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In this paper, we propose an infeasible arc-search interior-point algorithm for solving nonlinear programming problems. Most algorithms based on interior-point methods are categorized as line search since they compute a next iterate on a straight line determined by a search direction which approximates the central path. The proposed arc-search interior-point algorithm uses an arc for the approximation. We discuss convergence properties of the proposed algorithm. We also conduct numerical experiments on the CUTEst benchmark problems and compare the performance of the proposed arc-search algorithm with that of a line-search algorithm. Numerical results indicate that the proposed arc-search algorithm reaches the optimal solution using fewer iterations but longer times than a line-search algorithm. A modification that leads to a faster arc-search algorithm is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, a numerical algorithm based on a new kind of tension B-spline, named hyperbolic-trigonometric tension B-spline method, is applied for solving Burgers-Huxley equation. This method is generated over the space span {sin(tt),cos(tt),sinh(tt),cosh(tt),1,t,...,tn-?5},n =?5, where t is the tension parameter. Properties of it are the same in most of the properties of the usual polynomial B-splines and benefit from some other advantages, as well. Therefore, in this paper, we apply three methods consisting of trigonometric method, hyperbolic tension B-spline method, and our new hyperbolic-trigonometric tension B-spline method, to solve Burgers-Huxley equation. The convergence analysis is discussed. Then, we use some numerical examples to illustrate the accuracy and implementation of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The solution of a large-scale Sylvester matrix equation plays an important role in control and large scientific computations. In this paper, we are interested in the large Sylvester matrix equation with large dimensionA and small dimension B, and a popular approach is to use the global Krylov subspace method. In this paper, we propose three new algorithms for this problem. We first consider the global GMRES algorithm with weighting strategy, which can be viewed as a precondition method. We present three new schemes to update the weighting matrix during iterations. Due to the growth of memory requirements and computational cost, it is necessary to restart the algorithm effectively. The deflation strategy is efficient for the solution of large linear systems and large eigenvalue problems; to the best of our knowledge, little work is done on applying deflation to the (weighted) global GMRES algorithm for large Sylvester matrix equations. We then consider how to combine the weighting strategy with deflated restarting, and propose a weighted global GMRES algorithm with deflation for solving large Sylvester matrix equations. In particular, we are interested in the global GMRES algorithm with deflation, which can be viewed as a special case when the weighted matrix is chosen as the identity. Theoretical analysis is given to show rationality of the new algorithms. Numerical experiments illustrate the numerical behavior of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two new algorithms for solving bilevel pseudomonotone variational inequality problems in real Hilbert spaces. The advantages of our algorithms are that they only need to calculate one projection on the feasible set in each iteration, and do not require the prior information of the Lipschitz constant of the cost operator. Furthermore, two new algorithms are derived to solve variational inequality problems. We establish the strong convergence of the proposed algorithms under some suitable conditions imposed on parameters. Finally, several numerical results and applications in optimal control problems are reported to illustrate the efficiency and advantages of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, the block generalized product-type bi-conjugate gradient (GPBi-CG) method for solving large, sparse nonsymmetric linear systems of equations with multiple right-hand sides is proposed. The new algorithm is based on the block BiCG process. We analyze the convergence behavior of this method and present a bound for the residual norm of block GPBi-CG according to the residual norm of Bl-GMRES method. In addition, we prove that convergence is guaranteed when A is positive real. The numerical experiments show the efficiency of the new method and confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a second-order decoupled algorithm with different subdomain time steps for the non-stationary Stokes/Darcy model. It is based on the second-order spectral deferred correction method in time and the finite element method in space. We provide the stability and convergence results of our decoupled scheme. Last, some numerical experiments are given to illustrate the accuracy and effectiveness of our decoupled scheme. [ABSTRACT FROM AUTHOR]

