# Fatemeh Panjeh Ali BeikVali-e-Asr University Of Rafsanjan | VRU · Department of Mathematics

Fatemeh Panjeh Ali Beik

PhD

## About

71

Publications

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415

Citations

Citations since 2017

Introduction

My main research interest is studying the performance of (preconditioned) iterative methods to solve linear and multi-linear (tensor) operator equations.
Home page: http://beik.faculty.vru.ac.ir/

Additional affiliations

October 2020 - present

September 2016 - May 2017

May 2015 - October 2020

Education

September 2007 - December 2010

September 2005 - September 2007

September 1999 - September 2005

## Publications

Publications (71)

We consider the iterative solution of a class of linear systems with
double saddle point structure.
Several block preconditioners
for Krylov subspace methods are described and analyzed. We derive some bounds
for the eigenvalues of preconditioned matrices and present results of
numerical experiments using test problems from two different
application...

We review the use of block diagonal and block lower/upper triangular splittings for constructing iterative methods and preconditioners for solving stabilized saddle point problems. We introduce new variants of these splittings and obtain new results on the convergence of the associated stationary iterations and new bounds on the eigenvalues of the...

We analyze two types of block preconditioners for a class of saddle point problems arising from the modeling of liquid crystal directors using finite elements. Spectral properties of the preconditioned matrices are investigated, and numerical experiments are performed to assess the behavior of preconditioned iterations using both exact and inexact...

We consider the solution of linear discrete ill-posed systems of equations with a certain tensor product structure. Two aspects of this kind of problems are investigated: They are transformed to large linear systems of equations and the conditioning of the matrix of the latter system is analyzed. Also, the distance of this matrix to symmetry and sk...

We study the performance of some preconditioning techniques for a class of block three-by-three linear systems of equations arising from finite element discretizations of the coupled Stokes-Darcy flow problem. In particular, we investigate preconditioning techniques including block preconditioners, constraint preconditioners, and augmented Lagrangi...

In the present paper, we are interested in developing iterative Krylov subspace methods in tensor structure to solve a class of multilinear systems via the Einstein product. In particular, we develop tensor variants of the GMRES and Golub–Kahan bidiagonalization processes in tensor framework. We further consider the case that mentioned equation may...

We mainly study the solvability of tensor absolute value equation (TAVE) in the form $\mathscr{A}x^{m-1}-\mathscr{B} x^{m-1}=b$. Under certain conditions, some new sufficient conditions are provided which ensure that the mentioned problem has at least one solution. Furthermore, it reveals that the equation is unsolvable in some special cases. Brief...

We revisit the implementation of the Krylov subspace method based on the Hessenberg process for general linear operator equations. It is established that at each step, the computed approximate solution by the corresponding approach can be regarded as the minimizer of a certain norm of residual corresponding to the obtained approximate solution of s...

We consider a class of iterative methods based on block splitting (BBS) to solve absolute value equations $Ax-|x|=b$. Recently, several works were devoted to deriving sufficient conditions for the convergence of iterative methods of this type under certain assumptions including $\nu:=\|A^{-1}\|<1$. However, the BBS-type iterative methods tend to co...

We consider the absolute value equations (AVEs) with a certain tensor product structure. Two aspects of this kind of AVEs are discussed in detail: the solvability and approximate solution. More precisely, first, some sufficient conditions are provided which guarantee the unique solvability of this kind of AVEs. Furthermore, a new iterative method i...

We study the performance of some preconditioning techniques for a class of block three-by-three linear systems of equations arising from finite element discretizations of the coupled Stokes-Darcy flow problem. In particular, we investigate preconditioning techniques including block preconditioners, constraint preconditioners, and augmented Lagrangi...

We establish a new iterative method for solving a class of large and sparse linear systems of equations with three-by-three block coefficient matrices having saddle point structure. Convergence properties of the proposed method are studied in details and its induced preconditioner is examined for accelerating the convergence speed of generalized mi...

Recently, Zhang et al. [Applied Mathematics Letters 104 (2020) 106287] proposed a preconditioner to improve the convergence speed of three types of Jacobi iterative methods for solving multi-linear systems. In this paper, we consider the Jacobi-type method which works better than the other two ones and apply a new preconditioner. The convergence of...

We consider the absolute value equations (AVEs) with a certain tensor product structure. Two aspects of this kind of AVEs are discussed in detail: the solvability and approximate solution. More precisely, first, some sufficient conditions are provided which guarantee the unique solvability of this kind of AVEs. Furthermore, a new iterative method i...

In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for tensors corresponding to T-product. Furthermore, we discuss on the solution of least-squares problem associated...

This paper mainly deals with applying a general class of preconditioners to accelerate the convergence speed of some iterative schemes for solving multi-linear systems whose coefficient tensors are strong M-tensors. Theoretical results are established to analyze the performance of a general class of preconditioners extracted from the majorization m...

