurate approximations to gravitational potential and gravity gradient tensor. The accuracy is much higher and more efficient than traditional finite difference methods.

On Chebyshev accelerated iteration methods for two-by-two block linear systems

In this paper, we consider applying the Chebyshev acceleration technique to improve the convergence behavior of some iteration methods for solving the two-by-two block linear systems with semidefinite sub-blocks. Practical problem independent approximations about the eigenvalue bounds of the preconditioned matrices facilitate parameter free implementations of the concerned Chebyshev accelerated methods. To analyze their convergence behaviors, we discuss in detail the spectral properties of the corresponding preconditioned matrices, especially their eigenvector distributions, which lead to almost problem independent iteration error bounds. Numerical experiments of the complex symmetric linear systems demonstrate the efficiency of the new Chebyshev acceleration methods compared with the recently published Chebyshev accelerated PMHSS method and some preconditioned Krylov subspace methods.

Banded Preconditioners for Riesz Space Fractional Diffusion Equations

In this paper, we consider numerical methods for Toeplitz-like linear systems arising from the one- and two-dimensional Riesz space fractional diffusion equations. We apply the Crank–Nicolson technique to discretize the temporal derivative and apply certain difference operator to discretize the space fractional derivatives. For the one-dimensional problem, the corresponding coefficient matrix is the sum of an identity matrix and a product of a diagonal matrix and a symmetric Toeplitz matrix. We transform the linear systems to symmetric linear systems and introduce symmetric banded preconditioners. We prove that under mild assumptions, the eigenvalues of the preconditioned matrix are bounded above and below by positive constants. In particular, the lower bound of the eigenvalues is equal to 1 when the banded preconditioner with diagonal compensation is applied. The preconditioned conjugate gradient method is applied to solve relevant linear systems. Numerical results are presented to verify the theoretical results about the preconditioned matrices and to illustrate the efficiency of the proposed preconditioners.

A COMINRES‐QLP method for solving complex symmetric linear systems

AbstractSolving complex symmetric systems efficiently is a significantly important issue in the applications of electromagnetics since the discretization of the numerical model using the finite element method results in complex symmetric systems when computing the electromagnetic field. However, such complex symmetric systems are usually much more difficult to be solved because of the property of non‐Hermitian. Therefore, efficient numerical methods for the solution of complex symmetric systems are demanded. In this study, we proposed an iterative method called COMINRES‐QLP which is specially designed for complex symmetric linear systems. Moreover, we proposed a new stopping criterion which could be very useful in practical computations when the condition number of the coefficient matrix is unknown.

Exploiting Lower Precision Arithmetic in Solving Symmetric Positive Definite Linear Systems and Least Squares Problems

Exploiting Lower Precision Arithmetic in Solving Symmetric Positive Definite Linear Systems and Least Squares Problems

Two-Level Nyström--Schur Preconditioner for Sparse Symmetric Positive Definite Matrices

Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite linear systems of equations where the system matrix is preordered to doubly bordered block diagonal form (for example, using a nested dissection ordering). We investigate the use of randomized methods to construct high quality preconditioners. In particular, we propose a new and efficient approach that employs Nystr\"om's method for computing low rank approximations to develop robust algebraic two-level preconditioners. Construction of the new preconditioners involves iteratively solving a smaller but denser symmetric positive definite Schur complement system with multiple right-hand sides. Numerical experiments on problems coming from a range of application areas demonstrate that this inner system can be solved cheaply using block conjugate gradients and that using a large convergence tolerance to limit the cost does not adversely affect the quality of the resulting Nystr\"om--Schur two-level preconditioner.

Optimized Structured Preconditioner for a Class of Complex Symmetric Linear Systems

An optimized structured (OS) preconditioner for a class of complex symmetric linear systems is proposed. The OS-preconditioner results in a fast Krylov subspace solver, which is robust with respect to the mesh-size. The computational complexity of OS-preconditioner is analyzed and it is shown that all eigenvalues of the corresponding preconditioned matrix are positive real and distributed on[$\frac{1}{2}$+$\frac{\varepsilon }{{2\sqrt {1 + {\varepsilon ^2}} }}$,1]. Numerical experiments confirming the theoretical derivations are presented to verify the effectiveness and stability of the OS-preconditioner.

Lopsided modified Euler-extrapolated Hermitian and skew-Hermitian splitting method for a class of complex symmetric linear systems

In this paper, a lopsided modified Euler-extrapolated Hermitian and skew-Hermitian splitting (LME-HS) iteration method is introduced for solving the complex symmetric linear systems. Under a loose restriction on parameter $\theta$, we demonstrate that LME-HS iteration method is convergent. Moreover, we present the optimal parameter ${\theta}^{*}$ of the LME-HS method and discuss the spectral properties of corresponding preconditioned matrix. Finally, the numerical experiments are used to verify the effectiveness of the proposed method.

Equivalent low-order angular flux nonlinear finite difference equation of MOC transport calculation

The key issue in accelerating method of characteristics (MOC) transport calculations is in obtaining a completely equivalent low-order neutron transport or diffusion equation. Herein, an equivalent low-order angular flux nonlinear finite difference equation is proposed for MOC transport calculations. This method comprises three essential features: (1) the even parity discrete ordinates method is used to build a low-order angular flux nonlinear finite difference equation, and different boundary condition treatments are proposed; (2) two new defined factors, i.e., the even parity discontinuity factor and odd parity discontinuity factor, are strictly defined to achieve equivalence between the low-order angular flux nonlinear finite difference method and MOC transport calculation; (3) the energy group and angle are decoupled to construct a symmetric linear system that is much easier to solve. The equivalence of this low-order angular flux nonlinear finite difference equation is analyzed for two-dimensional (2D) pin, 2D assembly, and 2D C5G7 benchmark problems. Numerical results demonstrate that a low-order angular flux nonlinear finite difference equation that is completely equivalent to the pin-resolved transport equation is established.