We consider a non-convex quadratic program (QP) with linear and convex quadratic constraints that arises from a broad range of applications and is known to be NP-hard. In this paper, we first prove that the alternative direction method converges to a local solution of the underlying QP problem. We then propose a new branch-and-cut algorithm that finds a globally optimal solution to the underlying QP problem within a pre-specified 𝜖-tolerance by integrating the alternative direction method with semidefinite relaxation and disjunctive cut techniques. We establish the global convergence of the algorithm and estimate its complexity. Preliminary numerical results demonstrate that the proposed algorithm can effectively find a globally optimal solution to medium-scale QP instances in which the number of negative eigenvalues of the Hessian matrix in the objective function is less than or equals 20. [ABSTRACT FROM AUTHOR]

Symmetric nonnegative matrix factorization (symNMF) is a variant of nonnegative matrix factorization (NMF) that allows handling symmetric input matrices and has been shown to be particularly well suited for clustering tasks. In this paper, we present a new model, dubbed off-diagonal symNMF (ODsymNMF), that does not take into account the diagonal entries of the input matrix in the objective function. ODsymNMF has three key advantages compared to symNMF. First, ODsymNMF is theoretically much more sound as there always exists an exact factorization of size at most n(n − 1)/2 where n is the dimension of the input matrix. Second, it makes more sense in practice as diagonal entries of the input matrix typically correspond to the similarity between an item and itself, not bringing much information. Third, it makes the optimization problem much easier to solve. In particular, it will allow us to design an algorithm based on coordinate descent that minimizes the component-wise ℓ1 norm between the input matrix and its approximation. We prove that this norm is much better suited for binary input matrices often encountered in practice. We also derive a coordinate descent method for the component-wise ℓ2 norm, and compare the two approaches with symNMF on synthetic and document datasets. [ABSTRACT FROM AUTHOR]

Recently, zeroing dynamics (ZD) method has shown wonderful effect in solving time-varying problems. Generally, the continuous-time ZD model should be discretized as a discrete-time algorithm for numerical computation. A good choice is to apply Zhang et al. discretization (ZeaD) formulas, which are effective time-discretization formulas. Actually, ZeaD formulas can also be applied to obtaining various discrete-time algorithms, not only the ZD algorithms. In the previous work, some piecemeal ZeaD formulas have been proposed and investigated. However, the relationship between the time-instant number and precision of ZeaD formulas is not found, which is emphatically investigated in this paper. Specifically, the ZeaD formulas from two to nine instants are investigated, and the general ZeaD formula groups are studied. Two-instant and three-instant ZeaD formula groups have linear precision at most. Four-instant ZeaD formula group has quadratic precision at most. Five-instant and six-instant ZeaD formula groups have cubic precision at most. Seven-instant and eight-instant ZeaD formula groups have quartic precision at most. Nine-instant ZeaD formula group has quintic precision at most. Theoretical analyses are presented to substantiate the relationship. Moreover, the ZeaD formulas as well as ZD method are applied to solving time-varying quadratic optimization problem, and the numerical results verify the effectiveness of ZeaD formulas. [ABSTRACT FROM AUTHOR]

The recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Padé methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems. [ABSTRACT FROM AUTHOR]

Quaternion equality constrained least squares (QLSE) problems have attracted extensive attention in the field of mathematical physics due to its applicability as an extremely effective tool. However, the knowledge gap among numerous QLSE problems has not been settled now. In this paper, by using quaternion SVD (Q-SVD) and the equivalence of the QLSE problem and Karush-Kuhb-Tucker (KKT) equation, we obtain some equations about the matrices in the general solution of the QLSE problem. Using these equations, an equivalent form of the solution of the QLSE problem is obtained. Then, applying the special structure of real representation of quaternion, we propose a real structure-preserving algorithm based on Q-SVD. At last, we give numerical example, which illustrates the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

The split feasibility problem is to find a point x∗ with the property that x∗∈ C and Ax∗∈ Q, where C and Q are nonempty closed convex subsets of real Hilbert spaces X and Y, respectively, and A is a bounded linear operator from X to Y. The split feasibility problem models inverse problems arising from phase retrieval problems and the intensity-modulated radiation therapy. In this paper, we introduce a new inertial relaxed CQ algorithm for solving the split feasibility problem in real Hilbert spaces and establish weak convergence of the proposed CQ algorithm under certain mild conditions. Our result is a significant improvement of the recent results related to the split feasibility problem. [ABSTRACT FROM AUTHOR]

