Mathematical and Computational Applications

14 pages, 293 KiB

Open AccessArticle

Dual Methods for Optimal Allocation of Telecommunication Network Resources with Several Classes of Users

by Igor Konnov, Aleksey Kashuba and Erkki Laitinen

Math. Comput. Appl. 2018, 23(2), 31; https://doi.org/10.3390/mca23020031 - 17 Jun 2018

Cited by 2 | Viewed by 3231

We consider a general problem of optimal allocation of limited resources in a wireless telecommunication network. The network users are divided into several different groups (or classes), which correspond to different levels of service. The network manager must satisfy these different users’ requirements. [...] Read more.

We consider a general problem of optimal allocation of limited resources in a wireless telecommunication network. The network users are divided into several different groups (or classes), which correspond to different levels of service. The network manager must satisfy these different users’ requirements. This approach leads to a convex optimization problem with balance and capacity constraints. We present several decomposition type methods to find a solution to this problem, which exploit its special features. We suggest applying first the dual Lagrangian method with respect to the total capacity constraint, which gives the one-dimensional dual problem. However, calculation of the value of the dual cost function requires solving several optimization problems. Our methods differ in approaches for solving these auxiliary problems. We consider three basic methods: Dual Multi Layer (DML), Conditional Gradient Dual Multilayer (CGDM) and Bisection (BS). Besides these methods we consider their modifications adjusted to different kind of cost functions. Our comparison of the performance of the suggested methods on several series of test problems show satisfactory convergence. Nevertheless, proper decomposition techniques enhance the convergence essentially. Full article

(This article belongs to the Special Issue Applied Modern Mathematics in Complex Networks)

33 pages, 7977 KiB

Open AccessFeature PaperReview

A Survey of Recent Trends in Multiobjective Optimal Control—Surrogate Models, Feedback Control and Objective Reduction

by Sebastian Peitz and Michael Dellnitz

Math. Comput. Appl. 2018, 23(2), 30; https://doi.org/10.3390/mca23020030 - 1 Jun 2018

Cited by 59 | Viewed by 9982

Abstract

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. [...] Read more.

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control, which results in new challenges such as expensive models or real-time applicability. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in accelerating multiobjective optimal control for complex problems where either PDE constraints are present or where a feedback behavior has to be achieved. In the first case, surrogate models yield significant speed-ups. Besides classical meta-modeling techniques for multiobjective optimization, a promising alternative for control problems is to introduce a surrogate model for the system dynamics. In the case of real-time requirements, various promising model predictive control approaches have been proposed, using either fast online solvers or offline-online decomposition. We also briefly comment on dimension reduction in many-objective optimization problems as another technique for reducing the numerical effort. Full article

(This article belongs to the Collection Numerical Optimization Reviews)

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18 pages, 3167 KiB

Open AccessFeature PaperArticle

4D Remeshing Using a Space-Time Finite Element Method for Elastodynamics Problems

by Serge Dumont, Franck Jourdan and Tarik Madani

Math. Comput. Appl. 2018, 23(2), 29; https://doi.org/10.3390/mca23020029 - 25 May 2018

Cited by 3 | Viewed by 4242

Abstract

In this article, a Space-Time Finite Element Method (STFEM) is proposed for the resolution of mechanical problems involving three dimensions in space and one in time. Special attention will be paid to the non-separation of the space and time variables because this kind [...] Read more.

In this article, a Space-Time Finite Element Method (STFEM) is proposed for the resolution of mechanical problems involving three dimensions in space and one in time. Special attention will be paid to the non-separation of the space and time variables because this kind of interpolation is well suited to mesh adaptation. For that purpose, we have developed a technique of 4D mesh generation adapted to space-time remeshing. A difficulty arose in the representation of 4D finite elements and meshes. This original technique does not require coarse-to-fine and fine-to-coarse mesh-to-mesh transfer operators and does not increase the size of the linear systems to be solved, compared to traditional finite element methods. Space-time meshes are composed of simplex finite elements. Computations are carried out in the context of the continuous Galerkin method. We have tested the method on a linearized elastodynamics problem. Our technique of mesh adaptation was validated on elementary examples and applied to a problem of mobile loading. The convergence and stability of the method are studied and compared with existing methods. This work is a first implementation of 4D space-time remeshing. A stability criterion for the method is established, as well as a convergence rate of about two. Using simplex elements, it is possible to develop a technique of mesh adaptation able to follow a mobile loading zone. Full article

(This article belongs to the Special Issue Mesh-Free and Finite Element-Based Methods for Structural Mechanics Applications)

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19 pages, 3142 KiB

Open AccessArticle

The Impact of the Implementation Cost of Replication in Data Grid Job Scheduling

by Babar Nazir, Faiza Ishaq, Shahaboddin Shamshirband and Anthony T. Chronopoulos

Math. Comput. Appl. 2018, 23(2), 28; https://doi.org/10.3390/mca23020028 - 25 May 2018

Cited by 6 | Viewed by 3576

Abstract

Data Grids deal with geographically-distributed large-scale data-intensive applications. Schemes scheduled for data grids attempt to not only improve data access time, but also aim to improve the ratio of data availability to a node, where the data requests are generated. Data replication techniques [...] Read more.