We study the performance of some preconditioning techniques for a class of block three-by-three linear systems of equations arising from finite element discretizations of the coupled Stokes-Darcy flow problem. In particular, we investigate preconditioning techniques including block preconditioners, constraint preconditioners, and augmented Lagrangi...

In this paper, a framework is proposed for left/right preconditioning of multi-linear system with Einstein product. More precisely, the inverse of preconditioned tensor is derived analytically. Some numerical results are disclosed to experimentally illustrate the feasibility of preconditioned Krylov subspace methods based on Hesseberg process and c...

In this paper, a framework is proposed for left/right preconditioning of multi-linear system with Einstein product. More precisely, the inverse of preconditioned tensor is derived analytically. Some numerical results are disclosed to experimentally illustrate the feasibility of preconditioned Krylov subspace methods based on Hesseberg process and c...

In this paper, we develop the idea of constructing iterative methods based on block splittings (BBS) to solve absolute value equations. The class of BBS methods incorporates the well-known Picard iterative method as a special case. Convergence properties of the mentioned schemes are proved under some sufficient conditions. Numerical experiments are...

In this talk some preconditioning techniques are presented for a class of linear systems with double Saddle point structure arising in finite element discretizations of coupled Stokes-Darcy flow and modeling of liquid crystals directors. We investigate different preconditionering techniques including block preconditioners, constraint preconditioner...

This paper is concerned with studying the performance of some global iterative schemes based on Hessenberg process to solve the Sylvester tensor equation (STE). Furthermore, we briefly mention the implementation of flexible versions. Under certain conditions, we devote part of the work to derive upper bounds for condition number of matrices with a...

This paper is concerned with the solution of severely ill-conditioned linear tensor equations. These kinds of equations may arise when discretizing partial differential equations in many space-dimensions by finite difference or spectral methods. The deblurring of color images is another application. We describe the tensor Golub–Kahan bidiagonalizat...

This paper mainly deals with applying a general class of preconditioners to accelerate the convergence speed of some iterative schemes for solving multi-linear systems whose coefficient tensors are strong M-tensor. Some results are established to compare the performance of different preconditioners extracted from the majorization matrix associated...

In the present paper, we are interested in developing iterative Krylov subspace methods in tensor structure to solve a class of multilinear systems via Einstein product. In particular, we develop global variants of the GMRES and Gloub--Kahan bidiagonalization processes in tensor framework. We further consider the case that mentioned equation may be...

In the proof of Theorem 3 in Salkuyeh

We establish a new iterative method for solving a class of large and sparse linear systems of equations with three-by-three block coefficient matrices having saddle point structure. Convergence properties of the proposed method are studied in details and its induced preconditioner is examined for accelerating the convergence speed of generalized mi...

This paper is concerned with solving ill-posed tensor linear equations. These kinds of equations may appear from finite difference discretization of high-dimensional convection-diffusion problems or when partial differential equations in many dimensions are discretized by collocation spectral methods. Here, we propose the Tensor Golub--Kahan bidiag...

We study different types of stationary iterative methods for solving a class of large, sparse linear systems with double saddle point structure. In particular, we propose a class of Uzawa-like methods including a generalized (block) Gauss-Seidel (GGS) scheme and a generalized (block) successive overrelaxation (GSOR) method. Both schemes rely on a r...

This paper is concerned with studying the performance of some global iterative schemes based on Hessenberg process to solve the Sylvester tensor equation (STE). Furthermore, we briefly mention the implementation of flexible versions. Under certain conditions, we devote part of the work to derive upper bounds for condition number of matrices with a...

We study the performance of a class of block triangular preconditioners for saddle point systems with nonsymmetric positive definite (1,1)-block. The presented results incorporate the established results in Zhang and Zhao (2018) where a (parameter-dependent) preconditioner was suggested to solve the mentioned saddle point systems. The performance o...

A generalized global Arnoldi method based on tensor format for ill-posed tensor equations

Najafi et al. (Appl Math Lett 33:1–5, 2014) have elaborated an approach to compute the inverse of arrowhead matrices. This paper concerns with offering an alternative simple and neat framework to obtain an explicit formula for the inverse of arrowhead matrices. More precisely, the adopted manner makes us capable to derive the inverse of block arrow...

We study different types of stationary iterative methods for solving a class of large, sparse linear systems with double saddle point structure. In particular, we propose a class of Uzawa-like methods including a generalized (block) Gauss-Seidel (GGS) scheme and a generalized (block) successive overrelaxation (GSOR) method. Both schemes rely on a r...

Recently in [Journal of Computational Physics, 321 (2016), 829–907], an approach has been developed for solving linear system of equations with nonsingular coefficient matrix. The method is derived by using a delayed over-relaxation step (DORS) in a generic (convergent) basic stationary iterative method. In this paper,we first prove the semi-conver...

