High Order IMEX Stochastic Galerkin Schemes for Linear Transport Equation with Random Inputs and Diffusive Scalings

In this paper,we consider the high order method for solving the linear transport equations under diffusive scaling and with random inputs.To tackle the randomness in the problem,the stochastic Galerkin method of the generalized polynomial chaos approach has been employed.Besides,the high order implicit-explicit scheme under the micro-macro decomposition framework and the discontinuous Galerkin method have been employed.We provide several numerical experiments to validate the accuracy and the stochastic asymptotic-preserving property.

The Jaffa Transform for Hessian Matrix Systems and the Laplace Equation

Hessian matrices are square matrices consisting of all possible combinations of second partial derivatives of a scalar-valued initial function. As such, Hessian matrices may be treated as elementary matrix systems of linear second-order partial differential equations. This paper discusses the Hessian and its applications in optimization, and then proceeds to introduce and derive the notion of the Jaffa Transform, a new linear operator that directly maps a Hessian square matrix space to the initial corresponding scalar field in nth dimensional Euclidean space. The Jaffa Transform is examined, including the properties of the operator, the transform of notable matrices, and the existence of an inverse Jaffa Transform, which is, by definition, the Hessian matrix operator. The Laplace equation is then noted and investigated, particularly, the relation of the Laplace equation to Poisson’s equation, and the theoretical applications and correlations of harmonic functions to Hessian matrices. The paper concludes by introducing and explicating the Jaffa Theorem, a principle that declares the existence of harmonic Jaffa Transforms, which are, essentially, Jaffa Transform solutions to the Laplace partial differential equation.

Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.

SOLVERS FOR SYSTEMS OF LARGE SPARSE LINEAR AND NONLINEAR EQUATIONS BASED ON MULTI-GPUS

Numerical treatment of engineering application problems often eventually results in a solution of systems of linear or nonlinear equations.The solution process using digital computational devices usually takes tremendous time due to the extremely large size encountered in most real-world engineering applications.So,practical solvers for systems of linear and nonlinear equations based on multi graphic process units（GPUs）are proposed in order to accelerate the solving process.In the linear and nonlinear solvers,the preconditioned bi-conjugate gradient stable（PBi-CGstab）method and the Inexact Newton method are used to achieve the fast and stable convergence behavior.Multi-GPUs are utilized to obtain more data storage that large size problems need.

An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

THE ANALYTIC SMOOTHING EFFECT OF LINEAR LANDAU EQUATION WITH SOFT POTENTIALS

In this work,we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in L^(2)(ℝ^(3))enjoys an analytic regularization effect,and that the evolution of the analytic radius is the same as the heat equations.

Geometric interpretation of several classical iterative methods for linear system of equations and diverse relaxation parameter of the SOR method

Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterative methods and provide a reference for further study and design. Finally, a new iterative method is designed named as the diverse relaxation parameter of the SOR method which, in particular, demonstrates the geometric characteristics. Many examples prove that the method is quite effective.

ASYNCHRONOUS RELAXED ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS OF EQUATIONS

In this paper, the asynchronous versions of classical iterative methods for solving linear systems of equations are considered. Sufficient conditions for convergence of asynchronous relaxed processes are given for H-matrix by which nor only the requirements of [3] on coefficient matrix are lowered, but also a larger region of convergence than that in [3] is obtained.

Novel method based on ant colony opti mization for solving ill-conditioned linear systems of equations

A novel method based on ant colony optimization （ACO）, algorithm for solving the ill-conditioned linear systems of equations is proposed. ACO is a parallelized bionic optimization algorithm which is inspired from the behavior of real ants. ACO algorithm is first introduced, a kind of positive feedback mechanism is adopted in ACO. Then, the solu- tion problem of linear systems of equations was reformulated as an unconstrained optimization problem for solution by an ACID algorithm. Finally, the ACID with other traditional methods is applied to solve a kind of multi-dimensional Hilbert ill-conditioned linear equations. The numerical results demonstrate that ACO is effective, robust and recommendable in solving ill-conditioned linear systems of equations.

A Numerical Method for Solving Ill-Conditioned Equation Systems Arising from Radial Basis Functions

Continuously differentiable radial basis functions (C∞-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C∞-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C∞-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.

A Mathod to Solve Systems of Fuzzy Linear Equations

Many systems of fuzzy linear equations do not have solutions when the solution concept is based on α cuts and interval arithmetic. In this paper,we establish the relations between the systems of fuzzy linear equations and the possibilistic linear programming problems and present an alternative method of solving the systems of fuzzy linear equations.