In this paper, two effective derivative-free methods are proposed for solving large-scale nonlinear monotone equations, in which the search directions are sufficiently descent and independent of the line search. The methods are the extensions of the conjugate gradient methods proposed by Bojari and Eslahchi (Numer. Algorithms 83, pp. 901–933, 2020) combined with the hyperplane projection technique. Our approaches are low storage memory and derivative-free, which makes them suitable for large-scale nonsmooth monotone nonlinear equations. Under proper assumptions, we analyze the global convergence property of the proposed methods. Finally, numerical experiments show that the proposed methods outperform some existing ones. [ABSTRACT FROM AUTHOR]

Since sparse unmixing has emerged as a promising approach to hyperspectral unmixing, some spatial-contextual information in the hyperspectral images has been exploited to improve the performance of the unmixing recently. The total variation (TV) has been widely used to promote the spatial homogeneity as well as the smoothness between adjacent pixels. However, the computation task for hyperspectral sparse unmixing with a TV regularization term is heavy. Besides, the convergence of the primal alternating direction method of multipliers (ADMM) for the hyperspectral sparse unmixing with a TV regularization term has not been explained in detail. In this paper, we design an efficient and convergent dual symmetric Gauss-Seidel ADMM (sGS-ADMM) for hyperspectral sparse unmixing with a TV regularization term. We also present the global convergence and local linear convergence rate analysis for this algorithm. As demonstrated in numerical experiments, our algorithm can obviously improve the efficiency of the unmixing compared with the state-of-the-art algorithm. More importantly, we can obtain images with higher quality. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the development of a sublinear sparse Fourier algorithm for high-dimensional data. In 11Adaptive Sublinear Time Fourier Algorithm" by Lawlor et al. (Adv. Adapt. Data Anal.5(01):1350003, 2013), an efficient algorithm with Θ (k log k) average-case runtime and Θ(k) average-case sampling complexity for the one-dimensional sparse FFT was developed for signals of bandwidth N, where k is the number of significant modes such that k ≪ N. In this work we develop an efficient algorithm for sparse FFT for higher dimensional signals, extending some of the ideas in Lawlor et al. (Adv. Adapt. Data Anal.5(01):1350003, 2013). Note a higher dimensional signal can always be unwrapped into a one-dimensional signal, but when the dimension gets large, unwrapping a higher dimensional signal into a one-dimensional array is far too expensive to be realistic. Our approach here introduces two new concepts: "partial unwrapping" and "tilting." These two ideas allow us to efficiently compute the sparse FFT of higher dimensional signals. [ABSTRACT FROM AUTHOR]

Minimizing a differentiable function restricted to the set of matrices with orthonormal columns finds applications in several fields of science and engineering. In this paper, we propose to solve this problem through a nonmonotone variation of the inexact restoration method consisting of two phases: restoration phase, aimed to improve feasibility, and minimization phase, aimed to decrease the function value in the tangent set of the constraints. For this, we give a suitable characterization of the tangent set of the orthogonality constraints, allowing us to (i) deal with the minimization phase efficiently and (ii) employ the Cayley transform to bring a point in the tangent set back to feasibility, leading to a SVD-free restoration phase. Based on previous global convergence results for the nonmonotone inexact restoration algorithm, it follows that any limit point of the sequence generated by the new algorithm is stationary. Moreover, numerical experiments on three different classes of the problem indicate that the proposed method is reliable and competitive with other recently developed methods. [ABSTRACT FROM AUTHOR]

In this paper, we combine the S-iteration process introduced by Agarwal et al. (J. Nonlinear Convex Anal., 8(1), 61–79 2007) with the proximal point algorithm introduced by Rockafellar (SIAM J. Control Optim., 14, 877–898 1976) to propose a new modified proximal point algorithm based on the S-type iteration process for approximating a common element of the set of solutions of convex minimization problems and the set of fixed points of nearly asymptotically quasi-nonexpansive mappings in the framework of CAT(0) spaces and prove the △-convergence of the proposed algorithm for solving common minimization problem and common fixed point problem. Our result generalizes, extends and unifies the corresponding results of Dhompongsa and Panyanak (Comput. Math. Appl., 56, 2572–2579 2008), Khan and Abbas (Comput. Math. Appl., 61, 109–116 2011), Abbas et al. (Math. Comput. Modelling, 55, 1418–1427 2012) and many more. [ABSTRACT FROM AUTHOR]