In [Appl. Math. Comput. 217 (2011) 5596-5602], Li et al. suggested an effective iterative method for solving large sparse saddle point problems with symmetric positive definite (1; 1)-block. Recently, Zhu et al. [Appl. Math. Comput. 242 (2014) 907-916] developed the method for the saddle point problems with (1; 1)-block being non-symmetric positive...

Recently, Tang et al. [Numer Algorithms. 2014;66(2):379–397] have offered a cyclic iterative method for determining the unique solution of the coupled matrix equations
Analogues to the gradient-based algorithm, the proposed algorithm relies on a fixed parameter whereas it has wider convergence region. Nevertheless, the application of the algorithm...

A class of quaternion matrices called generalized
-(anti-)bi-Hermitian matrices is defined which incorporates the
-(anti-)bi-Hermitian matrices mentioned by Yuan et al. [Linear Multilinear Algebra. 63;2015:1849–1863] as special cases. In the earlier referred work, Yuan et al. have derived explicit formulas for the least-squares
-(anti-)bi-Hermitia...

Recently, Salkuyeh and Fahim [Int. Comput. Math. 88 (2011), no. 5, 950–956] have proposed a two-step iterative refinement of the solution of an ill-conditioned linear system of equations. In this paper, we first present a generalized two-step iterative refinement procedure to solve ill-conditioned linear system of equations and study its convergenc...

This paper deals with studying some of well-known iterative methods in their tensor forms to solve a Sylvester tensor equation. More precisely, the tensor form of the Arnoldi process and Full Orthogonalization Method (FOM) are derived by using a product between two tensors. Then tensor forms of the Conjugate Gradient (CG) and Nested CG (NCG) algori...

This paper deals with scrutinizing the convergence properties of iterative methods to solve linear system of equations. Recently, several types of the preconditioners have been applied for ameliorating the rate of convergence of the Accelerated Overrelaxation (AOR) method. In this paper, we study the applicability of a general class of the precondi...

This paper deals with developing four efficient algorithms (including the conjugate gradient least-squares, least-squares with QR factorization, least-squares minimal residual and Paige algorithms) to numerically find the (least-squares) solutions of the following (in-) consistent quaternion matrix equation (Formula presented.)in which the coeffici...

This paper deals with developing a robust iterative algorithm to find the least-squares (P, Q)-orthogonal symmetric and skew-symmetric solution sets of the generalized coupled matrix equations. To this end, first, some properties of these type of matrices are established. Furthermore, an approach is offered to determine the optimal approximate (P,...

In this paper, two algorithms called weighted Gl-FOM (WGl-FOM) and weighted Gl-GMRES (WGl-GMRES) are proposed for solving the general coupled linear matrix equations. In order to accelerate the speed of convergence, a new inner product is used. Invoking the new inner product and a new matrix product, the weighted global Arnoldi algorithm is introdu...

Recently, some research has been devoted to finding the explicit forms of the η- Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an i...

This paper concerns with exploiting an oblique projection technique to solve a general class of large and sparse least squares problem over symmetric arrowhead matrices. As a matter of fact, we develop the conjugate gradient least squares (CGLS) algorithm to obtain the minimum norm symmetric arrowhead least squares solution of the general coupled m...

This paper concerns with exploiting an oblique projection technique to solve a general class of large and sparse least squares problem over symmetric arrowhead matrices. As a matter of fact, we develop the conjugate gradient least squares (CGLS) algorithm to obtain the minimum norm symmetric arrowhead least squares solution of the general coupled m...

This paper deals with the problem of finding the minimum norm least-squares solution of a quite general class of coupled linear matrix equations defined over field of complex numbers. To this end, we examine a gradient-based approach and present the convergence properties of the algorithm. The highlight of the elaborated results in the current work...

Recently, Chen and Lu have handled the well-known generalized minimal residual based on tensor format (GMRES − BTF) for solving the Sylvester tensor equation. Nevertheless, the construction and convergence of the presented algorithm have not been discussed theoretically. This fact inspirits us to theoretically analyze the construction of the GMRES...

This paper concerns with developing the LSQR algorithm to obtain the least squares quaternion solutions of the general coupled Quaternion matrix equations

The global generalized minimum residual (Gl-GMRES) method is examined for solving the generalized Sylvester matrix equation (Math Presented) Some new theoretical results are elaborated for the proposed method by employing the Schur complement. These results can be exploited to establish new convergence properties of the Gl-GMRES method for solving...

Recently, Ramadan et al. have focused on the following matrix equation:
[Formula: see text]
and propounded two gradient-based iterative algorithms for solving the above matrix equation over reflexive and Hermitian reflexive matrices, respectively. In this paper, we develop two new iterative algorithms based on a two-dimensional projection technique...

