A numerical algorithm for nonlinear multi-point boundary value problems

In this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method.     

A numerical algorithm for some singularly perturbed boundary value problems

A numerical algorithm is proposed to solve singularly perturbed linear two-point value problems. The method starts with a partial decoupling of the system to obtain two independent subsystems, fast and slow components. Each subsystem is then solved separately. A second-order finite difference scheme is used for this purpose. Numerical examples will be presented to show the efficiency of the method.     

A numerical algorithm for solving a class of linear nonlocal boundary value problems

In order to solve a class of linear nonlocal boundary value problems, a new reproducing kernel space satisfying nonlocal conditions is constructed carefully. This makes it easy to solve the problems. Furthermore, the exact solutions of the problems can be expressed in series form. The numerical results demonstrate that the new method is quite accurate and efficient for solving fourth-order nonlocal boundary value problems.     

A Bayesian model and numerical algorithm for CBM availability maximization

In this paper, we consider an availability maximization problem for a partially observable system subject to random failure. System deterioration is described by a hidden, continuous-time homogeneous Markov process. While the system is operational, multivariate observations that are stochastically related to the system state are sampled through condition monitoring at discrete time points. The objective is to design an optimal multivariate Bayesian control chart that maximizes the long-run expected average availability per unit time. We have developed an efficient computational algorithm in the semi-Markov decision process (SMDP) framework and showed that the availability maximization problem is equivalent to solving a parameterized system of linear equations. A numerical example is presented to illustrate the effectiveness of our approach, and a comparison with the traditional age-based replacement policy is also provided.     

On the weak convergence of Wright-Fisher model

We prove that the sequence of stochastic processes obtained from Wright-Fisher models by transforming the time scales and state spaces in the usual way converges weakly to a diffusion process on the time interval [0,∞). Convergence of fixation probabilities and fixation time distributions are obtained as corollaries. These results extend a theorem of Watterson, who proved convergence in distribution to a diffusion at any given single time point for these processes.     

A sixth-order dual preserving algorithm for the Camassa-Holm equation

The paper presents a sixth-order numerical algorithm for studying the completely integrable Camassa-Holm (CH) equation. The proposed sixth-order accurate method preserves both the dispersion relation and the Hamiltonians of the CH equation. The CH equation in this study is written as an evolution equation, involving only the first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step method that first solves the evolution equation by a sixth-order symplectic Runge-Kutta method and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The sixth-order symplectic Runge-Kutta time integrator is preferable for an equation that possesses a Hamiltonian structure. We illustrate the ability of the proposed scheme by examining examples involving peakon or peakon-like solutions. We compare the computed solutions with exact solutions or asymptotic predictions. We also demonstrate the ability of the symplectic time integrator to preserve the Hamiltonians. Finally, via a smooth travelling wave problem, we compare the accuracy, elapsed computing time, and rate of convergence among the proposed method, a second-order two-step algorithm, and a completely integrable particle method.     

A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element

A general numerical method is proposed to compute nearly singular integrals arising in the boundary integral equations (BIEs). The method provides a new implementation of the conventional distance transformation technique to make the result stable and accurate no matter where the projection point is located. The distance functions are redefined in two local coordinate systems. A new system denoted as (α,β) is introduced here firstly. Its implementation is simpler than that of the polar system and it also performs efficiently. Then a new distance transformation is developed to remove or weaken the near singularities. To perform integration on irregular elements, an adaptive integration scheme is applied. Numerical examples are presented for both planar and curved surface elements. The results demonstrate that our method can provide accurate results even when the source point is very close to the integration element, and can keep reasonable accuracy on very irregular elements. Furthermore, the accuracy of our method is much less sensitive to the position of the projection point than the conventional method.     

A study of a numerical algorithm for a nonlinear transport model

For a nonlinear transport model, we propose a simple and economical two-step algorithm that decreases the dimension of the system of nonlinear equations, as compared with implicit difference schemes. We prove theorems on necessary conditions for stability with respect to the initial data for the nonlinear problem and theorems on sufficient conditions for stability in the case of the linearized model. We also obtain theorems on approximation of the integral conservation law on a grid. The necessary condition obtained is a condition on the coefficients of the differential equation (which singles out an admissible class of equations) but not a condition on the ratio of the grid steps. Bibliography: 3 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 25–32.     

A bisection algorithm for the numerical Mountain Pass

We propose a constructive proof for the Ambrosetti-Rabinowitz Mountain Pass Theorem providing an algorithm, based on a bisection method, for its implementation. The efficiency of our algorithm, particularly suitable for problems in high dimensions, consists in the low number of flow lines to be computed for its convergence; for this reason it improves the one currently used and proposed by Y.S. Choi and P.J. McKenna in [3]. Susanna Terracini: This work is partially supported by M.I.U.R. project “Metodi Variazionali ed Equazioni Differenziali Nonlineari”.     

