Our purpose is to compute the multi-partially symmetric rank-one approximations of higher-order multi-partially symmetric tensors. A special case is the partially symmetric rank-one approximation for the fourth-order partially symmetric tensors, which is related to the biquadratic optimization problem. For the special case, we implement the neural network model by the ordinary differential equations (ODEs), which is a class of continuous-time recurrent neural network. Several properties of states for the network are established. We prove that the solution of the ODE is locally asymptotically stable by establishing an appropriate Lyapunov function under mild conditions. Similarly, we consider how to compute the multi-partially symmetric rank-one approximations of multi-partially symmetric tensors via neural networks. Finally, we define the restricted $M$-singular values and the corresponding restricted $M$-singular vectors of higher-order multi-partially symmetric tensors and design to compute them. Numerical results show that the neural network models are efficient.

In this paper, we propose a framework for studying optimal agency execution strategies in a Limit Order Book (LOB) under a Markov-modulated market environment. The Almgren-Chriss's market impact model [1] is extended to a more general situation where multiple venues are available for investors to submit trades. Under the assumption of risk-neutrality, a compact recursive formula is derived, using the value iterative method, to calculate the optimal agency execution strategy. The original optimal control problem is then converted to a constrained quadratic optimization problem, which can be solved by using the Quadratic Programming (QP) approach. Numerical examples are given to illustrate the efficiency and effective of our proposed methods.

In this paper, we propose and analyze a spectral deferred correction method for the fractional differential equation of order α. The proposed method is based on a well-known finite difference method of $(2−α)$-order, see [Sun and Wu, Appl. Numer. Math., 56(2), 2006] and [Lin and Xu, J. Comput. Phys., 225(2), 2007], for prediction of the numerical solution, which is then corrected through a spectral deferred correction method. In order to derive the convergence rate of the prediction-correction iteration, we first derive an error estimate for the $(2−α)$-order finite difference method on some non-uniform meshes. Then the convergence rate of orders $\mathcal{O}(τ^{(2−α)(p+1)})$ and  $\mathcal{O}(τ^{(2−α)+p})$ of the overall scheme is demonstrated numerically for the uniform mesh and the Gauss-Lobatto mesh respectively, where $τ$ is the maximal time step size and $p$ is the number of correction steps. The performed numerical test confirms the efficiency of the proposed method.

The two-dimensional, unsteady, viscous, incompressible and asymmetric flow of a micropolar fluid in a uniformly porous channel with expanding or contracting walls is investigated. We assume that the channel walls have different permeability and that the flow is driven by uniformly suction at the lower and upper walls. The corresponding governing equations of motion are reduced to nonlinear coupled ordinary difference equations by a similarity transformation. For the most difficult high Reynolds number case, we construct an asymptotic solution by the method of boundary layer correction. There exist boundary layers near the both walls. Furthermore, this asymptotic results can also be used to validate the numerical method over a range of Reynolds numbers. Finally, the influence of Reynolds number $R$, asymmetric parameter $a$, expansion ratio $α$ and micropolar parameter $n$ on the flow is discussed numerically. The results show that the streamwise and micro-rotation velocity are sensitive to the parameters.

Two iterative algorithms are proposed for the split fixed point problem. The first algorithm is shown to be weakly convergent and the second one to be strongly convergent. One feature of these algorithms is that the step sizes are chosen in such a way that no priori knowledge of the operator norms is required. A new idea is introduced in order to prove strong convergence of the second algorithm.

This paper reports a compact fourth order accurate cubic spline collocation procedure for numerical solution of transport equation with discontinuous coefficients. The proposed scheme is shown to be stable. A new patch up technique is developed near the interface where the coefficient has jump discontinuity. A numerical experiment is given to confirm the accuracy and efficiency of the proposed scheme.

In this paper, we present a numerical study of the effect of slip in the fluid-particle interaction. The motion of the particle is described by the Newton's second law and the flows are simulated by solving the incompressible Navier-Stokes equations with the Navier slip boundary condition. Numerical schemes are designed using the extended finite element method (XFEM) combined with the temporary arbitrary Lagrangian-Eulerian (tALE) technique. In this method, both the fluid dynamics and the motion of particle are efficiently computed on a fixed Cartesian mesh. With the XFEM, the discontinuities at the particle boundary are naturally captured by the Heaviside-enriched finite element basis functions. With the tALE technique, field variables at the previous time level are mapped onto the computational mesh at the current time level, hence regeneration or deformation of meshes can be avoided. To study the effect of the slip, we simulate the rotation of an ellipsoidal particle in a simple shear flow and compare with the analytic results from the theory of Jeffery orbit.

In this paper, we first propose a $Z_p$-eigenvalue of a tensor, which includes the $Z_1$- and $Z_2$-eigenvalue as its special case, and then present a $Z_p$-eigenvalue bound. In particular, we give a $Z$-spectral radius bound for an irreducible nonnegative tensor via the spectral radius of a nonnegative matrix. The proposed bounds improve some existing ones. Some numerical examples are given to show the validity of the proposed bounds.

In this paper, we study linear systems arising from time-space fractional Caputo-Riesz diffusion equations with time-dependent diffusion coefficients. The coefficient matrix is a summation of a block-lower-triangular-Toeplitz matrix (temporal component) and a block-diagonal-with-diagonal-times-Toeplitz-block matrix (spatial component). The main aim of this paper is to propose separable preconditioners for solving these linear systems, where a block ϵ-circulant preconditioner is used for the temporal component, while a block diagonal approximation is used for the spatial variable. The resulting preconditioner can be block-diagonalized in the temporal domain. Furthermore, the fast solvers can be employed to solve smaller linear systems in the spatial domain. Theoretically, we show that if the diffusion coefficient (temporal-dependent or spatial-dependent only) function is smooth enough, the singular values of the preconditioned matrix are bounded independent of discretization parameters. Numerical examples are tested to show the performance of proposed preconditioner.

We develop in this paper a space-time Petrov-Galerkin spectral method for linear and nonlinear time fractional diffusion equations (TFDEs) involving either a Caputo or Riemann-Liouville derivative. Our space-time spectral method is based on generalized Jacobi functions (GJFs) in time and Fourier-like basis functions in space. A complete error analysis is carried out for both linear and nonlinear TFDEs. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

In a series of papers, Chen [4–6] developed some efficient algorithms for computing the maximal eigenpairs for tridiagonal matrices. The key idea is to explicitly construct effective initials for the maximal eigenpairs and also to employ a self-closed iterative algorithm. In this paper, we extend Chen's algorithm to deal with large scale tridiagonal matrices with super-/sub-diagonal elements. By using appropriate scalings and by optimizing numerical complexity, we make the computational cost for each iteration to be $\mathcal{O}$($N$). Moreover, to obtain accurate approximations for the maximal eigenpairs, the total number of iterations is found to be independent of the matrix size, i.e., $\mathcal{O}$($1$) number of iterations. Consequently, the total cost for computing the maximal eigenpairs is $\mathcal{O}$($N$). The effectiveness of the proposed algorithm is demonstrated by numerical experiments.