Recently, the projected Jacobi (PJ) and projected Gauss-Seidel (PGS) iteration methods have been studied for solving the horizontal linear complementarity problems (HLCPs). To further improve the convergence rates of the PJ and PGS iteration methods, by using the successive overrelaxation (SOR) matrix splitting technique, a projected SOR iteration method is introduced in this paper to solve the HLCP. Convergence analyses are carefully studied when the system matrices are strictly diagonally dominant and irreducibly diagonally dominant. The newly obtained convergence results greatly extend the current convergence theory. Finally, two numerical examples are given to show the effectiveness of the proposed PSOR iteration method and its advantages over the recently proposed PJ and PGS iteration methods.

We further analyze the solution of a class of block two-by-two linear systems. Instead of using the preconditioned GMRES iteration methods, we propose a new approximation of the Schur complement based on the special structure of this kind of block two-by-two matrix, and construct a practical restrictive preconditioner accordingly. Subsequently, we propose a practical restrictively preconditioned conjugate gradient (RPCG) method to solve this class of linear systems. The convergence property of the practical RPCG method is similar to the RPCG method. Last, numerical experiments show that this method is more efficient than some classical preconditioned Krylov subspace iteration methods.

We develop and investigate a new block preconditioner for a class of double saddle point (DSP) problems arising from liquid crystal directors modeling using a finite element scheme. We analyze the spectral properties of the preconditioned matrix. Numerical results are provided to evaluate the behavior of preconditioned iterative methods using the new preconditioner.

Based on the quasi-Hermitian and skew-Hermitian splitting (QHSS) iteration method proposed by Bai for solving the large sparse non-Hermitian positive definite linear systems of strong skew-Hermitian parts, this paper introduces a parameterized QHSS (PQHSS) iteration method. The PQHSS iteration is essentially a two-parameter iteration which covers the standard QHSS iteration and can further accelerate the iterative process. In addition, two practical variants, viz., inexact and extrapolated PQHSS iteration methods are established to further improve the computational efficiency. The convergence conditions for the iteration parameters of the three proposed methods are presented. Numerical results illustrate the effectiveness and robustness of the PQHSS iteration method and its variants when used as linear solvers, as well as the PQHSS preconditioner for Krylov subspace iteration methods.

Recently, the tensor robust principal component analysis (TRPCA), aiming to recover the true low-rank tensor from noisy data, has attracted considerable attention. In this paper, we solve the TRPCA problem under the framework of the tensor singular value decomposition (t-SVD). Since the convex relaxation approaches have some limitations, we establish a new non-convex TRPCA model by introducing the non-convex tensor rank approximation based on the Laplace function via the weighted $l_p$-norm regularization. An efficient algorithm based on the alternating direction method of multipliers (ADMM) is developed to solve the proposed model. We further prove that the constructed sequence converges to the desirable Karush-Kuhn-Tucker point. Experimental results show that the proposed approach outperforms various latest approaches in the literature.

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Based on the preconditioner presented by He and Huang (Comput Math Appl 62: 87–92, 2011), we introduce a parameterized augmentation block preconditioner for solving the nonsymmetric saddle point problems with the singular (1,1)-block. The theoretical analysis gives the eigenvalue and eigenvector properties of the corresponding preconditioned matrix, and numerical results confirm the effectiveness of the preconditioner for accelerating the convergence rate of the generalized minimal residual (GMRES) method when solving the large sparse nonsymmetric saddle point problems.

To enhance the computational performance of the partially randomized extended Kaczmarz (PREK) method, we propose the multi-step PREK (MPREK) method. By iteratively updating at each step, we establish a non-smooth inner-outer iteration scheme to solve the large, sparse, and inconsistent linear systems. For the MPREK method, a proof of its convergence and an upper bound on the convergence rate are given. Moreover, we show that this upper bound can be lower than that of the PREK method and the multi-step randomized extended Kaczmarz (MREK) method for certain typical choices of the inner iteration step size. Numerical experiments also indicate that, for an appropriate choice of the number of inner iteration steps, the MPREK method has a more efficient computational performance.

