Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods

This paper discusses a general framework for the numerical solution of multi-order fractional delay differential equations (FDDEs) in noncanonical forms with irrational/rational multiple delays by the use of a spectral collocation method. In contrast to the current numerical methods for solving fractional differential equations, the proposed framework can solve multi-order FDDEs in a noncanonical form with incommensurate orders. The framework can also solve multi-order FDDEs with irrational multiple delays. Next, the framework is enhanced by the fractional Chebyshev collocation method in which a Chebyshev operation matrix is constructed for the fractional differentiation. Spectral convergence and small computational time are two other advantages of the proposed framework enhanced by the fractional Chebyshev collocation method. In addition, the convergence, error estimates, and numerical stability of the proposed framework for solving FDDEs are studied. The advantages and computational implications of the proposed framework are discussed and verified in several numerical examples.     

A scalable pairwise class interaction framework for multidimensional classification

We present a general framework for multidimensional classification that captures the pairwise interactions between class variables. The pairwise class interactions are encoded using a collection of base classifiers (Phase 1), for which the class predictions are combined in a Markov random field that is subsequently used for multidimensional inference (Phase 2); thus, the framework can be positioned between multilabel Bayesian classifiers and label transformation-based approaches. Our proposal leads to a general framework supporting a wide range of base classifiers in the first phase as well as different inference methods in the second phase. We describe the basic framework and its main properties, as well as strategies for ensuring the scalability of the framework. We include a detailed experimental evaluation based on a range of publicly available databases. Here we analyze the overall performance of the framework and we test the behavior of the different scalability strategies proposed. A comparison with other state-of-the-art multidimensional classifiers show that the proposed framework either outperforms or is competitive with the tested straw-men methods.     

Coupling ant colony systems with strong local searches

Ant colony system is a well known metaheuristic framework, and many efficient algorithms for different combinatorial optimization problems have been derived from this general framework. In this paper some directions for improving the original framework when a strong local search routine is available, are identified. In particular, some modifications able to speed up the method and make it competitive on large problem instances, on which the original framework tends to be weaker, are described. The resulting framework, called Enhanced Ant Colony System is tested on three well-known combinatorial optimization problems arising in the transportation field. Many new best known solutions are retrieved for the benchmarks available for these optimization problems.     

Measurement errors in multivariate measurement scales

Our aim is to construct a general measurement framework for analyzing the effects of measurement errors in multivariate measurement scales. We define a measurement model, which forms the core of the framework. The measurement scales in turn are often produced by methods of multivariate statistical analysis. As a central element of the framework, we introduce a new, general method of estimating the reliability of measurement scales. It is more appropriate than the classical procedures, especially in the context of multivariate analyses. The framework provides methods for various topics related to the quality of measurement, such as assessing the structural validity of the measurement model, estimating the standard errors of measurement, and correcting the predictive validity of a measurement scale for attenuation. A proper estimate of reliability is a requisite in each task. We illustrate the idea of the measurement framework with an example based on real data.     

A Python framework to create and simulate models with variable structure in common simulation environments

In modelling and simulation, model complexity increases with the complexity of real systems, often resulting in unfeasibly long simulation times. Variable-structure models, which can change their equation set during run-time, offer a solution. This article introduces an object-oriented approach that describes such models independent of a simulation environment. The problem of changing equations during run-time is addressed using a Python framework. Three tools are already integrated, namely, Dymola, OpenModelica and Matlab/Simulink. With this framework, existing models can be easily reused, and the advantages of different simulation tools can be leveraged. The framework is illustrated using a simple satellite launch example, along with instructions for how to use the framework. The main aim of the framework is to simulate complex models with a few mode changes and thus save simulation time. With two real-world examples and a scalability analysis, it is shown that the framework fulfils these requirements.     

Feasible descent algorithms for mixed complementarity problems

In this paper we consider a general algorithmic framework for solving nonlinear mixed complementarity problems. The main features of this framework are: (a) it is well-defined for an arbitrary mixed complementarity problem, (b) it generates only feasible iterates, (c) it has a strong global convergence theory, and (d) it is locally fast convergent under standard regularity assumptions. This framework is applied to the PATH solver in order to show viability of the approach. Numerical results for an appropriate modification of the PATH solver indicate that this framework leads to substantial computational improvements. Received April 9, 1998 / Revised version received November 23, 1998?Published online March 16, 1999     

