Dynamics of a population kinetics model with cosymmetry

The dynamics of a cosymmetric system of nonlinear parabolic equations is studied to model the population kinetics of three interacting species. A finite-difference scheme that preserves the cosymmetry of the underlying problem is designed. A method for computing a continuous family of equilibria is developed. Different scenarios of instability in the model are analyzed, and the evolution of nonstationary regimes and families of steady states are examined.…     

Staggered grids discretization in three-dimensional Darcy convection

We consider three-dimensional convection of an incompressible fluid saturated in a parallelepiped with a porous medium. A mimetic finite-difference scheme for the Darcy convection problem in the primitive variables is developed. It consists of staggered nonuniform grids with five types of nodes, differencing and averaging operators on a two-nodes stencil. The nonlinear terms are approximated using special schemes. Two problems with different boundary conditions are considered to study scenarios of instability of the state of rest. Branching off of a continuous family of steady states was detected for the problem with zero heat fluxes on two opposite lateral planes.     

Three-dimensional solutions of laminated cylinders

A mixed finite-difference scheme is presented for the free vibration analysis of simply supported laminated orthotropic circular cylinders. The study is based on the linear three-dimensional theory of orthotropic elasticity, and the governing equations are reduced to six first-order ordinary differential equations in the thickness coordinate. In the finite-difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented of the effects of variations in the lamination and geometric characteristics of circular cylinders on their vibration characteristics. Also, the accuracy and range of validity of two-dimensional Sanders—Budiansky type shell theories are investigated.     

Nonstandard finite-difference methods for predator–prey models with general functional response

Predator–prey systems with linear and logistic intrinsic growth rate of the prey are analyzed. The models incorporate the mutual interference between predators into the functional response which stabilizes predator–prey interactions in the system. Positive and elementary stable nonstandard (PESN) finite-difference methods, having the same qualitative features as the corresponding continuous predator–prey models, are formulated and analyzed. The proposed numerical techniques are based on a nonlocal modeling of the growth-rate function and a nonstandard discretization of the time derivative. This discretization approach leads to significant qualitative improvements in the behavior of the numerical solution. In addition, it allows for the use of an essentially implicit method for the cost of an explicit method. Applications of the PESN methods to specific predator–prey systems are also presented.     

Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation

A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a second-order accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these difficulties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finite-difference gradients and several design examples are presented.     

Computational Methods for Verification of Stochastic Hybrid Systems

Stochastic hybrid system (SHS) models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing sound computational methods for verification is challenging because of the interaction between the discrete and the continuous stochastic dynamics. In this paper, we propose a probabilistic method for verification of SHSs based on discrete approximations focusing on reachability and safety problems. We show that reachability and safety can be characterized as a viscosity solution of a system of coupled Hamilton-Jacobi-Bellman equations. We present a numerical algorithm for computing the solution based on discrete approximations that are derived using finite-difference methods. An advantage of the method is that the solution converges to the one for the original system as the discretization becomes finer. We also prove that the algorithm is polynomial in the number of states of the discrete approximation. Finally, we illustrate the approach with two benchmarks: a navigation and a room heater example, which have been proposed for hybrid system verification.     

Approximation of the controls for the linear beam equation

This article deals with the approximation of the boundary controls of a 1-D linear equation modeling the transversal vibrations of a hinged beam using a finite-difference space semi-discrete scheme. Due to the high frequency numerical spurious oscillations, the semi-discrete model is not uniformly controllable with respect to the mesh size and the convergence of the approximate controls corresponding to initial data in the finite energy space cannot be guaranteed. In this paper we analyze how do the initial data to be controlled and their discretization affect the result of the approximation process. We prove that the convergence of the scheme is ensured if the continuous initial data are sufficiently regular or if the highest frequencies of their discretization have been filtered out. In both cases, the minimal weighted \(L^2\)-norm discrete controls are shown to be convergent to the corresponding continuous one when the mesh size tends to zero.     

Mixed finite-difference scheme for analysis of simply supported thick plates

A mixed finite-difference scheme is presented for the stress and free vibration analysis of simply supported nonhomogeneous and layered orthotropic thick plates. The analytical formulation is based on the linear, three-dimensional theory of orthotropic elasticity and a Fourier approach is used to reduce the governing equations to six first-order ordinary differential equations in the thickness coordinate. The governing equations possess a symmetric coefficient matrix and are free of derivatives of the elastic characteristics of the plate. In the finite difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to highly accurate results, even when a small number of finite difference intervals is used.     