Data Grids deal with geographically-distributed large-scale data-intensive applications. Schemes scheduled for data grids attempt to not only improve data access time, but also aim to improve the ratio of data availability to a node, where the data requests are generated. Data replication techniques manage large data by storing a number of data files efficiently. In this paper, we propose centralized dynamic scheduling strategy-replica placement strategies (CDSS-RPS). CDSS-RPS schedule the data and task so that it minimizes the implementation cost and data transfer time. CDSS-RPS consists of two algorithms, namely (a) centralized dynamic scheduling (CDS) and (b) replica placement strategy (RPS). CDS considers the computing capacity of a node and finds an appropriate location for the job. RPS attempts to improve file access time by using replication on the basis of number of accesses, storage capacity of a computing node, and response time of a requested file. Extensive simulations are carried out to demonstrate the effectiveness of the proposed strategy. Simulation results demonstrate that the replication and scheduling strategies improve the implementation cost and average access time significantly. Full article

(This article belongs to the Special Issue Applied Modern Mathematics in Complex Networks)

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21 pages, 391 KiB

Open AccessArticle

Nonlinear Elimination of Drugs in One-Compartment Pharmacokinetic Models: Nonstandard Finite Difference Approach for Various Routes of Administration

by Oluwaseun Egbelowo

Math. Comput. Appl. 2018, 23(2), 27; https://doi.org/10.3390/mca23020027 - 23 May 2018

Cited by 10 | Viewed by 8668

Abstract

The motivation for this study is to introduce and motivate the use of nonstandard finite difference (NSFD) schemes, capable of solving one-compartment pharmacokinetic models. These models are modeled by both linear and nonlinear ordinary differential equations. “Exact” finite difference schemes, which are a [...] Read more.

The motivation for this study is to introduce and motivate the use of nonstandard finite difference (NSFD) schemes, capable of solving one-compartment pharmacokinetic models. These models are modeled by both linear and nonlinear ordinary differential equations. “Exact” finite difference schemes, which are a special NSFD, are provided for the linear models while we apply the NSFD rules, based on Mickens’ idea of transferring nonlinear models into discrete schemes. The method used was compared with other established methods to verify its efficiency and accuracy. One-compartment pharmacokinetic models are considered for different routes of administration: I.V. bolus injection, I.V. bolus infusion and extravascular administration. Full article

(This article belongs to the Special Issue Computational Methods in Interdisciplinary Applications of Nonlinear Dynamics)

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14 pages, 429 KiB

Open AccessArticle

A New Analytical Procedure to Solve Two Phase Flow in Tubes

by Terry E. Moschandreou

Math. Comput. Appl. 2018, 23(2), 26; https://doi.org/10.3390/mca23020026 - 23 May 2018

Cited by 1 | Viewed by 4914

Abstract

A new formulation for a proposed solution to the 3D Navier-Stokes Equations in cylindrical co-ordinates coupled to the continuity and level set convection equation is presented in terms of an additive solution of the three principle directions in the radial, azimuthal and z [...] Read more.

A new formulation for a proposed solution to the 3D Navier-Stokes Equations in cylindrical co-ordinates coupled to the continuity and level set convection equation is presented in terms of an additive solution of the three principle directions in the radial, azimuthal and z directions of flow and a connection between the level set function and composite velocity vector for the additive solution is shown. For the case of a vertical tube configuration with small inclination angle, results are obtained for the level set function defining the interface in both the radial and azimuthal directions. It is found that the curvature dependent part of the problem alone induces sinusoidal azimuthal interfacial waves whereas when the curvature together with the equation for the composite velocity is considered oscillating radial interfacial waves occur. The implications of two extremes indicate the importance of looking at the full equations including curvature. Full article

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13 pages, 860 KiB

Open AccessArticle

The Construction of a Model-Robust IV-Optimal Mixture Designs Using a Genetic Algorithm

by Wanida Limmun, Boonorm Chomtee and John J. Borkowski

Math. Comput. Appl. 2018, 23(2), 25; https://doi.org/10.3390/mca23020025 - 17 May 2018

Cited by 3 | Viewed by 3860

Abstract

Among the numerous alphabetical optimality criteria is the IV-criterion that is focused on prediction variance. We propose a new criterion, called the weighted IV-optimality. It is similar to IV-optimality, because the researcher must first specify a model. However, unlike IV-optimality, a suite of [...] Read more.

