Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition

The hybrid spectral difference methods (HSD) for the Laplace and Helmholtz equations in exterior domains are proposed. We consider the fictitious domain method with the absorbing boundary conditions (ABCs). The HSD method is a finite difference version of the hybridized Galerkin method, and it consists of two types of finite difference approximations; the cell finite difference and the interface finite difference. The fictitious domain is composed of two subregions; the Cartesian grid region and the boundary layer region in which the radial grid is imposed. The boundary layer region with the radial grid makes it easy to implement the discrete radial ABC. The discrete radial ABC is a discrete version of the Bayliss–Gunzburger–Turkel ABC without pertaining any radial derivatives. Numerical experiments confirming efficiency of our numerical scheme are provided.     

Uniform boundary controllability of a discrete 1-D wave equation

A numerical scheme for the controlled discrete 1-D wave equation is considered. We prove the convergence of the boundary controls of the discrete equations to a control of the continuous wave equation when the mesh size tends to zero when time and space steps coincide. This positive result is in contrast with previous negative ones for space semi-discretizations.     

A novel artificial bee colony algorithm with Powell's method

Artificial bee colony (ABC) algorithm is a relatively new optimization technique which has been shown to be competitive to other population-based algorithms. However, there is still an insufficiency in ABC regarding its solution search equation, which is good at exploration but poor at exploitation. To address this concerning issue, we first propose a modified search equation which is applied to generate a candidate solution in the onlookers phase to improve the search ability of ABC. Further, we use the Powell's method as a local search tool to enhance the exploitation of the algorithm. The new algorithm is tested on 22 unconstrained benchmark functions and 13 constrained benchmark functions, and are compared with some other ABCs and several state-of-the-art algorithms. The comparisons show that the proposed algorithm offers the highest solution quality, fastest global convergence, and strongest robustness among all the contenders on almost all test functions.     

On The Eigenvalues of the Spectral Second Order Differentiation Operator and Application to the Boundary Observability of the Wave Equation

The behaviour of the eigenvalues of the spectral second-order differentiation operator is studied and the results are used to investigate the boundary observability of the one dimensional wave equation approximated with a spectral Galerkin method. New explicit estimates of the discrete eigenvalues are given. These estimates improve the previous results on the subject especially for the portion of eigenvalues converging exponentially to those of the continuous problem. Although the boundary observability property of the discretized wave equation is not uniform with respect to the discretization parameter, we show that a uniform observability estimate can be obtained by filtering out the highest eigenmodes.      

再辐射边界条件在FDTD中的应用

目前的吸收边界条件存在编程复杂,占有内存空间与计算量过大等缺点。再辐射边界条件通过在截断边界处引入与入射波等效电磁流符号相反的源,使得穿过边界的电磁场得到有效的衰减。与传统的吸收边界条件相比,再辐射边界条件具有吸收性能优越,设计思路清晰,编程简单,内存占有少,计算时间短的优点。推导了在离散网格中再辐射边界的反射系数与时间及空间步长的关系,通过时域有限差分方法（FDTD）进行了验证,与理论值基本相符。     

An Artificial Bee Colony algorithm with guide of global & local optima and asynchronous scaling factors for numerical optimization

Artificial Bee Colony (ABC) algorithm is a wildly used optimization algorithm. However, ABC is excellent in exploration but poor in exploitation. To improve the convergence performance of ABC and establish a better searching mechanism for the global optimum, an improved ABC algorithm is proposed in this paper. Firstly, the proposed algorithm integrates the information of previous best solution into the search equation for employed bees and global best solution into the update equation for onlooker bees to improve the exploitation. Secondly, for a better balance between the exploration and exploitation of search, an S-type adaptive scaling factors are introduced in employed bees’ search equation. Furthermore, the searching policy of scout bees is modified. The scout bees need update food source in each cycle in order to increase diversity and stochasticity of the bees and mitigate stagnation problem. Finally, the improved algorithms is compared with other two improved ABCs and three recent algorithms on a set of classical benchmark functions. The experimental results show that the our proposed algorithm is effective and robust and outperform than other algorithms.     

Nonreflecting boundary condition for the Helmholtz equation

To solve the Helmholtz equation in an infinite three-dimensional domain a spherical artificial boundary is introduced to restrict the computational domain Ω. To determine the nonreflecting boundary condition on ∂Ω, we start with a finite number of spherical harmonics for the Helmholtz equation. With a precise choice of (primary) nodes on the sphere, the theorem on Gauss-Jordan quadrature establishes the discrete orthogonality of the spherical harmonics when summed over these nodes. An approximate nonreflecting boundary condition for the Helmholtz equation follows readily upon solving the exterior Dirichlet problem. The accuracy of the boundary condition is determined using a point source, and the computational results are presented for the scattering of a wave from a sphere.     