In this paper, we propose a hybrid iterative method to approximate a common solution of split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces. We prove that sequences generated by the proposed hybrid iterative method converge strongly to a common solution of these problems. Further, we discuss some applications of the main result. We also discuss a numerical example to demonstrate the applicability of the iterative method. The method and results presented in this paper extend and unify the corresponding known results in this area. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the modified proximal point algorithm for common fixed points of asymptotically quasi-nonexpansive mappings in CAT(0) spaces and also prove some convergence theorems of the proposed algorithm to a common fixed point of asymptotically quasi-nonexpansive mappings and a minimizer of a convex function. The main results in this paper improve and generalize the corresponding results given by some authors. Moreover, we then give numerical examples to illustrate and show efficiency of the proposed algorithm for supporting our main results. [ABSTRACT FROM AUTHOR]

In this paper, novel representations of generalized inverses of rational matrices are developed. Therefore, a unified approach for the computation of {1,2,3} and {1,2,4} inverses and Moore-Penrose inverse of a given matrix A is considered. Full-rank QDR decomposition of a rational matrix is utilized to avoid the square roots of rational expressions in the evaluations, making the given algorithm very suitable for symbolic computations of generalized matrix inverses. Furthermore, we developed an algorithm for symbolic computation of the Moore-Penrose inverse of a polynomial matrix using the full-rank QDR decomposition, therefore maximizing the potential of using square root–free polynomial entries. Introduced algorithms are illustrated via numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we present PARASOF, an algorithm for the solution of linear systems with BABD matrices on massively parallel computing systems like graphic processing units or GPUs. This algorithm is compared with the state-of-the-art algorithms, in particular SOF, from which it is inspired and takes the same stability properties. We detail its design and implementation issues and give the main figures of its theoretical and experimental performances. [ABSTRACT FROM AUTHOR]

The LP-Newton method solves linear programming (LP) problems by repeatedly projecting a current point onto a certain relevant polytope. In this paper, we extend the algorithmic framework of the LP-Newton method to conic programming (CP) problems via a linear semi-infinite programming (LSIP) reformulation. In this extension, we produce a sequence by projection onto polyhedral cones constructed from LP problems obtained by finitely relaxing the LSIP problem equivalent to the CP problem. We show global convergence of the proposed algorithm under mild assumptions. To investigate its efficiency, we apply our proposed algorithm and a primal-dual interior-point method to second-order cone programming problems and compare the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, an efficient algorithm is presented by adopting the extrapolation technique to improve the accuracy of finite difference schemes for two-dimensional space-fractional diffusion equations with non-smooth solution. The popular fractional centered difference scheme is revisited and the stability and error estimation of numerical solution are given in maximum norm. Based on the analysis of leading singularity of exact solution for the underlying problem, the extrapolation technique and numerical correction method are exploited to enhance the accuracy and convergence rate of the computation. Two numerical examples are provided to validate the theoretical prediction and efficiency of the algorithm. It is shown that, by using the proposed algorithm, both accuracy and convergence rate of numerical solutions can be significantly improved and the second-order accuracy can even be recovered for the equations with large fractional orders. [ABSTRACT FROM AUTHOR]

In the last few decades, Runge-Kutta-Nyström (RKN) methods have made significant progress and the study of RKN-type methods for solving highly oscillatory differential equations has received a great deal of attention. In this paper, from the point of view of geometric integration, the oscillation-preserving behaviour of the existing RKN-type methods in the literature are analysed in detail. To this end, it is convenient to introduce the concept of oscillation preservation for RKN-type methods. It turns out that if both the internal stages and the updates of an RKN-type method respect the qualitative and global features of the highly oscillatory solution, then the method is oscillation preserving. Since the internal stages of standard RKN and adapted RKN (ARKN) methods are inimical to the oscillation-preserving structure, neither ARKN methods nor the symplectic and symmetric RKN methods, and standard RKN methods are oscillation preserving. Other concerns relating to oscillation preservation are also considered. In particular, we are concerned with the computational issues for efficiently solving semi-discrete wave equations such as semi-discrete Klein-Gordon (KG) equations and damped sine-Gordon equations. The results of numerical experiments show the importance of the oscillation-preserving property for an RKN-type method and the remarkable superiority of oscillation-preserving integrators when applied to nonlinear multi-frequency highly oscillatory systems. [ABSTRACT FROM AUTHOR]