Among the numerous alphabetical optimality criteria is the IV-criterion that is focused on prediction variance. We propose a new criterion, called the weighted IV-optimality. It is similar to IV-optimality, because the researcher must first specify a model. However, unlike IV-optimality, a suite of “reduced” models is also proposed if the original model is misspecified via over-parameterization. In this research, weighted IV-optimality is applied to mixture experiments with a set of prior weights assigned to the potential mixture models of interest. To address the issue of implementation, a genetic algorithm was developed to generate weighted IV-optimal mixture designs that are robust across multiple models. In our examples, we assign models with p parameters to have equal weights, but weights will vary based on varying p. Fraction-of-design-space (FDS) plots are used to compare the performance of an experimental design in terms of the prediction variance properties. An illustrating example is presented. The result shows that the GA-generated designs studied are robust across a set of potential mixture models. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization)

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17 pages, 3299 KiB

Open AccessFeature PaperArticle

Integration of Direction Fields with Standard Options in Finite Element Programs

by Herbert Moldenhauer

Math. Comput. Appl. 2018, 23(2), 24; https://doi.org/10.3390/mca23020024 - 7 May 2018

Cited by 6 | Viewed by 5216

Abstract

The two-dimensional differential equation y’ = f(x,y) can be interpreted as a direction field. Commercial Finite Element (FE) programs can be used for this integration task without additional programming, provided that these programs have options for the calculation of orthotropic heat conduction problems. [...] Read more.

The two-dimensional differential equation y’ = f(x,y) can be interpreted as a direction field. Commercial Finite Element (FE) programs can be used for this integration task without additional programming, provided that these programs have options for the calculation of orthotropic heat conduction problems. The differential equation to be integrated with arbitrary boundaries is idealized as an FE model with thermal 2D elements. Its orthotropic thermal conductivities are specified as k₁ = 1 and k₂ = 0. In doing so, k₁ is parallel to y´, and k₂ is oriented perpendicular to this. For this extreme case, it is shown that the isotherms are identical to the solution of y’ = f(x,y). The direction fields, for example, can be velocity vectors in fluid mechanics or principal stress directions in structural mechanics. In the case of the latter, possibilities for application in the construction of fiber-reinforced plastics (FRP) arise, since fiber courses, which follow the local principal stress directions, make use of the superior stiffness and strength of the fibers. Full article

(This article belongs to the Special Issue Mesh-Free and Finite Element-Based Methods for Structural Mechanics Applications)

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14 pages, 2541 KiB

Open AccessFeature PaperArticle

A Simple Spectral Observer

by Lizeth Torres, Javier Jiménez-Cabas, José Francisco Gómez-Aguilar and Pablo Pérez-Alcazar

Math. Comput. Appl. 2018, 23(2), 23; https://doi.org/10.3390/mca23020023 - 1 May 2018

Cited by 8 | Viewed by 4451

Abstract

The principal aim of a spectral observer is twofold: the reconstruction of a signal of time via state estimation and the decomposition of such a signal into the frequencies that make it up. A spectral observer can be catalogued as an online algorithm [...] Read more.

The principal aim of a spectral observer is twofold: the reconstruction of a signal of time via state estimation and the decomposition of such a signal into the frequencies that make it up. A spectral observer can be catalogued as an online algorithm for time-frequency analysis because is a method that can compute on the fly the Fourier Transform (FT) of a signal, without having the entire signal available from the start. In this regard, this paper presents a novel spectral observer with an adjustable constant gain for reconstructing a given signal by means of the recursive identification of the coefficients of a Fourier series. The reconstruction or estimation of a signal in the context of this work means to find the coefficients of a linear combination of sines a cosines that fits a signal such that it can be reproduced. The design procedure of the spectral observer is presented along with the following applications: (1) the reconstruction of a simple periodical signal, (2) the approximation of both a square and a triangular signal, (3) the edge detection in signals by using the Fourier coefficients, (4) the fitting of the historical Bitcoin market data from 1 December 2014 to 8 January 2018 and (5) the estimation of a input force acting upon a Duffing oscillator. To round out this paper, we present a detailed discussion about the results of the applications as well as a comparative analysis of the proposed spectral observer vis-à-vis the Short Time Fourier Transform (STFT), which is a well-known method for time-frequency analysis. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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15 pages, 811 KiB

Open AccessFeature PaperArticle

Solution of Optimal Harvesting Problem by Finite Difference Approximations of Size-Structured Population Model

by Johanna Pyy, Anssi Ahtikoski, Alexander Lapin and Erkki Laitinen

Math. Comput. Appl. 2018, 23(2), 22; https://doi.org/10.3390/mca23020022 - 26 Apr 2018

Cited by 3 | Viewed by 3338

Abstract

We solve numerically a forest management optimization problem governed by a nonlinear partial differential equation (PDE), which is a size-structured population model. The formulated problem is supplemented with a natural constraint for a solution to be non-negative. PDE is approximated by an explicit [...] Read more.

