Proposed Discretization Method Research Articles

A novel discrete differential evolution algorithm combining transfer function with modulo operation for solving the multiple knapsack problem

In this paper, an efficient method for solving multiple knapsack problem (MKP) using discrete differential evolution is proposed. Firstly, an integer programming model of MKP suitable for discrete evolutionary algorithm is established. Secondly, a new method for discretizing continuous evolutionary algorithm is proposed based on the combination of transfer function and modulo operation. Therefrom, a new discrete differential evolution algorithm (named TMDDE) is proposed. Thirdly, the algorithm GROA is proposed to eliminate infeasible solutions of MKP. On this basis, a new method for solving MKP using TMDDE is proposed. Finally, the performance of TMDDE using S-shaped, U-shaped, V-shaped, and Taper-shaped transfer functions combined with modulo operation is compared, respectively. It is pointed out that T3-TMDDE which used Taper-shaped transfer function T3 is the best. The comparison results of solving 30 MKP instances show that the performance of T3-TMDDE is better than five advanced evolutionary algorithms. It not only indicates that TMDDE is more competitive for solving MKP, but also demonstrates that the proposed discretization method is an effective method.

General Discretization Method for Enhanced Kinesthetic Haptic Stability.

The stability of haptic simulation systems has been studied for a safer interaction with virtual environments. In this work, the passivity, uncoupled stability, and fidelity of such systems are analyzed when a viscoelastic virtual environment is implemented using a general discretization method that can also represent methods such as backward difference, Tustin, and zero-order-hold. Dimensionless parametrization and rational delay are considered for device independent analysis. Aiming at expanding the virtual environment dynamic range, equationsto find optimum damping values for maximize stiffness are derived and it is shown that by tuning the parameters for a customized discretization method, the virtual environment dynamic range will supersede the ranges offered by methods such as backward difference, Tustin and zero-order-hold. It is also shown that minimum time delay is required for stable Tustin implementation and that specific delay ranges must be avoided. The proposed discretization method is numerically and experimentally evaluated.

Rough-Set-Theory-Based Classification with Optimized k-Means Discretization

The discretization of continuous attributes in a dataset is an essential step before the Rough-Set-Theory (RST)-based classification process is applied. There are many methods for discretization, but not many of them have linked the RST instruments from the beginning of the discretization process. The objective of this research is to propose a method to improve the accuracy and reliability of the RST-based classifier model by involving RST instruments at the beginning of the discretization process. In the proposed method, a k-means-based discretization method optimized with a genetic algorithm (GA) was introduced. Four datasets taken from UCI were selected to test the performance of the proposed method. The evaluation of the proposed discretization technique for RST-based classification is performed by comparing it to other discretization methods, i.e., equal-frequency and entropy-based. The performance comparison among these methods is measured by the number of bins and rules generated and by its accuracy, precision, and recall. A Friedman test continued with post hoc analysis is also applied to measure the significance of the difference in performance. The experimental results indicate that, in general, the performance of the proposed discretization method is significantly better than the other compared methods.

Reliable Distributed Fuzzy Discretizer for Associative Classification of Big Data

Data Mining is an essential task because the digital world creates huge data daily. Associative classification is one of the data mining task which is used to carry out classification of data, based on the demand of knowledge users. Most of the associative classification algorithms are not able to analyze the big data which are mostly continuous in nature. This leads to the interest of analyzing the existing discretization algorithms which converts continuous data into discrete values and the development of novel discretizer Reliable Distributed Fuzzy Discretizer for big data set. Many discretizers suffer the problem of over splitting the partitions. Our proposed method is implemented in distributed fuzzy environment and aims to avoid over splitting of partitions by introducing a novel stopping criteria. Proposed discretization method is compared with existing distributed fuzzy partitioning method and achieved good accuracy in the performance of associative classifiers.

Convex optimization of nonlinear inequality with higher order derivatives

This paper is devoted to the Mayer problem on the optimization of nonlinear inequalities containing higher-order derivatives. We formulate the conditions of optimality for discrete and differential problems with higher-order inequality constraints. Discrete and differential problems play a substantial role in the formulation of optimal conditions in the form of Euler–Lagrange inclusions and ‘transversality’ conditions. The basic concept of obtaining optimal conditions is the proposed discretization method and equivalence results. Combining this approach and passing to the limit in the discrete-approximation problem, we establish sufficient optimality conditions for higher-order differential inequality. Moreover, to demonstrate this approach, the optimization of second-order polyhedral differential inequality is considered and a numerical example is given to illustrate the theoretical results.