Given function values on a uniform grid in a domain Ω in ℝ d , one is often interested in extending the values to a larger grid on a box B containing Ω. In particular, we are interested in "periodic extensions." For such extensions the discrete Fourier transform (DFT) of the resulting grid values on B is expected to provide good efficient approximation to the underlying function on Ω. This paper presents two different extension algorithms. The first method is a natural approach to this problem, aiming at achieving the fastest decay of the DFT coefficients of the extended data.The second is a fast algorithm which is appropriate for the univariate case and for limited cases of multivariate scenarios. It is shown that if a "good" periodic extension exists, the proposed method will find an extension with similar properties. [ABSTRACT FROM AUTHOR]

In this paper, we define s-nonstationary iterative process and obtain its properties. We prove, that for any one-point iterative process without memory, there exists an s-nonstationary process of the same order, but of higher efficiency by the criteria of Traub and Ostrowski. We supply constructions of s-nonstationary processes for Newton's, Halley's, and Chebyshev's methods, obtain their properties and, for some of them, also their geometric interpretation. The algorithms we present can be transformed into computer programs in a straightforward manner. Additionally, we illustrate numerical examples, as demonstrations for the methods we present. [ABSTRACT FROM AUTHOR]

In this paper, we present some novel observations for the semidefinite linear complementarity problems, abbreviated as SDLCPs. Based on these new results, we establish the modulus-based matrix splitting iteration methods, which are obtained by reformulating equivalently SDLCP as an implicit fixed-point matrix equation. The convergence of the proposed modulus-based matrix splitting iteration methods has been analyzed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SDLCPs. [ABSTRACT FROM AUTHOR]

In this paper, by basing on the inexact subgradient and projection methods presented by Santos et al. (Comput. Appl. Math. 30: 91–107, 2011), we develop subgradient projection methods for solving strongly monotone equilibrium problems with pseudomonotone equilibrium constraints. The problem usually is called monotone bilevel equilibrium problems. We show that this problem can be solved by a simple and explicit subgradient method. The strong convergence for the proposed algorithms to the solution is guaranteed under certain assumptions in a real Hilbert space. Numerical illustrations are given to demonstrate the performances of the algorithms. [ABSTRACT FROM AUTHOR]

We develop a fast divide-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical scheme does not have a Toeplitz structure. In this paper, we derive a fast approximation of the coefficient matrix by the means of a finite sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires O (N log 2 N) memory and O (N log 3 N) computational complexity with N being the numbers of unknowns. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient improvement of gift wrapping algorithm for determining the convex hull of a finite set of points in ℝ n space, applying the best restricted area technique inspired from the Method of Orienting Curves (this method was used successfully in computational geometry by An and Trang in Numerical Algorithms59, 347–357 (2012), Optimization62, 975–988 (2013)). The numerical experiments on the sets of random points in two- and three-dimensional space show that the running time of our algorithm is faster than the gift wrapping algorithm and the newest modified one. [ABSTRACT FROM AUTHOR]

