In this paper we introduce a conforming discontinuous Galerkin (CDG) finite element method for solving the heat equation. Unlike the rest of discontinuous Galerkin (DG) methods, the numerical-flux is not introduced to the computation in the CDG method. Additionally, the numerical-trace is not the average (or some other simple average used in other DG methods), but a lifted polynomial from the solution on nearby four triangles in 2D, or eight tetrahedra in 3D. We show a two-order superconvergence in space approximation when using the CDG method with a backward Euler time discretization, on triangular and tetrahedral meshes, for solving the heat equation. Numerical tests are reported which confirm the theory.
To solve the large sparse complex symmetric linear equations more efficiently, we introduce a new matrix and establish two quasi-combining real and imaginary parts iteration methods, which will be simply called the QCRI1 and QCRI2 iteration methods. We give the upper bounds of the spectral radiuses of the two methods and discuss their convergence conditions that make these upper bounds less than 1. In addition, the theoretical quasi-optimal parameters minimizing the upper bound of the spectral radius of the iteration matrix of the QCRI1 method are presented. Meanwhile, the inexact versions of the proposed methods are also provided, and their convergence properties are given. Finally, numerical results illustrate the effectiveness of our methods.
In this paper, we investigate the numerical method for the two-dimensional time-fractional Zakharov-Kuznetsov (ZK) equation. By the method of order reduction, the model is first transformed into an equivalent system. A nonlinear difference scheme is then proposed to solve the equivalent model with -th order accuracy in time and second-order accuracy in space, where is the fractional order and the grading parameter . The existence of the numerical solution is carefully studied by the renowned Browder fixed point theorem. With the help of the Grönwall inequality and some crucial skills, we analyze the unconditional stability and convergence of the proposed scheme based on the energy method. Finally, numerical experiments are given to illustrate the correctness of our theoretical analysis.
The positive definite homogeneous multivariate form plays an important role in the automatic control and medical imaging, and the definiteness of this form can be identified by a special structure tensor. In this paper, we first state the equivalence between the positive definite multivariate form and the corresponding tensor and account for the links between the positive definite tensor with a strong -tensor. Then, based on diagonal dominance, some criteria are presented to test strong -tensors. Furthermore, with these relations, we provide an iterative scheme to identify the positive definite multivariate homogeneous form and prove its theoretically valid. The advantages of the results obtained are illustrated by numerical examples.
In this paper, based on the H2N2 method (a method for approximating the Caputo fractional derivative of order derived by the quadratic Hermite and Newton interpolation polynomials), a direct finite difference scheme with second-order accuracy in space and th order accuracy in time is constructed for the fractional sine-Gordon equation. The stability and convergence of the difference scheme are analyzed theoretically. Further, to improve the computational efficiency, a fast algorithm based on the sum-of-exponentials method is adopted to construct a fast difference scheme. In addition, the initial singularity of the solution is also discussed. Numerical outcomes verify the validity of the direct and fast difference schemes and the correctness of theoretical results.
In certain queueing circumstances, a bi-level vacation policy according to which the server can switch over from the working vacation (WV) to the complete vacation (CV) has been noticed. In the WV (CV) mode, customers may exhibit reneging behavior due to the slow (absence of) service. This study examines the queueing metrics of a Markovian queueing system incorporating the concepts of the bi-level hybrid vacation strategy, state-dependent rates, discouraged customers, and waiting servers. The continued fraction, the probability generating function (PGF), and the confluent hypergeometric function are used to derive the transient queue size distributions. As a special case of the transient results, we derive the steady-state results. The analytical results for the average system size, variance, average reneging rate, system throughput, etc. are established. Furthermore, the cost function framed is optimized to provide the optimal decision parameters and respective minimum cost. Numerical simulation and parameter sensitivity have been performed by taking illustrations.
This paper investigates two different Leslie matrix solutions for the reduced biquaternion matrix equation . Through the permutation matrices, the complex decomposition of reduced biquaternion matrices, and the Kronecker product, by leveraging the specific attributes of Leslie matrices, we transform the constrained reduced biquaternion matrix equation into an unconstrained form. Consequently, we derive the necessary and sufficient conditions for the existence of solutions in the form of Leslie matrices to the reduced biquaternion matrix equation and provide a general expression for such solutions. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.
