A NUMERICAL EMBEDDING METHOD FOR SOLVING THE NONLINEAR COMPLEMENTARITY PROBLEM(I) ——THEORY

AbstractIn this paper, we extend the numerical embedding method for solving the smooth equations to the nonlinear complementarity problem. By using the nonsmooth theory, we prove the existence and the continuation of the following path for the corresponding homotopy equations. Therefore the basic theory of the numerical embedding method for solving the nonlinear complementarity problem is established. In part II of this paper, we will further study the implementation of the method and give some numerical exapmles.     

Delay Induced Hopf Bifurcation of Small-World Networks

In this paper,the stability and the Hopf bifurcation of small-world networks with time delay are studied.By analyzing the change of delay,we obtain several sufficient conditions on stable and unstable properties.When the delay passes a critical value,a Hopf bifurcation may appear.Furthermore,the direction and the stability of bifurcating periodic solutions are investigated by the normal form theory and the center manifold reduction.At last,by numerical simulations,we further illustrate the effectiveness of theorems in this paper.     

ENERGY ESTIMATES FOR DELAY DIFFUSION-REACTION EQUATIONS

In this paper we consider nonlinear delay diffusion-reaction equations with initial and Dirichlet boundary conditions. The behaviour and the stability of the solution of such initial boundary value problems （IBVPs） are studied using the energy method. Simple numerical methods are considered for the computation of numerical approximations to the solution of the nonlinear IBVPs. Using the discrete energy method we study the stability and convergence of the numerical approximations. Numerical experiments are carried out to illustrate our theoretical results.     

Positivity and boundedness preserving schemes for the fractional reaction-difusion equation

In this paper,we design a semi-implicit scheme for the scalar time fractional reaction-difusion equation.We theoretically prove that the numerical scheme is stable without the restriction on the ratio of the time and space stepsizes,and numerically show that the convergence orders are 1 in time and 2 in space.As a concrete model,the subdifusive predator-prey system is discussed in detail.First,we prove that the analytical solution to the system is positive and bounded.Then,we use the provided numerical scheme to solve the subdifusive predator-prey system,and theoretically prove and numerically verify that the numerical scheme preserves the positivity and boundedness.     

SINGLY DIAGONALLY IMPLICIT RUNGE-KUTTA METHODS COMBINING LINE SEARCH TECHNIQUES FOR UNCONSTRAINED OPTIMIZATION

There exists a strong connection between numerical methods for the integration of ordinary differential equations and optimization problems. In this paper, we try to discover further their links. And we transform unconstrained problems to the equivalent ordinary differential equations and construct the LRKOPT method to solve them by combining the second order singly diagonally implicit Runge-Kutta formulas and line search techniques.Moreover we analyze the global convergence and the local convergence of the LRKOPT method. Promising numerical results are also reported.     

DOUBLE INERTIAL PROXIMAL GRADIENT ALGORITHMS FOR CONVEX OPTIMIZATION PROBLEMS AND APPLICATIONS

In this paper, we propose double inertial forward-backward algorithms for solving unconstrained minimization problems and projected double inertial forward-backward algorithms for solving constrained minimization problems. We then prove convergence theorems under mild conditions. Finally, we provide numerical experiments on image restoration problem and image inpainting problem. The numerical results show that the proposed algorithms have more efficient than known algorithms introduced in the li...     

非线性互补问题的逐次逼近类Broyden算法

In this paper, we present a new form of successive approximation Broyden-like algorithm for nonlinear complementarity problem based on its equivalent nonsmooth equations. Under suitable conditions, we get the global convergence on the algorithms. Some numerical results are also reported.     

Dispersion Analysis of Multi-symplectic Scheme for the Nonlinear Schrodinger Equations

In this paper,we study the dispersive properties of multi-symplectic discretizations for the nonlinear Schrodinger equations.The numerical dispersion relation and group velocity are investigated.It is found that the numerical dispersion relation is relevant when resolving the nonlinear Schrodinger equations.     

HIGH ACCURACY NUMERICAL METHOD OF THIN-FILM PROBLEMS IN MICROMAGNETICS

In this paper, a new high accuracy numerical method for the thin-film problems of micron and submicron size ferromagnetic elements is proposed. For the computation of stray field, we use the finite element method(FEM) by introducing a semi-discrete artificial boundary condition [1, 2]. In our numerical experiments about the domain patterns and their movement, we can see that the results are accordant to that of experiments and other numerical methods. Our method are very convenient to deal with arbitrary shape of thin films such as a polygon with high accuracy.     