A numerical algorithm for simulation of the Q-switched fiber laser using the travelling wave model

We develop a finite-difference scheme for approximation of a system of nonlinear PDEs describing the Q-switching process. We construct it by using staggered grids. The transport equations are approximated along characteristics, and quadratic nonlinear functions are linearized using a special selection of staggered grids. The stability analysis proves that a connection between time and space steps arises only due to approximation requirements in order to follow exactly the directions of characteristics. The convergence analysis of this scheme is done in two steps. First, some estimates of the uniform boundedness of the discrete solution are proved. This part of the analysis is done locally, in some neighborhood of the exact solution. Second, on the basis of the obtained estimates, the main stability inequality is proved. The second-order convergence rate with respect to the space and time coordinates follows from this stability estimate. Using the obtained convergence result, we prove that the local stability analysis in the selected neighborhood of the exact solution is sufficient.     

A perturb biogeography based optimization with mutation for global numerical optimization

Biogeography based optimization (BBO) is a new evolutionary optimization algorithm based on the science of biogeography for global optimization. We propose three extensions to BBO. First, we propose a new migration operation based sinusoidal migration model called perturb migration, which is a generalization of the standard BBO migration operator. Then, the Gaussian mutation operator is integrated into perturb biogeography based optimization (PBBO) to enhance its exploration ability and to improve the diversity of population. Experiments have been conducted on 23 benchmark problems of a wide range of dimensions and diverse complexities. Simulation results and comparisons demonstrate the proposed PBBO algorithm using sinusoidal migration model is better, or at least comparable to, the RCBBO based linear model, RCBBO-G, RCBBO-L and evolutionary algorithms from literature when considering the quality of the solutions obtained.     

A rain water infiltration model with unilateral boundary condition: qualitative analysis and numerical simulations

We present a rigorous mathematical treatment of a model describing rain water infiltration through the vadose zone in case of runoff of the excess water. The main feature of the mathematical problem emerging from the model lies on the boundary condition on the ground surface which is in the form of a unilateral constraint. Existence and uniqueness of a weak solution is proved under general assumptions. We present also the results of a numerical study comparing the proposed model with other models which approach in a different way the rain water infiltration problem. Copyright © 2006 John Wiley & Sons, Ltd.     

A detail preserving variational model for image Retinex

In this paper, we propose a detail preserving variational model for Retinex to simultaneously estimate the illumination and the reflectance from an observed image. Most previous models use the log-transform as pretreatment which results in loss of details in reflectance. From this observation, a detail preserving variational method is proposed for better decomposition. Different from the log-transform based models, the proposed model performs the decomposition directly in the image domain. Mathematically, we prove the existence of a solution for the proposed model. Numerically, we derive an efficient iterative algorithm by utilizing alternating direction method of multipliers (ADMM) method. Experimental results demonstrate the effectiveness of the proposed method. Compared with other closely related Retinex methods, the proposed method achieves competitive results on both subjective and objective assessments.     

A one-sweep numerical method for vector-matrix difference equations with two-point boundary conditions

A new method is proposed for reducing two-point boundaryvalue problems for vector-matrix systems of linear difference equations to initial-value problems. The method has the advantage that only one sweep is required, and memory requirements are minimal. Applications to potential theory are discussed.This research was supported by the National Institutes of Health under Grants Nos. GM-16197-01 and GM-16437-01 and by the Atomic Energy Commission under Contract No. AT(11-1)-113, Project No. 19.     

A sequentially optimal algorithm for numerical integration

For the class of functions of one variable, satisfying the Lipschitz condition with a fixed constant, an optimal passive algorithm for numerical integration (an optimal quadrature formula) has been found by Nikol'skii. In this paper, a sequentially optimal algorithm is constructed; i.e., the algorithm on each step makes use in an optimal way of all relevant information which was accumulated on previous steps. Using the algorithm, it is necessary to solve an integer program at each step. An effective algorithm for solving these problems is given.The author is indebted to Professor S. E. Dreyfus, Department of Industrial Engineering and Operations Research, University of California, Berkeley, California, for his helpful attention to this paper.     

Linear operators preserving the decomposable numerical range

Multilinear techniques are used to characterize unitary matrices in terms of a generalized numerical range. This characterization is then applied to analyze the structure of all linear operators on matrices which preserve this numerical range. The results generalize V. J. Pellegrini's determination of all linear operators preserving the classical numerical range.     

A structure‐preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear‐quadratic infinite time horizon optimal control. We focus on small‐scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete‐time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure‐preserving doubling algorithm. Implementation issues such as the choice of the parameter γ in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill‐posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method. Copyright © 2015 John Wiley & Sons, Ltd.     

Linear operators preserving the decomposable numerical radius

The characterization of all linear operators on matrices which preserve the decomposable numerical radius is obtained. This result refines those of Tam. Marcus and Filippenko on the topic. The proof of the main theorem depends on a characterization of scalar multiples of unitary matrices in terms of decomposable numerical radius that is of independent interest.     

A numerical algorithm for the nonlinear Timoshenko beam system

An initial boundary value problem is considered for the dynamic beam system Its solution is found by means of an algorithm, the constituent parts of which are the finite element method, the implicit symmetric difference scheme used to approximate the solution with respect to the spatial and time variables, and also a Picard type iteration process for solving the system of nonlinear equations obtained by discretization. Errors of three parts of the algorithm are estimated and, as a result, its total error estimate is obtained. A numerical example is solved.