SOLUTIONS FOR A SYSTEM OF NONLINEAR RANDOM INTEGRAL AND DIFFERENTIAL EQUATIONS

In this paper we first prove a Darbao type fixed point theorem for a system of continuous random operators with random domains. Thenb, by using the theorem. wegive the existence criteria of solutions for a systems of nonlinear random Volterraintegral equations and for the Cauchy problem of a system of nonlinear random differential equations. The existence of extremal random solutions and random comparison results for these systems of random equations are also obtained Our theorems improve and generalize the corresponding results of Vaughn Lakshmikantham Lakshmidantham-Leela De blasi-Myjak and Ding

Quantum algorithms for matrix operations and linear systems of equations

Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations.Using the‘sender-receiver’model,we propose quantum algorithms for matrix operations such as matrix-vector product,matrix-matrix product,the sum of two matrices,and the calculation of determinant and inverse matrix.We encode the matrix entries into the probability amplitudes of the pure initial states of senders.After applying proper unitary transformation to the complete quantum system,the desired result can be found in certain blocks of the receiver’s density matrix.These quantum protocols can be used as subroutines in other quantum schemes.Furthermore,we present an alternative quantum algorithm for solving linear systems of equations.

ZEROS OF ENTIRE SOLUTIONS TO COMPLEX LINEAR DIFFERENCE EQUATIONS

In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the order of growth of entire solutions to complex linear difference equations.

Robust H_∞ Control for Uncertain Markovian Jump Linear Time-Delay Systems

This paper studies the robust stochastic stabilization and robust H∞ control for linear time-delay systems with both Markovian jump parameters and unknown norm-bounded parameter uncertainties. This problem can be solved on the basis of stochastic Lyapunov approach and linear matrix inequality (LMI) technique. Sufficient conditions for the existence of stochastic stabilization and robust H∞ state feedback controller are presented in terms of a set of solutions of coupled LMIs. Finally, a numerical example is included to demonstrate the practicability of the proposed methods.

ESTIMATES FOR THE HYPER-ORDER OF SOLUTIONS OF CERTAIN LINEAR DIFFERENTIAL EQUATIONS

In this article, the authors study the growth of certain second order linear differential equation f″＋A（z）f′＋B（z）f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar conditions, higher order differential equations will be considered.

LINEAR SINGULAR INTEGRAL EQUATION ON DOMAINS COMPOSED BY BALLS

For domains composed by balls in C^n, this paper studies the boundary behaviour of Cauchy type integrals with discrete holomorphic kernels and the corresponding linear singular integral equation on each piece of smooth lower dimensional edges on the boundary of the domain.

On the Growth of Solutions of a Class of Higher Order Linear Differential Equations with Meromorphic Coefficients

In this paper, we investigate the growth of solutions of higher order linear differential equations with meromorphic coefficients. Under certain conditions, we obtain precise estimation of growth order and hyper-order of solutions of the equation.

Cryptanalysis of Cryptosystems Based on General Linear Group

Advances in quantum computers threaten to break public key cryptosystems such as RSA, ECC, and EIGamal on the hardness of factoring or taking a discrete logarithm, while no quantum algorithms are found to solve certain mathematical problems on non-commutative algebraic structures until now. In this background, Majid Khan et al.proposed two novel public-key encryption schemes based on large abelian subgroup of general linear group over a residue ring. In this paper we show that the two schemes are not secure. We present that they are vulnerable to a structural attack and that, it only requires polynomial time complexity to retrieve the message from associated public keys respectively. Then we conduct a detailed analysis on attack methods and show corresponding algorithmic description and efficiency analysis respectively. After that, we propose an improvement assisted to enhance Majid Khan's scheme. In addition, we discuss possible lines of future work.

HIGH PERFORMANCE SPARSE SOLVER FOR UNSYMMETRICAL LINEAR EQUATIONS WITH OUT-OF-CORE STRATEGIES AND ITS APPLICATION ON MESHLESS METHODS

A new direct method for solving unsymmetrical sparse linear systems（USLS） arising from meshless methods was introduced. Computation of certain meshless methods such as meshless local Petrov-Galerkin （MLPG） method need to solve large USLS. The proposed solution method for unsymmetrical case performs factorization processes symmetrically on the upper and lower triangular portion of matrix, which differs from previous work based on general unsymmetrical process, and attains higher performance. It is shown that the solution algorithm for USLS can be simply derived from the existing approaches for the symmetrical case. The new matrix factorization algorithm in our method can be implemented easily by modifying a standard JKI symmetrical matrix factorization code. Multi-blocked out-of-core strategies were also developed to expand the solution scale. The approach convincingly increases the speed of the solution process, which is demonstrated with the numerical tests.