For the large-scale linear discrete ill-posed problem min ∥ A x − b ∥ or Ax = b with b contaminated by Gaussian white noise, the following Krylov solvers are commonly used: LSQR, and its mathematically equivalent CGLS (i.e., the Conjugate Gradient (CG) method applied to ATAx = ATb), CGME (i.e., the CG method applied to min ∥ A A T y − b ∥ or AATy = b with x = ATy), and LSMR (i.e., the minimal residual (MINRES) method applied to ATAx = ATb). These methods have intrinsic regularizing effects, where the number k of iterations plays the role of the regularization parameter. In this paper, we analyze the regularizing effects of CGME and LSMR and establish a number of results including the filtered SVD expansion of CGME iterates, which prove that the 2-norm filtering best possible regularized solutions by CGME and LSMR are less accurate than and at least as accurate as those by LSQR, respectively. We also prove that the semi-convergence of CGME and LSMR always occurs no later and sooner than that of LSQR, respectively. As a byproduct, using the analysis approach for CGME, we improve a fundamental result on the accuracy of the truncated rank k approximate SVD of A generated by randomized algorithms, and reveal how the truncation step damages the accuracy. Numerical experiments justify our results on CGME and LSMR. [ABSTRACT FROM AUTHOR]

Our contribution in this paper, we introduce and analyze two new hybrid algorithms by combining Mann iteration and inertial method for solving split fixed point problems of demicontractive mappings and equilibrium problems in a real Hilbert space. By using a new technique of choosing step size, our algorithms do not need any prior information on the operator norm. In fact, an inertial type algorithm was proposed in order to accelerate its convergence rate. We then prove weak and strong convergence of proposed methods under some control conditions. Moreover, some numerical experiments for image restoration problems and oligopolistic market equilibrium problems are also provided for supporting our main results. [ABSTRACT FROM AUTHOR]

The Alternating Direction Multipliers Method (ADMM) is a very popular algorithm for computing the solution of convex constrained minimization problems. Such problems are important from the application point of view, since they occur in many fields of science and engineering. ADMM is a powerful numerical tool, but unfortunately its main drawback is that it can exhibit slow convergence. Several approaches for its acceleration have been proposed in the literature and in this paper we present a new general framework devoted to this aim. In particular, we describe an algorithmic framework that makes possible the application of any acceleration step while still having the guarantee of convergence. This result is achieved thanks to a guard condition that ensures the monotonic decrease of the combined residual. The proposed strategy is applied to image deblurring problems. Several acceleration techniques are compared; to the best of our knowledge, some of them are investigated for the first time in connection with ADMM. Numerical results show that the proposed framework leads to a faster convergence with respect to other acceleration strategies recently introduced for ADMM. [ABSTRACT FROM AUTHOR]

In this paper, we propose an incremental bundle method with inexact oracle for solving the portfolio optimization with stochastic second-order dominance (SSD) constraints. We first relax the SSD problem as a stochastic semi-infinite programming (SIP) problem. For the particular case of SIP problem, we exploit the improvement function and the idea of incremental technique for dealing with the infinitely many constraints. In the stochastic model, as an adding-rules, the "inexact oracle" is introduced in this algorithm. Therefore, the algorithm does not need all the information about the constraints, but only needs the inexact information of one component function to update the bundle and produces the search direction. Our numerical results on solving the academic problems have shown advantages of the incremental bundle method over three existing algorithms. Finally, numerical results on a part of portfolio optimization problem are presented by using the FTSE100 Index. [ABSTRACT FROM AUTHOR]

The paper introduces and analyzes the convergence of a new iterative algorithm for approximating solutions of equilibrium problems involving strongly pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. The algorithm uses a stepsize sequence which is non-increasing, diminishing, and non-summable. This leads to the main advantage of the algorithm, namely that the construction of solution approximations and the proof of its convergence are done without the prior knowledge of the modulus of strong pseudomonotonicity and Lipschitz-type constants of bifunctions. The strongly convergent theorem is established under suitable assumptions. The paper also discusses the assumptions used in the formulation of the convergent theorem. Several numerical results are reported to illustrate the behavior of the algorithm with different sequences of stepsizes and also to compare it with others. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalized viscosity algorithm for finding a fixed point of an asymptotically nonexpansive mapping in the intermediate sense which is also a solution to a variational inequality problem of two inverse-strongly monotone operators in 2-uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are given under suitable assumptions imposed on the parameters. The results obtained in this paper improve and extend many recent ones in the literature. Three numerical examples are also given to show the efficiency and implementation of our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the product eigenvalue problem for the class of Cauchy-polynomial-Vandermonde (CPV) matrices arising in a rational interpolation problem. We present the explicit expressions of minors of CPV matrices. An algorithm is designed to accurately compute the bidiagonal decomposition for strictly totally positive CPV matrices and their additive inverses. We then illustrate the sign regularity of the bidiagonal decomposition to show that all the eigenvalues of a product involving such matrices are computed to high relative accuracy. Numerical experiments are given to confirm the claimed high relative accuracy. [ABSTRACT FROM AUTHOR]

