Convergence of SSOR methods for linear complementarity problems

In this paper, by applying the SSOR splitting, we propose two new iterative methods for solving the linear complementarity problem LCP (M,q). Convergence results for these two methods are presented when M is an H-matrix (and also an M-matrix). Finally, two numerical examples are given to show the efficiency of the presented methods.     

Isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure

The nonnegative inverse eigenvalue problem is that given a family of complex numbers λ={λ₁,…,λ_n}, find a nonnegative matrix of order n with spectrum λ. This problem is difficult and remains unsolved partially. In this paper, we focus on its generalization that the reconstructed nonnegative matrices should have some prescribed entries. It is easy to see that this new problem will come back to the common nonnegative inverse eigenvalue problem if there is no constraint of the locations of entries. A numerical isospectral flow method which is developed by hybridizing the optimization theory and steepest descent method is used to study the reconstruction. Moreover, an error estimate of the numerical iteration for ordinary differential equations on the matrix manifold is presented. After that, a numerical method for the nonnegative symmetric inverse eigenvalue problem with prescribed entries and its error estimate are considered. Finally, the approaches are verified by the numerical test results.     

Structure methods for solving the nearest correlation matrix problem

The nearest correlation matrix problem is to find a positive semidefinite matrix with unit diagonal, that is, nearest in the Frobenius norm to a given symmetric matrix A. This problem arises in the finance industry, where the correlations are between stocks. In this paper, we formulate this problem as a smooth unconstrained minimization problem, for which rapid convergence can be obtained. Other methods are also studied. Comparative numerical results are reported.     

Two-level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. Based on the P₁-P₁ triangular element and using the pressure projection stabilized finite element method, we solve a small Navier-Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H²), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.     

The construction of numerical integration rules of degree three for product regions

Numerical integration formulas in n-dimensional Euclidean space of degree three are discussed. In this paper, for the product regions a method is presented to construct numerical integration formulas of degree three with 2n real points and positive weights. The presented problem is a little different from those dealt with by other authors. All the corresponding one-dimensional integrals can be different from each other and they are also nonsymmetrical. In this paper an n-dimensional numerical integration problem is turned into n one-dimensional moment problems, which simplifies the construction process. Some explicit numerical formulas are given. Furthermore, a more generalized numerical integration problem is considered, which will shed light on the final solution to the third degree numerical integration problem.     

New numerical methods and some applied aspects of the p-regularity theory

The theory of p-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu.G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the p-factor method for solving ODEs with a small parameter.     

A solution of the affine quadratic inverse eigenvalue problem

The quadratic inverse eigenvalue problem (QIEP) is to find the three matrices M,C, and K, given a set of numbers, closed under complex conjugations, such that these numbers become the eigenvalues of the quadratic pencil P(λ)=λ²M+λC+K. The affine inverse quadratic eigenvalue problem (AQIEP) is the QIEP with an additional constraint that the coefficient matrices belong to an affine family, that is, these matrices are linear combinations of substructured matrices. An affine family of matrices very often arise in vibration engineering modeling and analysis. Research on QIEP and AQIEP are still at developing stage. In this paper, we propose three methods and the associated mathematical theories for solving AQIEP: A Newton method, an alternating projections method, and a hybrid method combining the two. Validity of these methods are illustrated with results on numerical experiments on a spring-mass problem and comparisons are made with these three methods amongst themselves and with another Newton method developed by Elhay and Ram (2002) [12]. The results of our experiments show that the hybrid method takes much smaller number of iterations and converges faster than any of these methods.     

Development of the parametric tolerance modeling and optimization schemes and cost-effective solutions

Most of previous research on tolerance optimization seeks the optimal tolerance allocation with process parameters such as fixed process mean and variance. This research, however, differs from the previous studies in two ways. First, an integrated optimization scheme is proposed to determine both the optimal settings of those process parameters and the optimal tolerance simultaneously which is called a parametric tolerance optimization problem in this paper. Second, most tolerance optimization models require rigorous optimization processes using numerical methods, since closed-form solutions are rarely found. This paper shows how the Lambert W function, which is often used in physics, can be applied efficiently to this parametric tolerance optimization problem. By using the Lambert W function, one can express the optimal solutions to the parametric tolerance optimization problem in a closed-form without resorting to numerical methods. For verification purposes, numerical examples for three cases are conducted and sensitivity analyses are performed.     

Modified regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. It is known that such problem is severely ill-posed. We propose a modified regularization method to solve it based on the solution given by the method of separation of variables. Convergence estimates are presented under two different a-priori bounded assumptions for the exact solution. Finally, numerical examples are given to show the effectiveness of the proposed numerical method.     

Zero-finder methods derived from Obreshkov’s techniques

In this paper two families of zero-finding iterative methods for solving nonlinear equations f(x)=0 are presented. The key idea to derive them is to solve an initial value problem applying Obreshkov-like techniques. More explicitly, Obreshkov’s methods have been used to numerically solve an initial value problem that involves the inverse of the function f that defines the equation. Carrying out this procedure, several methods with different orders of local convergence have been obtained. An analysis of the efficiency of these methods is given. Finally we introduce the concept of extrapolated computational order of convergence with the aim of numerically test the given methods. A procedure for the implementation of an iterative method with an adaptive multi-precision arithmetic is also presented.     