Multiplicative noise removal is a challenging problem in image denoising. In this paper, we develop a nonlocal matrix rank minimization method for the multiplicative noise removal problem. By utilizing the logarithm transformation, we convert the problem into an additive noise removal problem and propose a maximum a posteriori (MAP) estimation-based matrix rank minimization model for this kind of additive noise removal. A proximal alternating algorithm is designed to solve the matrix rank minimization model. The convergence of the algorithm is demonstrated by the famous Kurdyka-Łojasiewicz property. Taking advantage of the proposed matrix rank minimization model and its proximal alternating algorithm, a multiplicative noise removal method is finally developed. Numerical experiments illustrate that the proposed method can remove multiplicative noise in images much better than the existing state-of-the-art methods in terms of both image recovered measure quantities and visual qualities.

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The mathematical formulation of the mixed-cell-height circuit legalization (MCHCL) problem can be expressed by a linear complementarity problem (LCP) with the system matrix being a block two-by-two saddle point matrix. Based on the robust modulus-based matrix splitting (RMMS) iteration method and its two-step improvement (RTMMS) studied recently, the well-known Hermitian and skew-Hermitian splitting iteration method and the generalized successive overrelaxation iteration method for solving saddle point linear systems, two variants of robust two-step modulus-based matrix splitting (VRTMMS) iteration methods are proposed for solving the MCHCL problem. Convergence analyses of the proposed two iteration methods are studied in detail. Finally, five test problems are presented. Numerical results show that the proposed two VRTMMS iteration methods not only take full use of the sparse property of the circuit system but also speed up the computational efficiency of the existing RMMS and RTMMS iteration methods for solving the MCHCL problem.

Spectral regression discriminant analysis (SRDA) is one of the most popular methods for large-scale discriminant analysis. It is a stepwise algorithm composed of two steps. First, the response vectors are obtained from solving an eigenvalue problem. Second, the projection vectors are computed by solving a least-squares problem. However, the independent two steps can not guarantee the optimality of the two terms. In this paper, we propose a unified framework to compute both the response matrix and the projection matrix in SRDA, so that one can extract the discriminant information of classification tasks more effectively. The convergence of the proposed method is discussed. Moreover, we shed light on how to choose the joint parameter adaptively, and propose a parameter-free joint spectral regression discriminant analysis (JointSRDA-PF) method. Numerical experiments are made on some real-world databases, which show the numerical behavior of the proposed methods and the effectiveness of our strategies.

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In this work, by applying the minimum residual technique to the block-diagonal and anti-block-diagonal splitting (BAS) iteration scheme, an iteration method named minimum residual BAS (MRBAS) is proposed to solve a two-by-two block system of nonlinear equations arising from the reformulation of the system of absolute value equations (AVEs). The theoretical analysis shows that the MRBAS iteration method is convergent under suitable conditions. Numerical results demonstrate the feasibility and the effectiveness of the MRBAS iteration method.

Based on the block Arnoldi process and minimizing the Frobenius norm of the error, the block generalized minimal error (GMERR) method and its simpler version are proposed for solving large-scale linear systems of equations with multiple right-hand sides. However, little is known about the behavior of these methods when they are applied to the solution of linear discrete ill-posed problems with multiple right-hand sides contaminated by errors. In this paper, the regularizing properties of the block GMERR method and the simpler block GMERR method are examined. Both a regularized block GMERR method and a regularized simpler block GMERR method are developed for solving large-scale linear discrete ill-posed problems with multiple right-hand sides. Numerical experiments on typical test matrices show the efficiency of the proposed methods.

The two-dimensional (2D) space-fractional diffusion equations can be effectively discretized by an implicit finite difference scheme with the shifted Grünwald formula. The coefficient matrices of the discretized linear systems are equal to the sum of the identity matrix and a block-Toeplitz with a Toeplitz-block matrix. In this paper, one variant of the alternating direction implicit (ADI) iteration method is proposed to solve the discretized linear systems. By making use of suitable permutations, each iteration of the ADI iteration method requires the solutions of two linear subsystems whose coefficient matrices are block diagonal matrices with diagonal blocks being Toeplitz matrices. These two linear subsystems can be solved block by block by fast or superfast direct methods. Theoretical analyses show that the ADI iteration method is convergent. In particular, we derive a sharp upper bound about its asymptotic convergence rate and deduce the optimal value of its iteration parameter. Numerical results exhibit that the corresponding ADI preconditioner can improve the computational efficiency of the Krylov subspace iteration methods.