This paper considers the well-known problem of dealing with non-convexity during the minimization of a non-linear function f(x) by Newton-like methods. The proposal made here involves a curvilinear search along an approximation to the continuous steepest descent path defined by the solution of the differential equation The algorithm we develop and describe has some features in common with trust-region methods and we present some numerical experiments in which its performance is compared with other ODE-based and trust-region methods. [ABSTRACT FROM AUTHOR]

In this paper, a full-Newton step infeasible interior-point method for solving linear optimization problems is presented. In each iteration, the algorithm uses only one so-called feasibility step and computes the feasibility search directions by using a trigonometric kernel function with a double barrier term. Convergence of the algorithm is proved and it is shown that the complexity bound of the algorithm matches the currently best known iteration bound for infeasible interior-point methods. Finally, some numerical results are provided to illustrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we derive a novel expression for the computation of the outer inverse A T , S (2) . Based on this expression, we present a new Gauss-Jordan elimination method for computing A R (G) , N (G) (2) . The analysis of computational complexity indicates that our algorithm is more efficient than the existing Gauss-Jordan elimination algorithms for A R (G) , N (G) (2) in the literature for a large class of problems. Especially for the case when G is a Hermitian matrix, our algorithm has the lowest computational complexity among the existing algorithms. Finally, numerical experiments show that our method for the outer inverse A R (G) , N (G) (2) generally is more efficient than that of the other existing methods in the cases of matrices A with m < n or square matrices G with high rank or Hermitian matrices G in practice. [ABSTRACT FROM AUTHOR]

In this paper, we present a time two-grid algorithm based on the finite difference (FD) method for the two-dimensional nonlinear time-fractional mobile/immobile transport model. We establish the problem as a nonlinear fully discrete FD system, where the time derivative is discretized by the second-order backward difference formula (BDF) scheme, the Caputo fractional derivative is treated by means of L1 discretization formula, and the spatial derivative is approximated by the central difference formula. For solving the nonlinear FD system more efficiently, a time two-grid algorithm is proposed, which consists of two steps: first, the nonlinear FD system on a coarse grid is solved by nonlinear iterations; second, the Newton iteration is utilized to solve the linearized FD system on the fine grid. The stability and convergence in L2-norm are obtained for the two-grid FD scheme. Numerical results are consistent with the theoretical analysis. Meanwhile, numerical experiments show that the two-grid FD method is much more efficient than the general FD scheme for solving the nonlinear FD system. [ABSTRACT FROM AUTHOR]

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 bidiagonalization (GKB) algorithm and apply it in conjunction with Tikhonov regularization. The conditioning of the Stein tensor equation is examined. These results suggest how the tensor GKB process can be used to solve general linear tensor equations. Computed examples illustrate the feasibility of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The conjugate gradient (CG) algorithm is the most frequently used iterative method for solving linear systems Ax = b with a symmetric positive definite (SPD) matrix. In this paper we construct real symmetric positive definite matrices A of order n and real right-hand sides b for which the CG algorithm has a prescribed residual norm convergence curve. We also consider prescribing as well the A-norms of the error. We completely characterize the tridiagonal matrices constructed by the Lanczos algorithm and their inverses in terms of the CG residual norms and A-norms of the error. This also gives expressions and lower bounds for the ℓ2 norm of the error. Finally, we study the problem of prescribing both the CG residual norms and the eigenvalues of A. We show that this is not always possible. Our constructions are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