On the discrete and continuous time infinite-dimensional algebraic Riccati equations

The standard state space solution of the finite-dimensional continuous time quadratic cost minimization problem has a straightforward extension to infinite-dimensional problems with bounded or moderately unbounded control and observation operators. However, if these operators are allowed to be sufficiently unbounded, then a strange change takes place in one of the coefficients of the algebraic Riccati equation, and the continuous time Riccati equation begins to resemble the discrete time Riccati equation. To explain why this phenomenon must occur we discuss a particular hyperbolic PDE in one space dimension with boundary control and observation (a transmission line) that can be formulated both as a discrete time system and as a continuous time system, and show that in this example the continuous time Riccati equation can be recovered from the discrete time Riccati equation. A particular feature of this example is that the Riccati operator does not map the domain of the generator into the domain of the adjoint generator, as it does in the standard case.     

The analysis of discrete artificial bee colony algorithm with neighborhood operator on traveling salesman problem

The artificial bee colony (ABC) algorithm, inspired intelligent behaviors of real honey bee colonies, was introduced by Karabo?a for numerical function optimization. The basic ABC has high performance and accuracy, if the solution space of the problem is continuous. But when the solution space of the problem is discrete, the basic ABC algorithm should be modified to solve this class optimization problem. In this study, we focused on analysis of discrete ABC with neighborhood operator for well-known traveling salesman problem and different discrete neighborhood operators are replaced with solution updating equations of the basic ABC. Experimental computations show that the promising results are obtained by the discrete version of the basic ABC and which neighborhood operator is better than the others. Also, the results obtained by discrete ABC were enriched with 2- and 3-opt heuristic approaches in order to increase quality of the solutions.     

Artificial bee colony algorithm with gene recombination for numerical function optimization

Artificial bee colony (ABC) algorithm is a stochastic and population-based optimization method, which mimics the collaborative foraging behaviour of honey bees and has shown great potential to handle various kinds of optimization problems. However, ABC often suffers from slow convergence speed since its internal mechanism and solution search equation do well in exploration, but badly in exploitation. In order to solve this knotty issue, inspired by the natural phenomenon that the good individuals (solutions) always contain good genes (variables) and the effective combination of the superior genes from different good individuals could more easily produce better offspring, we introduce a novel gene recombination operator (GRO) into ABC to accelerate convergence. To be specific, in GRO, a part of good solutions in the current population are selected to produce candidate solutions by the gene combination. Especially, each good solution recombines with only one other good solution to generate only one candidate solution. In addition, GRO will be launched at the end of each generation. In order to validate its efficiency and effectiveness, GRO is embedded into nine versions of ABC, i.e., the original ABC, GABC, best-so-far ABC(BSFABC), MABC, CABC, ABCVSS, qABC, dABC and distABC, while yields GRABC, GRGABC, GRBSFABC, GRMABC, GRCABC, GRABCVSS, GRqABC, GRdABC and GRdistABC respectively. The experimental results on 22 benchmark functions demonstrate that GRO could enhance the exploitation ability of ABCs and accelerate convergence without loss of diversity.     

Finite difference method and a spurious reflection of waves in a layer of fluid

For a layer of fluid and boundary conditions considered in this paper analytical solutions of the Helmholtz equation exist and are well known. When the continuum is replaced by a discrete space of chosen points with a regular grid, analogous solutions may be calculated. In the case of two uniform grids with sudden mesh size change from one grid to the other, in the layer a deformation of discrete solutions occurs. The deformation assumes the form of a spurious wave reflected from the boundary of the two grids. Moreover, the transmitted wave amplitude is changed. If the wave propagates from a grid of smaller mesh size, then the transmitted wave has a larger amplitude than the incident wave. The explicit formulae obtained enable us to calculate the degree of deformation of discrete solutions for the finite difference method and the nonuniform grid.     

Computation of Nonlinear Backscattering Using a High-Order Numerical Method

The nonlinear Schrödinger equation (NLS) is the standard model for propagation of intense laser beams in Kerr media. The NLS is derived from the nonlinear Helmholtz equation (NLH) by employing the paraxial approximation and neglecting the backscattered waves. In this study we use a fourth-order finite-difference method supplemented by special two-way artificial boundary conditions (ABCs) to solve the NLH as a true boundary value problem. Our numerical methodology allows for a direct comparison of the NLH and NLS models and, apparently for the first time, for an accurate quantitative assessment of the backscattered signal.     

An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions

In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme.     

On the Accuracy and Stability of the Perfectly Matched Layer in Transient Waveguides

Energy transmitted along a waveguide decays less rapidly than in an unbounded medium. In this paper we study the efficiency of a PML in a time-dependent waveguide governed by the scalar wave equation. A straight forward application of a Neumann boundary condition can degrade accuracy in computations. To ensure accuracy, we propose extensions of the boundary condition to an auxiliary variable in the PML. We also present analysis proving stability of the constant coefficient PML, and energy estimates for the variable coefficients case. In the discrete setting, the modified boundary conditions are crucial in deriving discrete energy estimates analogous to the continuous energy estimates. Numerical stability and convergence of our numerical method follows. Finally we give a number of numerical examples, illustrating the stability of the layer and the high order accuracy of our proposed boundary conditions.     