The photoacoustic tomography (PAT) is a new biomedical imaging modality. It has great advantages in early diagnosis of the human disease and accurate monitoring of disease progression. In photoacoustic imaging, when a beam of short-pulsed laser illuminates the biological tissue, the photoacoustic effect leads to the emergence of acoustic waves in the tissue. The initial acoustic pressure in the tissue reveals the structures of the tissue. The purpose of the PAT reconstruction problem is to obtain the initial acoustic pressure in the tissue from the collected photoacoustic signal information. In this paper, we propose a rank minimization-based regularization model for the sparse-view photoacoustic image reconstruction problem. We design a proximal alternating iterative algorithm to solve the model and the convergence of the algorithm is demonstrated by utilizing the Kudyka-Łojasiewicz theory. The experimental results show that the proposed method is competitive with the existing state-of-the-art PAT reconstruction methods in terms of both reconstructed quantities and visual effects for the sparse-view PAT reconstruction problem.

We propose, in this paper, the preconditioned accelerated generalized successive overrelaxation (PAGSOR) iteration method for efficiently solving the large complex symmetric linear systems. To solve the nonlinear systems whose Jacobian matrices are complex and symmetric, treating the PAGSOR method as internal iteration, we construct a modified Newton-PAGSOR (MN-PAGSOR) method to provide an effective approach for solving a wide range of problems in various scientific and engineering fields. Based on the Hölder continuous condition we present the theoretical framework of the modified method, demonstrate its local convergence properties, and provide numerical experiments to validate its effectiveness in solving a class of nonlinear systems.

The eigenvalue assignment for the fractional order linear time-invariant control systems is addressed in this paper and the existence of the solution to this problem is also analyzed based on the controllability theory of the fractional order systems. According to the relationship between the solution to this problem and the solution to the nonlinear matrix equation, we propose a numerical algorithm via the matrix sign function method based on the rational iteration for solving this nonlinear matrix equation, which can circumvent the limitation of the assumption of linearly independent eigenvectors. Moreover, the proposed algorithm only needs to solve the linear system with multiple right-hand sides and it converges quadratically. Finally, the efficiency of the proposed approach is shown through numerical examples.

In this paper, according to the Shamanskii technology, an alternately linearized implicit (ALI) iteration method is proposed to compute the minimal nonnegative solution to the nonsymmetric coupled algebraic Riccati equation. Based on the ALI iteration method, we propose two modified alternately linearized implicit (MALI) iteration methods with double parameters. Further, we prove the monotone convergence of these iteration methods. Numerical examples demonstrate the effectiveness of the presented iteration methods.

With the advent of tensor-valued time series data, tensor autoregression appears in many fields, in which the coefficient estimation is confronted with the problem of dimensional disaster. Based on the tensor ring (TR) decomposition, an autoregression model with one order for tensor-valued responses is proposed in this paper. A randomized method, TensorSketch, is applied to the TR autoregression model for estimating the coefficient tensor. Convergence and some properties of the proposed methods are given. Finally, some numerical experiment results on synthetic data and real data are given to illustrate the effectiveness of the proposed method.

In this paper, we present a new convergence upper bound for the greedy Gauss-Seidel (GGS) method proposed by Zhang and Li [38]. The new convergence upper bound improves the upper bound of the GGS method. In addition, we also propose a novel greedy block Gauss-Seidel (RDBGS) method based on the greedy strategy of the GGS method for solving large linear least-squares problems. It is proved that the RDBGS method converges to the unique solution of the linear least-squares problem. Numerical experiments demonstrate that the RDBGS method has superior performance in terms of iteration steps and computation time.

We propose the modulus-based cascadic multigrid (MCMG) method and the modulus-based economical cascadic multigrid method for solving the quasi-variational inequalities problem. The modulus-based matrix splitting iterative method is adopted as a smoother, which can accelerate the convergence of the new methods. We also give the convergence analysis of these methods. Finally, some numerical experiments confirm the theoretical analysis and show that the new methods can achieve high efficiency and lower costs simultaneously.

For solving large inconsistent linear systems, we research a novel format to enhance the numerical stability and control the complexity of the model. Based on the idea of two subspace iterations, we propose the max-residual two subspace coordinate descent (M2CD) method. To accelerate the convergence rate, we further present the cyclic block coordinate descent (CBCD) method. The convergence properties of these methods are analyzed, and their effectiveness is illustrated by numerical examples.