The block changing minimal residual method based on the Hessenberg reduction algorithm (in short BCMRH) is a recent block Krylov method that can solve large linear systems with multiple right-hand sides. This method uses the block Hessenberg process with pivoting strategy to construct a trapezoidal Krylov basis and minimizes a quasi-residual norm by solving a least squares problem. In this paper, we describe the simpler BCMRH method which is a new variant that avoids the QR factorization to solve the least-squares problem. Another major difference between the classical and simpler variants of BCMRH is that the simpler one allows to check the convergence within each cycle of the block Hessenberg process by using a recursive relation that updates the residual at each iteration. This is not possible with the classical BCMRH where we can only compute an estimate of the residual norm. Experiments are described to compare the behavior of the new proposed method with that of the classical and simpler versions of the block GMRES method. These numerical experiments show the good performances of the simpler BCMRH method. [ABSTRACT FROM AUTHOR]

This paper is devoted to properties of sequence transformations and the corresponding recursive algorithms for their implementation, which were never considered before. We first give necessary conditions that are satisfied if the transformed sequence converges faster than the initial one. These conditions can be used for deciding if a method is worth to be used. They also serve as the basis for defining criteria for stopping the acceleration algorithm when the best possible precision is obtained. Then, prescribing the transformed sequence, we show how to obtain the initial sequence which produces it via the transformation or via its recursive algorithm. These results show that almost any behavior is possible for the transformed sequence. A similar problem about Padé-type approximants is studied. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the numerical solutions of the coupled fractional Klein-Gordon-Schrödinger equation. The numerical schemes are constructed by combining the Crank-Nicolson/leap-frog difference methods for the temporal discretization and the Galerkin finite element methods for the spatial discretization. We give a detailed analysis of the conservation properties in the senses of discrete mass and energy. Then the numerical solutions are shown to be unconditionally bounded in L2 −norm, H α 2 − semi-norm and L ∞ − norm, respectively. Based on the well-known Brouwer fixed-point theorem and the mathematical induction, the unique solvability of the discrete solutions is proved. Moreover, the schemes are proved to be unconditionally convergent with the optimal order O τ 2 + h r + 1 , where τ is the temporal step, h is the spatial grid size, and r is the order of the selected finite element space. Furthermore, by using the proposed decoupling and iterative algorithms, several numerical examples are included to support theoretical results and show the effectiveness of the schemes. Finally, the fast Krylov subspace solver with suitable circulant preconditioner is designed to effectively solve the Toeplitz-like linear systems. In each iterative step, this method can effectively reduce the memory requirement of above each finite element scheme from O M 2 to O(M), and the computational complexity from O M 3 to O (M log M) , where M is the number of grid nodes. Numerical tests are carried out to show that this fast algorithm is more practical than the traditional backslash and LU factorization/Cholesky decomposition methods, in terms of memory requirement and computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new iterative algorithm for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a step size; an advantage of the algorithm is the computation of only one value of the mapping and one projection onto the admissible set per one iteration. The convergence of the algorithm is established without the knowledge of the Lipschitz constant of the mapping. Meanwhile, R-linear convergence rate is obtained under strong monotonicity assumption of the mapping. Also, we generalize the method with Bregman projection. Finally, some numerical experiments are presented to show the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new extragradient-like algorithm for solving a variational inequality problem with a pseudo-monotone and Lipschitz continuous mapping in a Hilbert space. The iterative algorithm combines inertial ideas and hybrid extragradient ideas with the Armijo-like step size rule. Strong convergence of the algorithm is obtained and numerical experiments are provided. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new arc-search predictor-corrector infeasible-interior-point algorithm for linear complementarity problems over symmetric cones with the Cartesian P∗(κ)-property (P∗(κ)-SCLCP). The proposed algorithm is based on a wide neighborhood of the central path and searches the optimizers along the ellipses that approximate the entire the central path. The algorithm uses a commutative class of search directions, which includes the Nesterov-Todd direction and the xs and sx directions. To the best of our knowledge, this is using a new strategy in the complexity analysis; we improve the theoretical complexity bound of an arc-search infeasible-interior-point method for P∗(κ)-SCLCP. Some preliminary numerical results are provided to demonstrate the computational performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we define a backward error for the linear complementarity problem (LCP), and then present an expression of it which can be employed to examine the stability of algorithms solving the LCP. Some numerical examples are given to show the efficiency of the proposed theoretical results. [ABSTRACT FROM AUTHOR]