Improved adaptive coding learning for artificial bee colony algorithms

Applied Intelligence - Recently, the artificial bee colony (ABC) algorithm has become increasingly popular in the field of evolutionary computing and manystate- of-the-art ABC variants (ABCs) have...     

An adaptive artificial bee colony algorithm based on objective function value information

Artificial bee colony (ABC) algorithm is a novel biological-inspired optimization algorithm, which has many advantages compared with other optimization algorithm, such as less control parameters, great global optimization ability and easy to carry out. It has proven to be more effective than some evolutionary algorithms (EAs), particle swarm optimization (PSO) and differential evolution (DE) when testing on both benchmark functions and real issues. ABC, however, its solution search equation is poor at exploitation. For overcoming this insufficiency, two new solution search equations are proposed in this paper. They apply random solutions to take the place of the current solution as base vector in order to get more useful information. Exploitation is further improved on the basis of enhancing exploration by utilizing the information of the current best solution. In addition, the information of objective function value is introduced, which makes it possible to adjust the step-size adaptively. Owing to their respective characteristics, the new solution search equations are combined to construct an adaptive algorithm called MTABC. The methods our proposed balance the exploration and exploitation of ABC without forcing severe extra overhead in respect of function evaluations. The performance of the MTABC algorithm is extensively judged on a set of 20 basic functions and a set of 10 shifted or rotated functions, and is compared favorably with other improved ABCs and several state-of-the-art algorithms. The experimental results show that the proposed algorithm has a higher convergence speed and better search ability for almost all functions.     

Optimal design for 2D wave equations

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. ?veràk in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.     

Thermal boundary conditions for thermal lattice Boltzmann simulations

Consistent 2D and 3D thermal boundary conditions for thermal lattice Boltzmann simulations are proposed. The boundary unknown energy distribution functions are made functions of known energy distribution functions and correctors, where the correctors at the boundary nodes are obtained directly from the definition of internal energy density. This boundary condition can be easily implemented on the wall and corner boundary using the same formulation. The discrete macroscopic energy equation is also derived for a steady and fully developed channel flow to assess the effect of the boundary condition on the solutions, where the resulting second order accurate central difference equation predicts continuous energy distribution across the boundary, provided the boundary unknown energy distribution functions satisfy the macroscopic energy level. Four different local known energy distribution functions are experimented with to assess both this observation and the applicability of the present formulation, and are scrutinized by calculating the 2D thermal Poiseuille flow, thermal Couette flow, thermal Couette flow with wall injection, natural convection in a square cavity, and 3D thermal Poiseuille flow in a square duct. Numerical simulations indicate that the present formulation is second order accurate and the difference of adopting different local known energy distribution functions is, as expected, negligible, which are consistent with the results from the derived discrete macroscopic energy equation.     

Memetic search in artificial bee colony algorithm

Artificial bee colony (ABC) optimization algorithm is relatively a simple and recent population based probabilistic approach for global optimization. ABC has been outperformed over some Nature Inspired Algorithms (NIAs) when tested over benchmark as well as real world optimization problems. The solution search equation of ABC is significantly influenced by a random quantity which helps in exploration at the cost of exploitation of the search space. In the solution search equation of ABC, there is a enough chance to skip the true solution due to large step size. In order to balance between diversity and convergence capability of the ABC, a new local search phase is integrated with the basic ABC to exploit the search space identified by the best individual in the swarm. In the proposed phase, ABC works as a local search algorithm in which, the step size that is required to update the best solution, is controlled by Golden Section Search approach. The proposed strategy is named as Memetic ABC (MeABC). In MeABC, new solutions are generated around the best solution and it helps to enhance the exploitation capability of ABC. MeABC is established as a modified ABC algorithm through experiments over 20 test problems of different complexities and 4 well known engineering optimization problems.     

Artificial Bee Colony Algorithm Based on Adaptive Search Equation and Extended Memory

From the perspective of psychology, a modified artificial bee colony algorithm (ABC, for short) based on adaptive search equation and extended memory (ABCEM, for short) for global optimization is proposed in this paper. In the proposed ABCEM algorithm, an extended memory factor is introduced into store employed bees’ and onlooker bees’ historical information comprising recent food sources, personal best food sources, and global best food sources, and the solution search equation for the employed bees is equipped with adaptive ability. Moreover, a parameter is employed to describe the importance of the extended memory. Furthermore, the extended memory is added to two solution search equations for the employed bees and the onlookers to improve the quality of food source. To evaluate the proposed algorithm, experiments are conducted on a set of numerical benchmark functions. The results show that the proposed algorithm can balance the exploration and exploitation, and can improve the accuracy of optima solutions and convergence speed compared with other current improved ABCs for global optimization in most of the tested functions.