A COMINRES-QLP Method for Solving Complex Symmetric Linear Systems

A COMINRES-QLP Method for Solving Complex Symmetric Linear Systems

On preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting method for complex symmetric linear systems

In this paper, we introduce a preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting (PE-SHSS) iteration method for solving a class of complex symmetric system of linear equations. The convergence properties of the PE-SHSS iteration method are investigated under suitable restrictions. In addition, the spectral properties of the corresponding preconditioned matrix are discussed. Finally, three numerical examples are also used to verify the effectiveness of the PE-SHSS iteration method.

Three-dimensional forward modelling of gravity field vector and its gradient tensor using the compact difference schemes

SUMMARY The traditional gravity forward modelling methods for solving partial differential equations (PDEs) only can yield second-order accuracy. When computing the gravity field vector and gradient tensor from the obtained potential, those numerical differentiation approaches will inevitably lose accuracy. To mitigate this issue, we propose an efficient and accurate 3-D forward modelling algorithm based on a fourth-order compact difference scheme. First, a 19-point fourth-order compact difference scheme with general meshsizes in x-, y- and z-directions is adopted to discretize the governing 3-D Poisson’s equation. The resulting symmetric positive-definite linear systems are solved by the pre-conditioned conjugate gradient algorithm. To obtain the first-order (i.e. the gravity field vector) and second-order derivatives (i.e. the gravity gradient tensor) with fourth-order accuracy, we seek to solve a sequence of tridiagonal linear systems resulting from the above mentioned finite difference discretization by using fast Thomas algorithm. Finally, two synthetic models and a real topography relief are used to verify the accuracy of our method. Numerical results show that our method can yield a nearly fourth-order accurate approximation not only to the gravitational potential, but also to the gravity field vector and its gradient tensor, which clearly demonstrates its superiority over the traditional PDE-based methods.

Compact quasi‐Newton preconditioners for symmetric positive definite linear systems

SummaryIn this paper, preconditioners for the conjugate gradient method are studied to solve the Newton system with symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of Symmetric Rank one (SR1) and Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) low‐rank updates. We develop conditions under which the SR1 update maintains the preconditioner symmetric positive definite. Spectral analysis of the SR1 preconditioned Jacobians shows an improved eigenvalue distribution as the Newton iteration proceeds. A compact matrix formulation of the preconditioner update is developed which reduces the cost of its application and is more suitable to parallel implementation. Some notes on the implementation of the corresponding Inexact Newton method are given and some numerical results on a number of model problems illustrate the efficiency of the proposed preconditioners.

Two lopsided TSCSP (LTSCSP) iteration methods for solution of complex symmetric positive definite linear systems

In this paper we make two new lopsided methods (LTSCSP1 and LTSCSP2) based on the two-scale-splitting (TSCSP) method and show that the convergence speed of TSCSP iteration method can be increased under some conditions without adding any additional parameter. The convergence analysis of the new methods in detail is given. Then we will obtain the quasi-optimal parameter to minimize the spectral radius of iteration matrix for the new methods. The inexact version of these methods is derived. To illustrate the effectiveness of the proposed framework, several numerical examples are given.

Parallel solution of saddle point systems with nested iterative solvers based on the Golub‐Kahan Bidiagonalization

SummaryThe Golub‐Kahan bidiagonalization is widely used in the singular value decomposition of rectangular matrices and has been generalized to an iterative solver for symmetric indefinite linear systems with a two‐by‐two block structure. In this work, we present a scalability study of this generalized solver as implemented in a recent release of the parallel numerical library PETSc (Portable, Extensible Toolkit for Scientific Computation). We present an improved solver performance for the two‐dimensional (2D) Stokes equations as compared to previous work. Furthermore, we investigate the performance of different parallel inner solvers in the outer Golub‐Kahan iteration for a three‐dimensional Stokes problem. The study includes parallel sparse direct solvers and multigrid methods. When increasing the number of cores for a fixed total problem size, the solver exhibits good speedups of up to 50% at the 1024 core count. For the tests in which the total problem size grows while the workload in each core stays constant, the parallel performance of the solver scales almost linearly with the increase in the core counts. In particular, the computation time increases only by about 15% when the number of cores increases from 80 to 1024 for a 2D test case.

Construction of a Preconditioner for General Elliptic Problems Using Riesz Map

The current work aspires to design and study the construction of an efficient preconditioner for linear symmetric systems in a Hilbert space setting. Compliantly to Josef Málek and Zdeněk Strakoš’s work [Preconditioning and the Conjugate Gradient Method in the Context of Solving[Formula: see text] PDEs, Vol. 1 (SIAM, USA).], we shed new light on the dependence of algebraic preconditioners with the resolution steps of partial differential equations (PDEs) and describe their impact on the final numerical solution. The numerical strength and efficiency of the proposed approach is demonstrated on a two-dimensional examples.

Spectral estimates for saddle point matrices arising in weak constraint four‐dimensional variational data assimilation

SummaryWe consider the large sparse symmetric linear systems of equations that arise in the solution of weak constraint four‐dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3 × 3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2 × 2 block structure, or further to symmetric positive definite systems. In this article, we analyse how sensitive the spectra of these matrices are to the number of observations of the underlying dynamical system. We also obtain bounds on the eigenvalues of the matrices. Numerical experiments are used to confirm the theoretical analysis and bounds.