Recently, the gradient-based iterative algorithms have been widely exploited for finding the (least-squares) solutions of the different kinds of (coupled) linear matrix equations. Nevertheless, so far, the convergence of the propounded gradient-based algorithms has been studied for the case where the mentioned (coupled) linear matrix equations have...

In this paper, we focus on the following coupled linear matrix equations (Formula presented.),with (Formula presented.).where (Formula presented.) and (Formula presented.) (for (Formula presented.)) are given matrices with appropriate dimensions defined over complex number field. Our object is to obtain the solution groups X=((Formula presented.))...

Recently, Shen et al. [Preconditioned iterative methods for solving weighted linear least squares problems, Appl. Math. Mech.-Engl. Ed. 33(3) (2012), pp. 375–384] have considered four kinds of preconditioned generalized accelerated overrelaxation (GAOR) methods for solving the linear systems based on a class of weighted least-squares problems and e...

Linear matrix equations play an important role in many areas, such as control theory, system theory, stability theory and some other fields of pure and applied mathematics. In the present paper, we consider the generalized coupled Sylvester-transpose and conjugate matrix equations
[Formula: see text]
where [Formula: see text] is a group of unknown...

Recently, Wang et al. have propounded some precon-ditioned GAOR methods. It has been proved that applying one of the preconditioners leads to the superior convergence rate than other mentioned preconditioners. In the present paper we examine a new type of preconditioner which outperforms those proposed by the authors in the above referred work.

More recently, Chen and Lu [Math. Probl. Eng., DOI: 10.1155/2013/819479] have proposed an iterative algorithm to solve the Sylveter tensor equation. More precisely, the gradient-based iterative algorithm has been developed for finding the unique solution of the mentioned Sylveter tensor equation. In this paper we demonstrate that how an oblique pro...

Let A : R-mxn -> R-mxn be a symmetric positive definite linear operator. In this paper, we propose an iterative algorithm to solve the general matrix equation A(X) = C which includes the Lyapunov matrix equation and Sylvester matrix equation as special cases. It is proved that the sequence of the approximate solutions, obtained by the presented alg...

In this paper, we consider a class of general coupled linear matrix equations over the complex number field. The mentioned coupled linear matrix equations contain the unknown complex matrix groups \(X=(X_1,X_2,\ldots ,X_q)\) and \(Z=(Z_1,Z_2,\ldots ,Z_q)\) . The conjugate and transpose of the unknown matrices \(X_i\) and \(Z_i\) , \(i\in I[1,q]\) ,...

In this paper, we present an iterative algorithm for solving the following coupled Sylvester-transpose matrix equations over the generalized centro-symmetric matrix group (X 1, X 2, …, X q ). The solvability of the problem can be determined by the proposed algorithm, automatically. If the coupled Sylvester-transpose matrix equations are consistent...

Recently, Dehghan and Hajarian [Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices, Comput. Appl. Math. 31 no. 2, (2012), 353–371] have focused on the following coupled linear matrix equations A 1 XB 1 + C 1 X T D 1 = M 1 , A 2 XB 2 + C 2 X T D 2 = M 2 , and propounded two gradient-based iterati...

Consider the linear system Ax = b where the coefficient matrix A is an M-matrix. Here, it is proved that the rate of convergence of the Gauss-Seidel method is faster than the mixed- type splitting and AOR (SOR) iterative methods for solving M- matrix linear systems. Furthermore, we improve the rate of con- vergence of the mixed-type splitting itera...

In the present paper, we propose the global full orthogonalization method (Gl-FOM) and global generalized minimum residual (Gl-GMRES) method for solving large and sparse general coupled matrix equations ∑j=1pAijXjBij=Ci,i=1,…,p, where Aij∈Rm×m, Bij∈Rn×n, Ci∈Rm×n,i,j=1,2,…,p, are given matrices and Xi∈Rm×n, i=1,2,…,p, are the unknown matrices. To do...

In this paper, we present a new generalized form for the Mixed-Type splitting iterative method. The method will be called the generalized Mixed-Type (GMixed-Type) splitting iterative method. In order to improve the rate of the convergence of the GMixed-Type splitting iterative method, a preconditioned matrix is employed.

## Projects

Projects (3)

The main goal is to consider a general class of splitting and working with preconditioners extracted from the coefficient tensor.

The first goal is obtaining sufficient conditions for solvability of the mentioned equation which are computationally cheap to verify. The second purpose is proposing some new iterative methods and examining some ideas for preconditioning the main problem.

The main goal is to propose constant preconditioners to accelerate the convergence of iterative methods such as Krylov subspace (e.g., GMRES, CMRH, etc) methods for solving a class of tensor equations with Einstein product.