Fireworks algorithm framework for Big Data optimization

This paper presents a novel optimization framework based on the Fireworks Algorithm for Big Data Optimization problems. Indeed, the proposed framework is composed of two optimization algorithms. A single objective Fireworks Algorithm and a multi-objective Fireworks Algorithm are proposed for solving the Big Optimization of Signals problem “Big-OPT” which belongs to the Big Data Optimization problems class. The single objective Fireworks Algorithm adopts a modified search mechanism to ensure rapidity and preserve the explorative capacities of the basic Fireworks Algorithm. Afterward, the algorithm is extended to handle multi-objective optimization of Big-OPT with a supplementary special sparks phase and a novel strategy for next generation selection. To validate the performance of the framework, extensive tests on six EEG datasets are performed. The framework is also compared with several approaches from recent state of the art. The study concludes the competitive performance of the proposed framework in comparison with the other techniques reported in this paper.     

Students’ conceptions of mathematics as sensible: Towards the SCOMAS framework

In this article I describe the development of a framework for considering students’ conceptions about the sensible nature of mathematics. I begin by using extant literature on conceptions of mathematics to develop a framework of action-oriented indicators that students’ conceive of mathematics as sensible. I then use classroom data to modify and illustrate the framework. The result is a coding framework, grounded in the literature, which can be used to assess the enacted conceptions of mathematics as sensible of a group of students. This work also provides a conceptual framework, grounded in classroom data, of the dimensions of these conceptions.     

GENERALIZED MATRIX MULTISPLITTING RELAXATION METHODS AND THEIR CONVERGENCE

In this paper, we set up a general framework of parallel matrix mullisplitting relaxation methods for solving large scale system of linear equations. We investigate the convergence properties of this framework and give several sufficient conditions ensuring it to converge as well as diverge. At last, we conclude a necessary and sufficient condition for the convergence of this framework when the coefficient matrix is an L-matrix.     

Online surrogate multiobjective optimization algorithm for contaminated groundwater remediation designs

This paper proposes an online surrogate model-assisted multiobjective optimization framework to identify optimal remediation strategies for groundwater contaminated with dense non-aqueous phase liquids. The optimization involves three objectives: minimizing the remediation cost and duration and maximizing the contamination removal rate. The proposed framework adopts a multiobjective feasibility-enhanced particle swarm optimization algorithm to solve the optimization model and uses an online surrogate model as a substitute for the time-consuming multiphase flow model for calculating contamination removal rates during the optimization process. The resulting approach allows decision makers to find a balance among the remediation cost, remediation duration and contamination removal rate for remediating contaminated groundwater. The new algorithm is compared with the nondominated sorting genetic algorithm II, which is an extensively applied and well-known algorithm. The results show that the Pareto solutions obtained by the new algorithm have greater diversity and stability than those obtained by the nondominated sorting genetic algorithm II, indicating that the new algorithm is more applicable than the nondominated sorting genetic algorithm II for optimizing remediation strategies for contaminated groundwater. Additionally, the surrogate model and Pareto optimal set obtained by the proposed framework are compared with those of the offline surrogate model-assisted multiobjective optimization framework. The results indicate that the surrogate model accuracy and Pareto front achieved by the proposed framework outperform those of the offline surrogate model-assisted optimization framework. Thus, we conclude that the proposed framework can effectively enhance the surrogate model accuracy and further extend the comprehensive performance of Pareto solutions.     

Unsupervised construction of fuzzy measures through self-organizing feature maps and its application in color image segmentation

The paper presents a framework for the segmentation of multi-dimensional images, e.g., color, satellite, multi-sensory images, based on the employment of the fuzzy integral, which undertakes the classification of the input features. The framework makes use of a self-organizing feature map, whereby the coefficients of the fuzzy measure are determined. This process is unsupervised and therefore constitutes one of the main contributions of the paper.The performance of the framework is shown by successfully realizing the segmentation of color images in two different applications. First, the features of the framework and its parameterization are analyzed by segmenting different images used as benchmark in image processing. Finally, the framework is applied in the segmentation of different images taken under difficult illumination conditions. The images serve the development of an automated cashier system, where the weak segmentation constitutes the first step for the identification of different market items. The presented framework succeeds in the segmentation of all these color images.     