In this paper, we first study carefully the positive solutions to defined on a complete non-compact Riemannian manifold (M, g) with , which can be regarded as Lichnerowicz-type equations, and according to the different parameter values in the equation, seven cases are discussed to obtain the gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on (M, g). For the case , this improves considerably the previous related results. Moreover, we also obtain the Liouville-type result for these equations when and establish the Harnack inequality as consequences.
A new subclass of H-matrices named -type matrices is introduced. The relationships between -type matrices and other subclasses of H-matrices are studied. Moreover, the infinite norm bounds for the inverse of -type matrices are provided. As applications, error bounds of the linear complementarity problems (LCPs) for -type matrices and strictly diagonally dominant () matrices strictly diagonally dominant (are also presented, which improve some existing bounds. Numerical examples are presented to demonstrate the effectiveness of the obtained results.
We analyze the numerical solution of a non-linear evolutionary variational inequality, which is encountered in the investigation of quasi-static contact problems. The study of this article encompasses both the semi-discrete and fully discrete schemes, where we employ the backward Euler method for time discretization and utilize the lowest order Crouzeix-Raviart non-conforming finite-element method for spatial discretization. By assuming appropriate regularity conditions on the solution, we establish a priori error analysis for these schemes, achieving the optimal convergence order for linear elements. To illustrate the numerical convergence rates, we provide numerical results on a two-dimensional test problem.
This work deals with developing two fast randomized algorithms for computing the generalized tensor singular value decomposition (GTSVD) based on the tensor product (T-product). The random projection method is utilized to compute the important actions of the underlying data tensors and use them to get small sketches of the original data tensors, which are easier to handle. Due to the small size of the tensor sketches, deterministic approaches are applied to them to compute their GTSVD. Then, from the GTSVD of the small tensor sketches, the GTSVD of the original large-scale data tensors is recovered. Some experiments are conducted to show the effectiveness of the proposed approach.
Dual quaternions are essential for the precise formation flying of satellite clusters and for the Relative Navigation and Positioning (RNP). In this paper, we investigate dual quaternion matrices within the contexts of the precise formation and the RNP. We begin by reformulating the graph model of the formation flying problem using dual quaternion unit gain graphs. Following this, we study the dual quaternion incidence matrix to characterize the balance of these unit gain graphs. We also show that the Perron-Frobenius theorem holds for balanced dual quaternion unit gain graphs. As an application, we study a pose graph optimal problem in the RNP.
In this paper, the discrete-time linear quadratic (LQ) optimal control problem for a stochastic system with random coefficients is studied. Unlike the classical LQ optimal control problem, there exists a great difficulty in the LQ optimal control problem when the coefficient matrices of the stochastic system and weighting matrices in the cost functional are not assumed to be deterministic. Therefore, two innovative points are mentioned. First, we mainly consider structural changes and innovations in discrete-time LQ optimal control problems once the coefficients are randomized. Second, the stochastic system of this article includes nonhomogeneous terms. Interestingly, we show that the maximum principle leads to a Riccati equation. Specifically speaking, the fully coupled forward-backward stochastic difference equations (FBSDEs) are used to characterize the optimal control. Through decoupling the FBSDEs, we derive the expression corresponding to the Riccati equation with nonhomogeneous terms and get a state feedback representation of the optimal control. Finally, we construct the expression of the value function.
We extend the fourth-order, two-stage multiderivative Runge-Kutta (MDRK) scheme to the flux reconstruction (FR) framework by writing both stages in terms of a time-averaged flux and then using the approximate Lax-Wendroff (LW) procedure to compute the time-averaged flux. Numerical flux is carefully constructed to enhance Fourier CFL stability and accuracy. A subcell-based blending limiter is developed for the MDRK scheme which ensures that the limited scheme is provably admissibility preserving. Along with being admissibility preserving, the blending scheme is constructed to minimize dissipation errors using Gauss-Legendre (GL) solution points and performing the MUSCL-Hancock (MH) reconstruction on subcells. The accuracy enhancement of the blending scheme is numerically verified on compressible Euler equations, with test cases involving shocks and small-scale structures.