ZOH discretization effect on single-input sliding mode control systems with matched uncertainties

In this paper, discretization behaviors of equivalent control based sliding mode control systems with matched uncertainties are studied. Upper bounds for system steady states are established. Some inherent dynamical periodic properties of the systems subject to matched constant and periodic uncertainties are explored. Simulations are presented to verify the theoretical results.     

Discretization behaviors of equivalent control based sliding-mode control systems

In this note, discretization behaviors of the equivalent control based sliding-mode control (SMC) systems are studied. Some inherent dynamical properties of the discretized second-order systems are first explored. Upper bounds for the system steady states are established. The system's steady-state behaviors are discussed. The analysis for the second-order systems is then extended to higher order systems. Simulations are presented to verify the theoretical results.     

An axis treatment for flow equations in cylindrical coordinates based on parity conditions

A novel axis treatment using parity conditions is presented for flow equations in cylindrical coordinates that are represented in azimuthal Fourier modes. The correct parity states of scalars and the velocity vector are derived such that symmetry conditions for each Fourier mode of the respective variable can be determined. These symmetries are then used to construct finite-difference and filter stencils at and near the axis, and an interpolation scheme for the computation of terms premultiplied by 1/r. A grid convergence study demonstrates that the axis treatment retains the formal accuracy of the spatial discretization scheme employed. Two further test cases are considered for evaluation of the axis treatment, the propagation of an acoustic pulse and direct numerical simulation of a fully turbulent supersonic axisymmetric wake. The results demonstrate the applicability of the axis treatment for non-axisymmetric flows.     

Euler's Discretization of Single Input Sliding-Mode Control Systems

In this note, the dynamical behaviors of Euler's discretization of the equivalent control based sliding-mode control (SMC) systems are studied. We show that if the discretized SMC system is asymptotically stable, then every trajectory converges to a period-2 cycle. Bounds for the steady states are derived, which allow one to estimate the maximum chattering amplitude for a given value of the time step. The analysis and results are verified with several computer simulations. [All rights reserved Elsevier].     

基于节点生长k-均值聚类算法的强化学习方法

处理连续状态强化学习问题,主要方法有两类:参数化的函数逼近和自适应离散划分.在分析了现有对连续状态空间进行自适应划分方法的优缺点的基础上,提出了一种基于节点生长k均值聚类算法的划分方法,分别给出了在离散动作和连续动作两种情况下该强化学习方法的算法步骤.在离散动作的MountainCar问题和连续动作的双积分问题上进行仿真实验.实验结果表明,该方法能够根据状态在连续空间的分布,自动调整划分的精度,实现对于连续状态空间的自适应划分,并学习到最佳策略.     

On the numerical solution of space–time fractional diffusion models

A flexible numerical scheme for the discretization of the space–time fractional diffusion equation is presented. The model solution is discretized in time with a pseudo-spectral expansion of Mittag–Leffler functions. For the space discretization, the proposed scheme can accommodate either low-order finite-difference and finite-element discretizations or high-order pseudo-spectral discretizations. A number of examples of numerical solutions of the space–time fractional diffusion equation are presented with various combinations of the time and space derivatives. The proposed numerical scheme is shown to be both efficient and flexible.     

Recurrent learning algorithms for designing optimal controllers ofcontinuous systems

Proposes a recurrent learning algorithm for designing the controllers of continuous dynamical systems in optimal control problems. The controllers are in the form of unfolded recurrent neural nets embedded with physical laws from classical control techniques. The learning algorithm is characterized by a double forward-recurrent-loops structure for solving both temporal recurrent and structure recurrent problems. The first problem results from the nature of general optimal control problems, where the objective functions are often related to (evaluated at) some specific time steps or system states only, causing missing learning signals at some steps or states. The second problem is due to the high-order discretization of continuous systems by the Runge-Kutta method that we perform to increase accuracy. This discretization transforms the system into several identical interconnected subnetworks, like a recurrent neural net expanded in the time axis. Two recurrent learning algorithms with different convergence properties are derived; first- and second-order learning algorithms. Their computations are local and performed efficiently as net signal propagation. We also propose two new nonlinear control structures for the 2D guidance problem and the optimal PI control problem. Under the training of the recurrent learning algorithms, these controllers can be easily tuned to be suboptimal for given objective functions. Extensive computer simulations show the controllers' optimization and generalization abilities     