We solve numerically a forest management optimization problem governed by a nonlinear partial differential equation (PDE), which is a size-structured population model. The formulated problem is supplemented with a natural constraint for a solution to be non-negative. PDE is approximated by an explicit or implicit in time finite difference scheme, whereas the cost function is taken from the very beginning in the finite-dimensional form used in practice. We prove the stability of the constructed nonlinear finite difference schemes on the set of non-negative vectors and the solvability of the formulated discrete optimal control problems. The gradient information is derived by constructing discrete adjoint state equations. The projected gradient method is used for finding the extremal points. The results of numerical testing for several real problems show good agreement with the known results and confirm the theoretical statements. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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28 pages, 1824 KiB

Open AccessArticle

Optimal Control Analysis of a Mathematical Model for Breast Cancer

by Segun Isaac Oke, Maba Boniface Matadi and Sibusiso Southwell Xulu

Math. Comput. Appl. 2018, 23(2), 21; https://doi.org/10.3390/mca23020021 - 24 Apr 2018

Cited by 54 | Viewed by 7890

Abstract

In this paper, a mathematical model of breast cancer governed by a system of ordinary differential equations in the presence of chemotherapy treatment and ketogenic diet is discussed. Several comprehensive mathematical analyses were carried out using a variety of analytical methods to study [...] Read more.

In this paper, a mathematical model of breast cancer governed by a system of ordinary differential equations in the presence of chemotherapy treatment and ketogenic diet is discussed. Several comprehensive mathematical analyses were carried out using a variety of analytical methods to study the stability of the breast cancer model. Also, sufficient conditions on parameter values to ensure cancer persistence in the absence of anti-cancer drugs, ketogenic diet, and cancer emission when anti-cancer drugs, immune-booster, and ketogenic diet are included were established. Furthermore, optimal control theory is applied to discover the optimal drug adjustment as an input control of the system therapies in order to minimize the number of cancerous cells by considering different controlled combinations of administering the chemotherapy agent and ketogenic diet using the popular Pontryagin’s maximum principle. Numerical simulations are presented to validate our theoretical results. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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18 pages, 4630 KiB

Open AccessArticle

Impact of Thermal Radiation and Heat Source/Sink on Eyring–Powell Fluid Flow over an Unsteady Oscillatory Porous Stretching Surface

by Abdullah Dawar, Zahir Shah, Muhammad Idrees, Waris Khan, Saeed Islam and Taza Gul

Math. Comput. Appl. 2018, 23(2), 20; https://doi.org/10.3390/mca23020020 - 9 Apr 2018

Cited by 35 | Viewed by 4577

Abstract

The main intention of this article is to examine the heat transmission of the flow of Eyring–Powell fluid over an unstable oscillatory porous stretching surface. The effect of thermal radiation on the fluid flow is investigated, where the flow is actuated by the [...] Read more.

The main intention of this article is to examine the heat transmission of the flow of Eyring–Powell fluid over an unstable oscillatory porous stretching surface. The effect of thermal radiation on the fluid flow is investigated, where the flow is actuated by the unbounded flexible surface which is extended occasionally to and fro on its plane. The rudimentary leading equations are changed to differential equations through the use of applicable similarity variables. An optimal and numerical approach was used to find the solution to the modeled problems. The convergence of the homotopy analysis method (HAM) is shown numerically. The homotopy analysis method predictions of the structures formed are in close agreement with the obtained results from the numerical method. Comparisons between HAM and numerical methods are shown graphically as well as numerically. The convergence of this method is shown numerically. The impacts of the skin friction and heat flux are shown through a table. The influence of the porosity, oscillation, thermal radiation, and heat absorption/generation are the main focus of this work. The consequences of emerging parameters are demonstrated through graphs. Full article

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13 pages, 775 KiB

Open AccessFeature PaperArticle

How Am I Driving? Using Genetic Programming to Generate Scoring Functions for Urban Driving Behavior

by Roberto López, Luis Carlos González Gurrola, Leonardo Trujillo, Olanda Prieto, Graciela Ramírez, Antonio Posada, Perla Juárez-Smith and Leticia Méndez

Math. Comput. Appl. 2018, 23(2), 19; https://doi.org/10.3390/mca23020019 - 3 Apr 2018

Cited by 4 | Viewed by 4748

Abstract

Road traffic injuries are a serious concern in emerging economies. Their death toll and economic impact are shocking, with 9 out of 10 deaths occurring in low or middle-income countries; and road traffic crashes representing 3% of their gross domestic product. One way [...] Read more.