Optimal Operation of a Single Unit with an Adjustable Blade in an Interbasin Water Transfer Pumping Station Based on Successive Approximation Discretization for Blade Angle

In the mathematical model of the optimal operation of a single pump unit with a fully adjustable blade in the Chinese South-to-North Water Diversion Project, the decision variable, namely, blade angle, was uniformly dispersed in its feasible region in a fixed step size in consideration of the requirements of the pumping head and matching motor power. 1D dynamic programming was applied to solve the original model. When the obtained blade for each time period was set as the middle reference value and the discrete region of the blade was reduced to two times of the step size in the previous time, the blade angle was correspondingly reduced and dispersed in this new discrete region, thus eliminating unnecessary optimization space. Then, 1D dynamic programming was applied again to optimize the blade angle of the single pump unit further. After a series of successive approximation discretization of the blade angle and corresponding solutions of the obtained mathematical model, the optimization process was considered completed when the given control precision met the requirement. A case study showed that under typical operating conditions, the total cost saving percentage of water pumping quantity reached 0.048%–0.463%, with an average saving rate of 0.192%. The actual total water pumping quantity of the single unit decreased by 2153 m3 on the average. The proposed discretization method exerted a better optimization effect and needed a smaller computational amount compared with traditional one-time uniform discretization in the original feasible region of the blade angle.

New nonlocal forward model for diffuse optical tomography.

The forward model in diffuse optical tomography (DOT) describes how light propagates through a turbid medium. It is often approximated by a diffusion equation (DE) that is numerically discretized by the classical finite element method (FEM). We propose a nonlocal diffusion equation (NDE) as a new forward model for DOT, the discretization of which is carried out with an efficient graph-based numerical method (GNM). To quantitatively evaluate the new forward model, we first conduct experiments on a homogeneous slab, where the numerical accuracy of both NDE and DE is compared against the existing analytical solution. We further evaluate NDE by comparing its image reconstruction performance (inverse problem) to that of DE. Our experiments show that NDE is quantitatively comparable to DE and is up to 64% faster due to the efficient graph-based representation that can be implemented identically for geometries in different dimensions. The proposed discretization method can be easily applied to other imaging techniques like diffuse correlation spectroscopy which are normally modeled by the diffusion equation.

A combined scheme of the local spectral element method and the generalized plane wave discontinuous Galerkin method for the anisotropic Helmholtz equation

In this paper we first consider a special class of nonhomogeneous and anisotropic Helmholtz equation with variable coefficients and the source term. Combined with local spectral element method, a generalized plane wave discontinuous Galerkin method for the discretization of such Helmholtz equation is designed. Then we define new generalized plane wave basis functions for two-dimensional anisotropic Helmholtz equation with the variable coefficient. Besides, the error estimates of the approximation solutions generated by the proposed discretization method are derived. Especially, the orders of the condition number ρ of the anisotropic matrix in the error estimates are optimal. Finally, numerical results verify the validity of the theoretical results, and indicate that the new method possesses high accuracy and is slightly affected by the pollution effect for the large wavenumbers.

Generalized plane wave discontinuous Galerkin methods for nonhomogeneous Helmholtz equations with variable wave numbers

ABSTRACT In this paper we are concerned with numerical method for nonhomogeneous Helmholtz equations with variable wave numbers. We derive generalized plane wave basis functions for three-dimensional homogeneous Helmholtz equations with variable wave numbers. Then, by combining the local spectral element method, we design a generalized plane wave discontinuous Galerkin method for the discretization of such nonhomogeneous Helmholtz equations (in both dimensions two and three dimensions). We show that the approximation solutions generated by the proposed discretization method yield error estimates with high accuracies. Numerical results indicate that the resulting approximate solutions generated by the new method possess high accuracy and verify the validity of the theoretical results.

A unified direct approach for discretizing fractional-order differentiator in delta-domain

Fractional-order differentiator is a principal component of the fractional-order controller. Discretization of fractional-order differentiator is essential to implement the fractional-order controller digitally. Discretization methods generally include indirect approach and direct approach to find the discrete-time approximation of fractional-order differentiator in the [Formula: see text]-domain as evident from the existing literature. In this paper, a direct approach is proposed for discretization of fractional-order differentiator in delta-domain instead of the conventional [Formula: see text]-domain as the delta operator unifies both analog system and digital system together at a high sampling frequency. The discretization of fractional-order differentiator is accomplished in two stages. In the first stage, the generating function is framed by reformulating delta operator using trapezoidal rule or Tustin approximation and in the next stage, the fractional-order differentiator has been approximated by expanding the generating function using continued fraction expansion method. The proposed method has been compared with two well-known direct discretization methods taken from the existing literature. Two examples are presented in this context to show the efficacy of the proposed discretization method using simulation results obtained from MATLAB.