Discrete Multi‐Material Interface Reconstruction for Volume Fraction Data

Material interface reconstruction (MIR) is the task of constructing boundary interfaces between regions of homogeneous material, while satisfying volume constraints, over a structured or unstructured spatial domain. In this paper, we present a discrete approach to MIR based upon optimizing the labeling of fractional volume elements within a discretization of the problem's original domain. We detail how to construct and initially label a discretization, and introduce a volume conservative swap move for optimization. Furthermore, we discuss methods for extracting and visualizing material interfaces from the discretization. Our technique has significant advantages over previous methods: we produce interfaces between multiple materials that are continuous across cell boundaries for time‐varying and static data in arbitrary dimension with bounded error.     

Adaptive logarithmic discretization for numerical renormalization group methods

The problem of the logarithmic discretization of an arbitrary positive function (such as the density of states) is studied in general terms. Logarithmic discretization has arbitrary high resolution around some chosen point (such as Fermi level) and it finds application, for example, in the numerical renormalization group (NRG) approach to quantum impurity problems (Kondo model), where the continuum of the conduction band states needs to be reduced to a finite number of levels with good sampling near the Fermi level. The discretization schemes under discussion are required to reproduce the original function after averaging over different interleaved discretization meshes, thus systematic deviations which appear in the conventional logarithmic discretization are eliminated. An improved scheme is proposed in which the discretization-mesh points themselves are determined in an adaptive way; they are denser in the regions where the function has higher values. Such schemes help in reducing the residual numeric artefacts in NRG calculations in situations where the density of states approaches zero over extended intervals. A reference implementation of the solver for the differential equations which determine the full set of discretization coefficients is also described.     

Rough Set理论中连续属性的离散化方法

Rough Set(RS)理论是一种新的处理不精确、不完全与不相容知识的数学工具.传 统的RS理论只能对数据库中的离散属性进行处理,而绝大多数现实的数据库既包含了离散 属性,又包含了连续属性.文中针对传统RS理论的这一缺陷,利用决策表相容性的反馈信 息,提出了一种领域独立的基于动态层次聚类的连续属性离散化算法.该方法为RS理论处 理离散与连续属性提供了一种统一的框架,从而极大地拓广了RS理论的应用范围.通过一 些例子将本算法与现有方法进行了比较分析,得到了令人鼓舞的结果.     

Effect of data discretization on the classification accuracy in a high‐dimensional framework

We investigate discretization of continuous variables for classification problems in a high‐ dimensional framework. As the goal of classification is to correctly predict a class membership of an observation, we suggest a discretization method that optimizes the discretization procedure using the misclassification probability as a measure of the classification accuracy. Our method is compared to several other discretization methods as well as result for continuous data. To compare performance we consider three supervised classification methods, and to capture the effect of high dimensionality we investigate a number of feature variables for a fixed number of observations. Since discretization is a data transformation procedure, we also investigate how the dependence structure is affected by this. Our method performs well, and lower misclassification can be obtained in a high‐dimensional framework for both simulated and real data if the continuous feature variables are first discretized. The dependence structure is well maintained for some discretization methods. © 2012 Wiley Periodicals, Inc.     

A second-order finite volume scheme for three dimensional truncated pyramidal quantum dot

Three dimensional truncated pyramidal quantum dots are simulated numerically to compute the energy states and the wave functions. The simulation of the hetero-structures is realized by using a novel finite volume scheme to solve the Schrödinger equation. The simulation benefits greatly from the finite volume scheme in threefold. Firstly, the BenDaniel-Duke hetero-junction interface condition is ingeniously embedded into the scheme. Secondly, the scheme uses uniform meshes in discretization and leads to simple computer implementation. Thirdly, the scheme is efficient as it achieves second-order convergence rates over varied mesh sizes. The scheme has successfully computed all the confined energy states and visualized the corresponding wave functions. The results further predict the relation of the energy states and wave functions versus the height of the truncated pyramidal quantum dots.