Road traffic injuries are a serious concern in emerging economies. Their death toll and economic impact are shocking, with 9 out of 10 deaths occurring in low or middle-income countries; and road traffic crashes representing 3% of their gross domestic product. One way to mitigate these issues is to develop technology to effectively assist the driver, perhaps making him more aware about how her (his) decisions influence safety. Following this idea, in this paper we evaluate computational models that can score the behavior of a driver based on a risky-safety scale. Potential applications of these models include car rental agencies, insurance companies or transportation service providers. In a previous work, we showed that Genetic Programming (GP) was a successful methodology to evolve mathematical functions with the ability to learn how people subjectively score a road trip. The input to this model was a vector of frequencies of risky maneuvers, which were supposed to be detected in a sensor layer. Moreover, GP was shown, even with statistical significance, to be better than six other Machine Learning strategies, including Neural Networks, Support Vector Regression and a Fuzzy Inference system, among others. A pending task, since then, was to evaluate if a more detailed comparison of different strategies based on GP could improve upon the best GP model. In this work, we evaluate, side by side, scoring functions evolved by three different variants of GP. In the end, the results suggest that two of these strategies are very competitive in terms of accuracy and simplicity, both generating models that could be implemented in current technology that seeks to assist the driver in real-world scenarios. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization)

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8 pages, 286 KiB

Open AccessArticle

A Family of 5-Point Nonlinear Ternary Interpolating Subdivision Schemes with C² Smoothness

by Muhammad Aslam

Math. Comput. Appl. 2018, 23(2), 18; https://doi.org/10.3390/mca23020018 - 23 Mar 2018

Cited by 2 | Viewed by 2810

Abstract

The occurrence of the Gibbs phenomenon near irregular initial data points is a widely known fact in curve generation by interpolating subdivision schemes. In this article, we propose a family of 5-point nonlinear ternary interpolating subdivision schemes. We provide the convergence analysis and [...] Read more.

The occurrence of the Gibbs phenomenon near irregular initial data points is a widely known fact in curve generation by interpolating subdivision schemes. In this article, we propose a family of 5-point nonlinear ternary interpolating subdivision schemes. We provide the convergence analysis and prove that this family of subdivision schemes is

C^{2}

continuous. Numerical results are presented to show that nonlinear schemes reduce the Gibbs phenomenon significantly while keeping the same order of smoothness. Full article

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15 pages, 1357 KiB

Open AccessArticle

An Improved Digital Signature Protocol to Multi-User Broadcast Authentication Based on Elliptic Curve Cryptography in Wireless Sensor Networks (WSNs)

by Hamed Bashirpour, Saman Bashirpour, Shahaboddin Shamshirband and Anthony T. Chronopoulos

Math. Comput. Appl. 2018, 23(2), 17; https://doi.org/10.3390/mca23020017 - 21 Mar 2018

Cited by 13 | Viewed by 5095

Abstract

In wireless sensor networks (WSNs), users can use broadcast authentication mechanisms to connect to the target network and disseminate their messages within the network. Since data transfer for sensor networks is wireless, as a result, attackers can easily eavesdrop deployed sensor nodes and [...] Read more.

In wireless sensor networks (WSNs), users can use broadcast authentication mechanisms to connect to the target network and disseminate their messages within the network. Since data transfer for sensor networks is wireless, as a result, attackers can easily eavesdrop deployed sensor nodes and the data sent between them or modify the content of eavesdropped data and inject false data into the sensor network. Hence, the implementation of the message authentication mechanisms (in order to avoid changes and injecting messages into the network) of wireless sensor networks is essential. In this paper, we present an improved protocol based on elliptic curve cryptography (ECC) to accelerate authentication of multi-user message broadcasting. In comparison with previous ECC-based schemes, complexity and computational overhead of proposed scheme is significantly decreased. Also, the proposed scheme supports user anonymity, which is an important property in broadcast authentication schemes for WSNs to preserve user privacy and user untracking. Full article

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Math. Comput. Appl., Volume 23, Issue 2 (June 2018) – 15 articles

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