An exploratory framework for handling the complexity of mathematical problem posing in small groups

The paper introduces an exploratory framework for handling the complexity of students’ mathematical problem posing in small groups. The framework integrates four facets known from past research: task organization, students’ knowledge base, problem-posing heuristics and schemes, and group dynamics and interactions. In addition, it contains a new facet, individual considerations of aptness, which accounts for the posers’ comprehensions of implicit requirements of a problem-posing task and reflects their assumptions about the relative importance of these requirements. The framework is first argued theoretically. The framework at work is illustrated by its application to a situation, in which two groups of high-school students with similar background were given the same problem-posing task, but acted very differently. The novelty and usefulness of the framework is attributed to its three main features: it supports fine-grained analysis of directly observed problem-posing processes, it has a confluence nature, it attempts to account for hidden mechanisms involved in students’ decision making while posing problems.     

连续化方法求解变分不等式问题

利用变分不等式问题的KKT条件,给出了连续化方法求解变分不等式问题的一般框架,该框架包含了现存的几种连续方法;并给出一种求解的基本算法,证明了基本算法的可行性及算法的收敛性;最后用数值试验验证了算法的稳定性和有效性。     

Measures of uncertainty for imprecise probabilities: An axiomatic approach

The aim of this paper is to formalize, within a broad range of theories of imprecise probabilities, the notion of a total, aggregate measure of uncertainty and its various disaggregations into measures of nonspecificity and conflict. As a framework for facilitating this aim, we introduce a system of well-justified axiomatic requirements for such measures. It is shown that these requirements can be equivalently defined for belief functions and credal sets. Some important consequences are then derived within this framework, which clarify the role of various uncertainty measures proposed in the literature. Moreover, some well-defined new open problems for future research also emerge from the introduced framework.     

Evolving timetabling heuristics using a grammar-based genetic programming hyper-heuristic framework

This paper introduces a Grammar-based Genetic Programming Hyper-Heuristic framework (GPHH) for evolving constructive heuristics for timetabling. In this application GP is used as an online learning method which evolves heuristics while solving the problem. In other words, the system keeps on evolving heuristics for a problem instance until a good solution is found. The framework is tested on some of the most widely used benchmarks in the field of exam timetabling and compared with the best state-of-the-art approaches. Results show that the framework is very competitive with other constructive techniques, and did outperform other hyper-heuristic frameworks on many occasions.     

Building a local conceptual framework for epistemic actions in a modelling environment with experiments

A local conceptual framework for the construction of mathematical knowledge in learning environments with experiments is developed. For this purpose, the mathematical modelling framework and the epistemic action model for abstraction in context are used simultaneously. In a case study, experiments of pre-service teachers with the motion of a ball on a circular billiard table are analysed within the local conceptual framework. The role of the experiments for epistemic actions of mathematical abstractions is described. In the case study, two different types of students’ approaches to the role of experiments can be distinguished.     

The quality of bank forecasts: The dollar-pound exchange rate, 1990–1993

A general framework for examining the quality of currency forecasts is described. This framework incorporates existing accuracy measures and modifies them to give components of accuracy that are appropriate for statistically derived and judgementally based forecasts. The framework is described and applied to one week ahead US$/UK£ forecasts from three major banks over a three year period between 1990 and 1993. The results suggest that, while overall forecast performance was poor, some aspects of the predictions could still be useful in a practical setting.     

Mixtures of truncated basis functions

In this paper we propose a framework, called mixtures of truncated basis functions (MoTBFs), for representing general hybrid Bayesian networks. The proposed framework generalizes both the mixture of truncated exponentials (MTEs) framework and the Mixture of Polynomials (MoPs) framework. Similar to MTEs and MoPs, MoTBFs are defined so that the potentials are closed under combination and marginalization, which ensures that inference in MoTBF networks can be performed efficiently using the Shafer-Shenoy architecture.Based on a generalized Fourier series approximation, we devise a method for efficiently approximating an arbitrary density function using the MoTBF framework. The translation method is more flexible than existing MTE or MoP-based methods, and it supports an online/anytime tradeoff between the accuracy and the complexity of the approximation. Experimental results show that the approximations obtained are either comparable or significantly better than the approximations obtained using existing methods.     

A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework on unstructured meshes

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes on two-dimensional unstructured triangular meshes [39], to the correction procedure via reconstruction (CPR) framework for solving nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes with straight or curved edges. This is an extension of our earlier work [4] in which the WENO limiter was designed for the CPR framework on regular meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper uses information only from the target cell and its immediate neighbors. Hence, it is particularly simple to implement and will not harm the conservativeness and compactness of the CPR framework. Since the CPR framework with this WENO limiter does not in general satisfy the positivity preserving property, we also extend the positivity-preserving limiters [36], [33] to the CPR framework. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of this procedure.