A New K ‐Ary Crisp Decision Tree Induction with Continuous Valued Attributes

The simplicity and interpretability of decision tree induction makes it one of the more widely used machine learning methods for data classification. However, for continuous valued (real and integer) attribute data, there is room for further improvement in classification accuracy, complexity, and tree scale. We propose a new K-ary partition discretization method with no more than K−1 cut points based on Gaussian membership functions and the expected class number. A new K-ary crisp decision tree induction is also proposed for continuous valued attributes with a Gini index, combining the proposed discretization method. Experimental results and non-parametric statistical tests on 19 real-world datasets showed that the proposed algorithm outperforms four conventional approaches in terms of both classification accuracy, tree scale, and particularly tree depth. Considering the number of nodes, the proposed methods decision tree tends to be more balanced than in the other four methods. The complexity of the proposed algorithm was relatively low.

A saddle point least squares approach for primal mixed formulations of second order PDEs

We present a Saddle Point Least Squares (SPLS) method for discretizing second order elliptic problems written as primal mixed variational formulations. A stability LBB condition and a data compatibility condition at the continuous level are automatically satisfied. The proposed discretization method follows a general SPLS approach and has the advantage that a discrete inf–sup condition is automatically satisfied for standard choices of the test and trial spaces. For the proposed iterative processes a nodal basis for the trial space is not required. Efficient preconditioning techniques that involve inversion only on the test space can be considered. Stability and approximation properties for two choices of discrete spaces are investigated. Applications of the new approach include discretization of second order problems with highly oscillatory coefficient, interface problems, and higher order approximation of the flux for elliptic problems with smooth coefficients.

On the discretization and control of an SEIR epidemic model with a periodic impulsive vaccination

This paper deals with the discretization and control of an SEIR epidemic model. Such a model describes the transmission of an infectious disease among a time-varying host population. The model assumes mortality from causes related to the disease. Our study proposes a discretization method including a free-design parameter to be adjusted for guaranteeing the positivity of the resulting discrete-time model. Such a method provides a discrete-time model close to the continuous-time one without the need for the sampling period to be as small as other commonly used discretization methods require. This fact makes possible the design of impulsive vaccination control strategies with less burden of measurements and related computations if one uses the proposed instead of other discretization methods. The proposed discretization method and the impulsive vaccination strategy designed on the resulting discretized model are the main novelties of the paper. The paper includes (i) the analysis of the positivity of the obtained discrete-time SEIR model, (ii) the study of stability of the disease-free equilibrium point of a normalized version of such a discrete-time model and (iii) the existence and the attractivity of a globally asymptotically stable disease-free periodic solution under a periodic impulsive vaccination. Concretely, the exposed and infectious subpopulations asymptotically converge to zero as time tends to infinity while the normalized subpopulations of susceptible and recovered by immunization individuals oscillate in the context of such a solution. Finally, a numerical example illustrates the theoretic results.

Using a discretized measure of academic performance to approximate primary and secondary effects in inequality of educational opportunity

This study proposes an easy-to-implement approximation for primary and secondary effects in the study of inequality of educational opportunity by discretizing the measure of academic performance. Relative to the widely-used Erikson–Jonsson model, our method is not subject to the potential limitations that are associated with the parametric configurations of the normal distribution of academic performance and the model form restriction for predicting educational choice. Besides, the proposed discretization method can be used to reveal the heterogeneous effect of academic performance on the likelihood of educational transition across the spectrum of performance. Using Monte Carlo simulation and survey data collected in China, we show that our method recovers the results based on the Erikson–Jonsson approach. With another simulation, we illustrate that the Erikson–Jonsson model might produce misleading results if academic performance is not normally distributed. The discretization approach, in contrast, does not suffer from this problem.

