This paper presents a new method for solving the basic problem in the "model-trust region" approach to large scale minimization: Compute a vector $x$ such that $1/2 x^THx +c^Tx $ = min, subject to the constraint $\| x \|_2 ≤a$. The method is a combination of the CG method and a projection and contraction (PC) method. The first (CG) method with $x_0 = 0$ as the starting point either directly offers a solution of the problem, or — as soon as the norm of the iteration is greater than $a$, — it gives a suitable starting point and a favourable choice of a crucial scaling parameter in the second (PC) method. Some numerical examples are given, which indicate that the method is applicable.

The Newton method for variational inequality problem is locally and quadratically convergent. By using a differentiable merit function, Taji, Fukushima and Ibaraki have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under some additional assumptions. In this paper we propose to present a trust region-type modification of Newton method for the strictly monotone variational inequality problem using the same merit function as that in [1]. It is then shown that our method is well defined and globally convergent and that, under the same assumptions as those in [1], our algorithm reduces to the basic Newton method and hence the rate of convergence is quadratic. Computational experience indicates the efficiency of the proposed method.  

In this paper, we show how to use wavelet to discretize the boundary integral equations which are both singular and ill-conditioned. By using an explicit diagonal preconditioning, the condition number of the corresponding matrix is bounded by a constant, while the sparse structure speeds up the iterative solving process. Using an iterative method, one thus obtains a fast numerical algorithm to solve the boundary integral equations.  

Problem of regularization of a singular system by derivative and proportional output feedback is studied. Necessary and sufficient conditions are obtained under which a singular system can be regularized into a closed-loop system that is regular and of index at most one. The reduced form is given that can easily explore the system properties as well as the feedback to be determined. The main results of the present paper are based on orthogonal transformations. Therefore, they can be implemented by numerically stable ways.  

With the help of symplecticity conditions of Partitioned Runge-Kutta methods, a simple way constructing symplectic methods is derived. Examples including several classes of high order symplectic Runge-Kutta methods are given, and showed up the relationship between existing high order Runge-Kutta methods.  

In this paper we study a cell entropy inequality for a class of the local relaxation approximation — The Relaxing Schemes for scalar conservation laws presented by Jin and Xin in [1], which implies convergence for the one-dimensional scalar case.  

This paper deals with the GPL-stability of the Implicit Runge-Kutta methods for the numerical solutions of the systems of delay differential equations. We focus on the stability behaviour of the Implicit Runge-Kutta (IRK) methods in the solutions of the following test systems with a delay term$$y'(t) = Ly(t) + My(t-\tau), t\ge 0,$$ $$y(t)=\Phi(t), t\le 0,$$where $L, M$ are $N \times N$ complex matrices, $\tau \gt 0$, $\Phi(t)$ is a given vector function. We shall show that the IRK methods are GPL-stable if and only if it is L-stable, when we use the IRK methods to the test systems above.

In this paper, the overlapping domain decomposition method, which is based on the natural boundary reduction and first suggested in [2], is applied to slove the exterior boundary value problem of harmonic equation over three-dimensional domain. The convergence and error estimates both for the continuous case and the discrete case are given. The contraction factor for the exterior spherical domain is also discussed. Moreover, numerical results are given which show that the accuracy and the convergence are in accord with the theoretical analyses.  

Some Jacobi approximations are investigated, which are used for numerical solutions of differential equations on the half line. The stability and convergence of the proposed schemes are proved. The main idea and techniques in this paper are also applicable to other problems on unbounded domains.