In this paper, we consider numerical methods for two-sided space variable-order fractional diffusion equations (VOFDEs) with a nonlinear source term. The implicit Euler (IE) method and a shifted Grünwald (SG) scheme are used to approximate the temporal derivative and the space variable-order (VO) fractional derivatives, respectively, which leads to an IE-SG scheme. Since the order of the VO derivatives depends on the space and the time variables, the corresponding coefficient matrices arising from the discretization of VOFDEs are dense and without the Toeplitz-like structure. In light of the off-diagonal decay property of the coefficient matrices, we consider applying the preconditioned generalized minimum residual methods with banded preconditioners to solve the discretization systems. The eigenvalue distribution and the condition number of the preconditioned matrices are studied. Numerical results show that the proposed banded preconditioners are efficient.

The electrocardiogram (ECG) segmentation needs to separate different waves from an ECG and cluster the waves simultaneously. Clusterwise regression is a useful approach that can segment and cluster the data simultaneously. In this paper, we apply the clusterwise regression method to segment the ECG. By modeling the ECG signal wave by the Gaussian mixture model (GMM) and introducing a weight function, we propose a minimization model that consists of the weighted sum of the negative log-likelihood and the total variation (TV) of the weight function. The TV of the weight function enforces the temporal consistency. A supervised algorithm is designed to solve the proposed model. Experimental results show the efficiency of the proposed method for the ECG segmentation.

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) iteration scheme and a novel minimum residual technique with the aid of a positive definite matrix, a novel minimum residual MHSS (NMRMHSS) iteration method was proposed for solving complex symmetric systems of linear equations. As is known, the NMRMHSS iteration is unconditional convergent; however, its numerical performance is degraded. In this work, we consider to improve the rate of the convergence of the NMRMHSS iteration method and inherit its theoretical property. To the end, we first combine the minimization technique of NMRMHSS with an accelerating method and obtain a fast and unconditional convergent iteration method. Then, the convergence is demonstrated, which indicates that the contraction factor of our method is smaller than that of NMRMHSS. Besides, the theoretical analysis shows that our method has more widespread application for solving complex symmetric linear systems. Finally, numerical results are reported to illustrate the numerical behavior of the proposed iteration method.

Multilinear systems play an important role in scientific calculations of practical problems. In this paper, we consider a tensor splitting method with a relaxed Anderson acceleration for solving the multilinear systems. The new method preserves the nonnegativity for every iterative step and improves the existing ones. Furthermore, the convergence analysis of the proposed method is given. The new algorithm performs effectively for numerical experiments.

An improved SSOR-like (ISSOR-like) preconditioner is proposed for the non-Hermitian positive definite linear system with a dominant skew-Hermitian part. The upper and lower bounds on the real and imaginary parts of the eigenvalues of the ISSOR-like preconditioned matrix and the convergence property of the corresponding ISSOR-like iteration method are discussed in depth. Numerical experiments show that the ISSOR-like preconditioner can effectively accelerate preconditioned GMRES.

The randomized block Kaczmarz (RBK) method is a randomized orthogonal projection iterative approach, which plays an important role in solving large-scale linear systems. A key point of this type of method is to select working rows effectively during iterations. However, in most of the RBK-type methods, one has to scan all the rows of the coefficient matrix in advance to compute probabilities or paving, or to compute the residual vector of the linear system in each iteration to determine the working rows. These are unfavorable for big data problems. To cure these drawbacks, we propose a semi-randomized block Kaczmarz (SRBK) method with simple random sampling for large-scale linear systems in this paper. The convergence of the proposed method is established. Numerical experiments on some real-world and large-scale data sets show that the proposed method is often superior to many state-of-the-art RBK-type methods for large linear systems.

For the tensor auto-regression (TAR) model, we consider combining the model with the multilinear systems in this work, to expand the previous methods of parameter estimation and prediction. Inspired by the SubCount Sketch (SCS) method, a Count Column Sketch (CCS) method is proposed for the parameter estimation. Compared with the SCS method, the proposed CCS method makes the parameter estimation for the underdetermined problems more efficient. At the same time, the convergence properties of the CCS method are also given. In the numerical experiments, we use the CCS method to analyze and predict the L-transform tensor auto-regressive (L-TAR) model. For underdetermined problems, the numerical results indicate that our approach is validated and less training time is required than other methods.

This paper further extends the shift-splitting (SS) and local shift-splitting (LSS) preconditioners to solve the general block two-by-two linear systems. We demonstrate that the eigenvalues of the corresponding preconditioned matrices cluster tightly around 2 by detailed spectral property analysis. Numerical experiments not only validate the theoretical results but also show the effectiveness and superiority of the SS and LSS preconditioners by comparing them with some existing preconditioners applied to the generalized minimal residual (GMRES) method for solving the block two-by-two linear systems.