A discretization method for predicting the equivalent elastic parameters of the graded lattice structure

In recent years, cellular materials have been widely studied and applied in aerospace and other fields due to the advantages of lightweight and multi-function. However, it is difficult to predict the equivalent elastic properties of the graded lattice structure and other non-uniform cellular materials because of the complex configuration and non-uniformity. A new discretization method for predicting the equivalent elastic parameters of the graded lattice structure is proposed based on the strain energy equivalent method and the discretization method in this paper. The graded lattice structure is discretized into lattice cells, the equivalent elastic properties are predicted by calculating the global equivalent elastic parameters with the parameters of lattice cells, and the calculation formulas are derived. After that, taking edge cube, face-centered cubic and body-centered cubic lattice as examples, the effectiveness and accuracy of the method are verified by theoretical calculation, numerical analysis, and experiment. The results show that the calculation errors of equivalent elastic parameters are between 4.5%–9.7%, and the errors can be significantly improved by reducing the graded factor. It proves that the proposed discretization method can predict the equivalent elastic parameters of the graded lattice structure effectively, and is suitable for different lattice structures.

Discretization of Nonlinear Non-affine Time Delay Systems Based on Second-order Hold

When calculating the sampled-date representation of nonlinear systems second-order hold (SOH) assumption can be applied to improving the precision of the discretization results. This paper proposes a discretization method based on Taylor series and the SOH assumption for the nonlinear systems with the time delayed non-affine input. The mathematical structure of the proposed discretization method is explored. This proposed discretization method can provide a precise and finite dimensional discretization model for the nonlinear time-delayed non-affine system by keeping the truncation order of the Taylor series. The performance of the proposed discretization method is evaluated by doing the simulation using a nonlinear system with the time-delayed non-affine input. Different input signals, time-delay values and sampling periods are considered in the simulation to investigate the proposed method. The simulation results demonstrate that the proposed method is practical and easy for time-delayed nonlinear non-affine systems. The comparison between SOH assumption with first-order hold (FOH) and zero-order hold (ZOH) assumptions is given to show the advantages of the proposed method.

A Plane-Wave Least-Squares Method for Time-Harmonic Maxwell's Equations in Absorbing Media

In this paper we discuss numerical methods for solving time-harmonic Maxwell equations in three-dimensional absorbing media. We propose a new variational problem on the space spanned by the plane-wave basis functions for the discretization of these types of equations. The variational problem is derived mathematically from a minimization problem, and the resulting stiffness matrix is Hermitian positive definite. A simple domain decomposition preconditioner is introduced for the system generated by the proposed variational formula. Numerical results indicate that the approximate solution generated by the proposed discretization method is clearly more accurate than that generated by the ultra weak variational formulation method, and the proposed preconditioner is effective.

Mixed Finite Elements for Spatial Regression with PDE Penalization

We study a class of models at the interface between statistics and numerical analysis. Specifically, we consider nonparametric regression models for the estimation of spatial fields from pointwise and noisy observations, which account for problem-specific prior information, described in terms of a partial differential equation governing the phenomenon under study. The prior information is incorporated in the model via a roughness term using a penalized regression framework. We prove the well-posedness of the estimation problem, and we resort to a mixed equal order finite element method for its discretization. Moreover, we prove the well-posedness and the optimal convergence rate of the proposed discretization method. Finally the smoothing technique is extended to the case of areal data, particularly interesting in many applications.

A Novel Discretization Method for Continuous Attributes: A Machine Learning Approach

Discretization plays a vital role where continuous attributes are transformed into discrete values for preprocessing tasks in data mining algorithms and machine learning classifiers. Most of the real data often come in mixed notations, i.e., continuous and discrete, while many machine learning algorithms require it in the form of discrete data only. In the past few decades, numerous techniques have been proposed as discretization techniques for continuous data, which are unable to provide better accuracy, understandability of models and also classifier confusion in the form of continuous data. In this paper, we propose a new discretization technique called ‘ZDisc’ based on the standard deviation normalization statistical technique for continuous attributes to generate less number of cut points. The proposed method has been compared with other state-of-the-art discretization techniques on benchmark continuous datasets. The experiment shows that the proposed discretization method performs competitive discretization schemes in terms of classifier accuracy and to minimize the classifi er confusion.

Fractional-order Chua’s system: discretization, bifurcation and chaos

In this paper we are interested in the fractional-order form of Chua’s system. A discretization process will be applied to obtain its discrete version. Fixed points and their asymptotic stability are investigated. Chaotic attractor, bifurcation and chaos for different values of the fractional-order parameter are discussed. We show that the proposed discretization method is different from other discretization methods, such as predictor-corrector and Euler methods, in the sense that our method is an approximation for the right-hand side of the system under study.