In this study, we introduce two finite-dimensional Lie-Poisson Hamiltonian systems related to the Manakov equation through the nonlinearization method. Additionally, we apply the separation of variables on the common level set of Casimir functions to analyze these systems, which are related to the non-hyperelliptic algebraic curve. Ultimately, we construct the action-angle variables for these systems based on the Hamilton-Jacobi theory and derive the Jacobi inversion problem for the Manakov equation.
A hybrid fifth-order finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme is applied for the solution of a multi-class traffic flow model (mcLWR model) with heterogeneous factors on the highroad. Since the hybrid HWENO scheme could avoid oscillations near discontinuities while maintaining efficiency and compactness, it can be a reliable numerical method and tool for the simulation and prediction of rapidly changing the traffic flow. The benchmark examples of Riemann problems and traffic signal control problems are given, and results indicate that the hybrid HWENO numerical scheme can obtain the fifth-order precision, and also has higher resolution and accuracy than the hybrid WENO scheme.
In this paper, we discuss a framework for efficient construction of rational approximations for the three-parameter Mittag-Leffler function (MLF) based on the global-Padé approximation technique. Obtaining the coefficients of these approximants is usually the most tedious part in their construction as it requires delicate matching to procure a linear system for these coefficients. The main focus of this paper is the derivation of a novel generalized system that can be used to obtain the coefficients without having to perform the matching task. In particular, we illustrate the use of the generalized system in constructing rational approximants of various degrees subject to choices of feasible parameters. Inequalities providing bounds on permissible choices of these parameters are also obtained. Numerical experiments are conducted to illustrate the accuracy and efficiency of the approximants.
A new Galerkin finite element for the biharmonic equation is constructed on 2D rectangular and 3D cuboid meshes. In this - () interpolated Galerkin finite element construction, all unknowns associated with the interior of each element are determined by the direct interpolation of the right-hand-side function, and the rest of the unknowns, associated with the boundary of each element, are determined by solving the remaining linear equations of the Galerkin projection. In comparison with the traditional finite element method in two dimensions, our method reduces the number of unknowns from to O(k). Additionally, the method reduces the condition number drastically as it requires only 1% of the computer time, compared with the standard finite elements, in several numerical tests. We prove the existence and uniqueness of the solution and the optimal order of convergence. We confirm the theory by numerical tests in two dimensions and three dimensions.
Deep learning requires solving a nonconvex optimization problem of a large size to learn a deep neural network (DNN). The current deep learning model is of a single-grade, that is, it trains a DNN end-to-end, by solving a single nonconvex optimization problem. When the layer number of the neural network is large, it is computationally challenging to carry out such a task efficiently. The complexity of the task comes from learning all weight matrices and bias vectors from one single nonconvex optimization problem of a large size. Inspired by the human education process which arranges learning in grades, we propose a multi-grade learning model: instead of solving one single optimization problem of a large size, we successively solve a number of optimization problems of small sizes, which are organized in grades, to learn a shallow neural network (a network having a few hidden layers) for each grade. Specifically, the current grade is to learn the leftover from the previous grade. In each of the grades, we learn a shallow neural network stacked on the top of the neural network, learned in the previous grades, whose parameters remain unchanged in training of the current and future grades. By dividing the task of learning a DDN into learning several shallow neural networks, one can alleviate the severity of the nonconvexity of the original optimization problem of a large size. When all grades of the learning are completed, the final neural network learned is a stair-shape neural network, which is the superposition of networks learned from all grades. Such a model enables us to learn a DDN much more effectively and efficiently. Moreover, multi-grade learning naturally leads to adaptive learning. We prove that in the context of function approximation if the neural network generated by a new grade is nontrivial, the optimal error of a new grade is strictly reduced from the optimal error of the previous grade. Furthermore, we provide numerical examples which confirm that the proposed multi-grade model outperforms significantly the standard single-grade model and is much more robust to noise than the single-grade model. They include three proof-of-concept examples, classification on two benchmark data sets MNIST and Fashion MNIST with two noise rates, which is to find classifiers, functions of 784 dimensions, and as well as numerical solutions of the one-dimensional Helmholtz equation.