Sparse time-frequency representation of nonlinear and nonstationary data

Adaptive data analysis provides an important tool in extracting hidden physical information from multiscale data that arise from various applications. In this paper, we review two data-driven time-frequency analysis methods that we introduced recently to study trend and instantaneous frequency of nonlinear and nonstationary data. These methods are inspired by the empirical mode decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t) cos(θ(t))}, where a is assumed to be less oscillatory than cos(θ(t)) and θ′ ? 0. This problem can be formulated as a nonlinear l ⁰ optimization problem. We have proposed two methods to solve this nonlinear optimization problem. The first one is based on nonlinear basis pursuit and the second one is based on nonlinear matching pursuit. Convergence analysis has been carried out for the nonlinear matching pursuit method. Some numerical experiments are given to demonstrate the effectiveness of the proposed methods.     

A multigrid method for the Cahn-Hilliard equation with obstacle potential

We present a multigrid finite element method for the deep quench obstacle Cahn-Hilliard equation. The non-smooth nature of this highly nonlinear fourth order partial differential equation make this problem particularly challenging. The method exhibits mesh-independent convergence properties in practice for arbitrary time step sizes. In addition, numerical evidence shows that this behaviour extends to small values of the interfacial parameter γ. Several numerical examples are given, including comparisons with existing alternative solution methods for the Cahn-Hilliard equation.     

Block methods for nonlinear Volterra integral equations

A method used to solve linear Fredholm equations is adapted to obtain a generalization of block methods for solving nonlinear Volterra integral equations. Conditions for such methods to be convergent andA-stable are given along with some numerical examples.     

The dynamics of a one-dimensional fox-rabies model

The fox-rabies model consists of two non-linear partial differential equations. In this model the fox population is divided into two species: susceptible (S) and infective (I). Finite-difference methods are used to compute the numerical solution of the initial/boundary-value problem.     

Non-polynomial sextic spline approach for the solution of fourth-order boundary value problems

In this paper a non-polynomial sextic spline function is applied to the numerical solution of a linear fourth-order two-point boundary-value problem occurring in a plate deflection theory. We have developed a non-polynomial sextic spline, which reduces to ordinary sextic spline as θ → 0. Spline relations and error estimates are given. Direct methods of order two, four and six have been obtained. Numerical results are provided to demonstrate the superiority of our methods.     

Stability of Runge–Kutta methods in the numerical solution of equation u′(t)=au(t)+a0u([t])

This paper is concerned with the stability analysis of the Runge–Kutta methods for the equation u′(t)=au(t)+a₀u([t]). The stability regions for the Runge–Kutta methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given.     

Interpolation Wavelets in Boundary Value Problems

We propose and validate a simple numerical method that finds an approximate solution with any given accuracy to the Dirichlet boundary value problem in a disk for a homogeneous equation with the Laplace operator. There are many known numerical methods that solve this problem, starting with the approximate calculation of the Poisson integral, which gives an exact representation of the solution inside the disk in terms of the given boundary values of the required functions. We employ the idea of approximating a given 2π-periodic boundary function by trigonometric polynomials, since it is easy to extend them to harmonic polynomials inside the disk so that the deviation from the required harmonic function does not exceed the error of approximation of the boundary function. The approximating trigonometric polynomials are constructed by means of an interpolation projection to subspaces of a multiresolution analysis (approximation) with basis 2π-periodic scaling functions (more exactly, their binary rational compressions and shifts). Such functions were constructed by the authors earlier on the basis of Meyer-type wavelets; they are either orthogonal and at the same time interpolating on uniform grids of the corresponding scale or only interpolating. The bounds on the rate of approximation of the solution to the boundary value problem are based on the property ofMeyer wavelets to preserve trigonometric polynomials of certain (large) orders; this property was used for other purposes in the first two papers listed in the references. Since a numerical bound of the approximation error is very important for the practical application of the method, a considerable portion of the paper is devoted to this issue, more exactly, to the explicit calculation of the constants in the order bounds of the error known earlier.     

Superconvergence and a posteriori error estimates of RT1 mixed methods for elliptic control problems with an integral constraint

In this paper, we investigate the superconvergence property and a posteriori error estimates of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Approximations of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h ². Moreover, we derive a posteriori error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.     

Oscillation of Numerical Solution in the Runge-Kutta Methods for Equation x＇（t） = ax（t） ＋ aox（[t]）

The paper de ls with oscillation of Runge-Kutta methods for equation x＇（t） = ax（t） ＋ aox（[t]）. The conditions of oscillation for the numerical methods are presented by considering the characteristic equation of the corresponding discrete scheme. It is proved that any nodes have the same oscillatory property as the integer nodes. Furthermore, the conditions under which the oscillation of the analytic solution is inherited by the numerical solution are obtained. The relationships between stability and oscillation are considered. Finally, some numerical experiments are given.