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Studies on human mobility from abundant trajectory data have become more and more popular with the development of location-based services. Prediction for locations people may visit in the future is a significant task, helping to make visiting recommendations and manage traffic conditions. Different from other time series prediction tasks, location prediction is temporally dependent as well as spatial-aware. In this paper, we propose a novel multi-graph attention network with sequence-to-sequence structures based on spatial–temporal long short-term memory to predict future locations. Specifically, we build three graphs with movements in geographic space and apply graph attention networks to explore the latent spatial associations among geographic regions. Additionally, we come up with spatial–temporal long short-term memory and use it to establish a sequence-to-sequence framework, which collects the temporal dependence as well as some spatial information from history trajectories. The predictions of future location are finally made by aggregating spatial–temporal contexts. [ABSTRACT FROM AUTHOR]

In this paper, we establish a modulus-based nonsmooth Newton's method for solving a class of horizontal nonlinear complementarity problems and prove the nearly quadratic convergence of the method under certain assumptions. Based on the analysis of convergence, we give the selection strategy for the approximate iteration vector and propose the improved modulus-based nonsmooth Newton's method. Finally, numerical experiments show that the new methods are superior to the modulus-based matrix iterative method regarding computational efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical solution of a large complex linear system with a saddle-point form obtained by the discretization of the time-harmonic eddy-current optimal control problem. A new Schur complement is proposed for this algebraic system, extending it to both the block-triangular preconditioner and the structured preconditioner. A theoretical analysis proves that the eigenvalues of block-triangular and structured preconditioned matrices are located in the interval [1/2, 1]. Numerical simulations show that two new preconditioners coupled with a Krylov subspace acceleration have good feasibility and effectiveness and are superior to some existing efficient algorithms. [ABSTRACT FROM AUTHOR]

For the discretized linear systems of the spatial fractional diffusion equations, we construct a class of a modified DTS iteration method and give its asymptotic convergence conditions. Then, we design a fast modified DTS preconditioner by replacing Toeplitz matrix T with the τ matrix to accelerate the convergence rates of GMRES method. Theoretically, we show that the spectrum of fast modified DTS preconditioned matrix is clustered around one. Numerical experiments verify the validity of the constructed fast modified DTS preconditioner for GMRES method. [ABSTRACT FROM AUTHOR]

To improve the MRPSS preconditioner developed by Cao et al. (BIT Numer Math 56:423–439, 2016), a generalized variant of the MRPSS (GMRPSS) preconditioner is proposed. The invertibility of the GMRPSS preconditioner, the convergence of the GMRPSS iteration method and some properties of the corresponding preconditioned matrix are presented. The degree of minimal polynomials of the preconditioned matrix is obtained. Simultaneously, the algorithm implementation and the selection of the parameter of the GMRPSS preconditioner are discussed. Numerical experiments are performed to illustrate the feasibility of the new preconditioner. [ABSTRACT FROM AUTHOR]

For the horizontal linear complementarity problem, we establish a linear method based on the sign patterns of the solution of the equivalent modulus equation under the assumption of strict complementarity. The new method is equivalent to solving two linear equations, avoiding parameters selection, so it is more convenient and practical. Numerical examples show that the new method is superior to the previous modulus-based matrix splitting iterative method, two-step modulus-based matrix splitting iterative method and modulus-based nonsmooth Newton's method in computing efficiency. The new method can be used as a solution scheme to the horizontal complementarity problem of large sparse matrices. [ABSTRACT FROM AUTHOR]

The random walk model describing the super-diffusion competition phenomenon of particles can derive the spatial fractional diffusion equation. For irregular diffusion and super-diffusion phenomena, the use of such equation can obtain more accurate and realistic results, so it has a wide application background in practice. The implicit finite-difference method derived from the shifted Grünwald scheme is used to discretize the spatial fractional diffusion equation. The coefficient matrix of discrete system is in the form of the sum of a diagonal matrix and a Toeplitz matrix. In this paper, a preconditioner is constructed, which transforms the coefficient matrix into the form of an identity matrix plus a diagonal matrix multiplied by Toeplitz matrix. On this basis, a new quasi-Toeplitz trigonometric transform splitting iteration format (abbreviated as QTTTS method) is proposed. We theoretically verify the unconditional convergence of the new method, and obtain the effective optimal form of the iteration parameter. Finally, numerical simulation experiments also demonstrate the accurateness and efficiency of the new method. [ABSTRACT FROM AUTHOR]