Classification of numerical 3-semigroups by means of L-shapes

We recall L-shapes, which are minimal distance diagrams, related to weighted 2-Cayley digraphs, and we give the number and the relation between minimal distance diagrams related to the same digraph. On the other hand, we consider some classes of numerical semigroups useful in the study of curve singularity. Then, we associate L-shapes to each numerical 3-semigroup and we describe some main invariants of numerical 3-semigroups in terms of their associated L-shapes. Finally, we give a characterization of the parameters of the L-shapes associated with a numerical 3-semigroup in terms of its generators, and we use it to classify the numerical 3-semigroups of interest in curve singularity.     

The oversemigroups of a numerical semigroup

We study the set of numerical semigroups containing a given numerical semigroup. As an application we prove characterizations of irreducible numerical semigroups that unify some of the existing characterizations for symmetric and pseudo-symmetric numerical semigroups. Finally we describe an algorithm for computing a minimal decomposition of a numerical semigroup in terms of irreducible numerical semigroups.     

The Oversemigroups of a Numerical Semigroup

Abstract. We study the set of numerical semigroups containing a given numerical semigroup. As an application we prove characterizations of irreducible numerical semigroups that unify some of the existing characterizations for symmetric and pseudo-symmetric numerical semigroups. Finally we describe an algorithm for computing a minimal decomposition of a numerical semigroup in terms of irreducible numerical semigroups.     

PROBABILISTIC NUMERICAL APPROACH FOR PDE AND ITS APPLICATION IN THE VALUATION OF EUROPEAN OPTIONS

1. IlltroductionThis paPer is aimed to give a probabilistic numerical aPproach for PDE. Probabilistic numerical method can get the solution one by one, whiCh differs from other nu-merical methods,such aJs the Anite e1ement and drite dffeence method, and rea1ize total parallel computing easily Another advallage of this method is that it suits for problems of highdimension becauseit is dimension indep endent.Consider the fOl1owing Cauchy problem of convectiondiffusion equations. FOr simplic…     

Asymptotic stability of the numerical solution for an integrodifferential reaction-diffusion system

This paper presents the study of the numerical solution of a reaction-diffusion system involving a reaction term of integral type arising from biological models. By means of a monotone approach we introduce upper and lower solutions and then we show the existence and the asymptotic behavior of nonnegative numerical solutions. To this end, we require the positivity of the numerical scheme and so we can use some properties of positive and M-matrices. Finally we give some sufficient conditions to verify the asymptotic stability of the numerical solution.     

Polynomial numerical hulls of matrix polynomials,II

In this article, we study some algebraic and geometrical properties of polynomial numerical hulls of matrix polynomials and joint polynomial numerical hulls of a finite family of matrices (possibly the coefficients of a matrix polynomial). Also, we study polynomial numerical hulls of basic A-factor block circulant matrices. These are block companion matrices of particular simple monic matrix polynomials. By studying the polynomial numerical hulls of the Kronecker product of two matrices, we characterize the polynomial numerical hulls of unitary basic A-factor block circulant matrices.     

Commutative Group Algebras of Summable p-Groups

We study numerical semigroups generated by generalized arithmetic sequences. We present a membership criterion for such a numerical semigroup, and by this we are able to answer fundamental questions concerning a numerical semigroup such as computing the Frobenius number and the type of the numerical semigroup, and decide whether the numercial semigroup is symmetric. Also for this kind of numerical semigroups, we compute the cardinality of a minimal presentation and determine whether they are complete intersections.     

直接法的数值稳定性

到目前为止,数值线代数方面最重要的进展是五十年代末Wilkinson提出的向后误差分析方法。但他给出的数值稳定性定义太严格,把不少实际上工作得很好的算法排斥在外。1975年Miller发现了这一问题。他举了Z(d)=d_1 d_2 d_1d_2这样很简单的问题说明Wilkinson的定义不够恰当,并给出了改进的数值稳定性定义。 设X是n维Euclid空间,Y是m维Euclid空间。I X,φ Y。一个数值计算问题P是三元组{I,φ,F},F是I到φ的一个映照,即对x∈J,存在唯一的y∈φ,使F(x)=y。问题P可有若干个算法求解。譬如用算法A来解。显然A是一个数值计算     

A sufficient condition for the stability of direct quadrature methods for Volterra integral equations

Within the theoretical framework of the numerical stability analysis for the Volterra integral equations, we consider a new class of test problems and we study the long-time behavior of the numerical solution obtained by direct quadrature methods as a function of the stepsize. Furthermore, we analyze how the numerical solution responds to certain perturbations in the kernel.     

Numerical solution for a sub-diffusion equation with a smooth kernel

In this paper we study the numerical solution of an initial value problem of a sub-diffusion type. For the time discretization we apply the discontinuous Galerkin method and we use continuous piecewise finite elements for the space discretization. Optimal order convergence rates of our numerical solution have been shown. We compare our theoretical error bounds with the results of numerical computations. We also present some numerical results showing the super-convergence rates of the proposed method.