The fractional diffusion equations can be discretized by applying the implicit finite difference scheme and the unconditionally stable shifted Grünwald formula. Hence, the generating linear system has a real Toeplitz structure when the two diffusion coefficients are non-negative constants. Through a similarity transformation, the Toeplitz linear system can be converted to a generalized saddle point problem. We use the generalization of a parameterized inexact Uzawa (GPIU) method to solve such a kind of saddle point problem and give a new algorithm based on the GPIU method. Numerical results show the effectiveness and accuracy for the new algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a new Uzawa-SOR method for fourth-order block saddle point problems. Then we study its convergence conditions through some lemmas and theorems. Finally, the numerical results are presented to prove the 4-USOR method we proposed has less workload per iteration and higher accuracy for large linear system than the other Uzawa-type methods for solving the fourth-order block saddle point problem, which shows the effectiveness and feasibility of our method. [ABSTRACT FROM AUTHOR]

The successive overrelaxation-like (SOR-like) method with the real parameters ω is considered for solving the augmented system. The new method is called the modified SOR-like (MSOR-like) method. The functional equation between the parameters and the eigenvalues of the iteration matrix of the MSOR-like method is given. Therefore, the necessary and sufficient condition for the convergence of the MSOR-like method is derived. The optimal iteration parameter ω of the MSOR-like method is derived. Finally, the proof of theorem and numerical computation based on a particular linear system are given, which clearly show that the MSOR-like method outperforms the SOR-like (Li, C. J., Li, B. J., and Evans, D. J. Optimum accelerated parameter for the GSOR method. Neural, Parallel & Scientific Computations, 7(4), 453-462 (1999)) and the modified symmetric SOR-like (MSSOR-like) methods (Wu, S. L., Huang, T. Z., and Zhao, X. L. A modified SSOR iterative method for augmented systems. Journal of Computational and Applied Mathematics, 228(4), 424-433 (2009)). [ABSTRACT FROM AUTHOR]

A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation (GAOR) methods, is proposed to solve linear systems based on a class of weighted linear least squares problems. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned iterative methods is better than that of the original methods. Furthermore, the effectiveness of the proposed methods is shown in the numerical experiment. [ABSTRACT FROM AUTHOR]

In many fields of science and engineering, partial differential equation (PDE) constrained optimal control problems are widely used. We mainly solve the optimization problem constrained by the time-periodic eddy current equation in this paper. We propose the three-block splitting (TBS) iterative method and proved that it is unconditionally convergent. At the same time, the corresponding TBS preconditioner is derived from the TBS iteration method, and we studied the spectral properties of the preconditioned matrix. Finally, numerical examples in two-dimensions is applied to demonstrate the advantages of the TBS iterative method and TBS preconditioner with the Krylov subspace method. [ABSTRACT FROM AUTHOR]

For the large sparse generalized saddle point problems with non-Hermitian (1,1) blocks, we introduce a constraint preconditioner, which is based on the positive definite and semidefinite splitting (PPS) iteration method. Then we discuss one trait of the PPS-based constraint preconditioner, such as invertibility. We give the convergence of conditions of the preconditioning iteration method. Moreover, numerical experiments are given to illustrate that PPS-based constraint preconditioner has an obvious advantage of efficiency. [ABSTRACT FROM AUTHOR]

In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix A and B satisfy certain conditions, the FPPSS iterative method is convergent in the parameter's value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime. [ABSTRACT FROM AUTHOR]

Based on previous research results, we propose a new preprocessing HSS iteration method (PHSS) for the generalized Lyapunov equation. At the same time, the corresponding inexact PHSS algorithm (IPHSS) is given from the angle of application. All the new methods presented in this paper have given the corresponding convergence proof. The numerical experiments are carried out to compare the new method with the existing methods, and the improvement effect is obvious. The feasibility and effectiveness of the proposed method are proved from two aspects of theory and calculation. [ABSTRACT FROM AUTHOR]