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Math. Comput. Appl., Volume 26, Issue 1 (March 2021) – 24 articles

Cover Story (view full-size image): Optimization algorithms demand a few thousand evaluations to arrive at a reasonably good solution, often requiring a few days of computational time. For solving problems with multiple conflicting objectives, the overall effort is even more. Ten surrogate modeling methods are suggested by independent or combined objective and constraint modeling. An adaptive switching of ten methods is able to solve two to five objective constrained problems faster and more accurately than some existing methods. View this paper
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Article
Application of Data Assimilation and the Relationship between ENSO and Precipitation
Math. Comput. Appl. 2021, 26(1), 24; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010024 - 12 Mar 2021
Viewed by 417
Abstract
Climate change in Thailand is related to the El Niño and Southern Oscillation (ENSO) phenomenon, in particular drought and heavy precipitation. The data assimilation method is used to improve the accuracy of the Ensemble Intermediate Coupled Model (EICM) that simulates the sea surface [...] Read more.
Climate change in Thailand is related to the El Niño and Southern Oscillation (ENSO) phenomenon, in particular drought and heavy precipitation. The data assimilation method is used to improve the accuracy of the Ensemble Intermediate Coupled Model (EICM) that simulates the sea surface temperature (SST). The four-dimensional variational (4D-Var) and three-dimensional variational (3D-Var) schemes have been used for data assimilation purposes. The simulation was performed by the model with and without data assimilation from satellite data in 2011. The result shows that the model with data assimilation is better than the model without data assimilation. The 4D-Var scheme is the best method, with a Root Mean Square Error (RMSE) of 0.492 and a Correlation Coefficient of 0.684. The relationship between precipitation in Thailand and the ENSO area in Niño 3.4 was consistent for seven months, with a correlation coefficient of −0.882. Full article
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Article
Finite Element Analysis of Laminar Heat Transfer within an Axial-Flux Permanent Magnet Machine
Math. Comput. Appl. 2021, 26(1), 23; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010023 - 10 Mar 2021
Viewed by 556
Abstract
Axial-Flux Permanent Magnet (AFPM) machines have gained popularity over the past few years due to their compact design. Their application can be found, for example, in the automotive and medical sectors. For typically considered materials, excessive heat can be generated, causing possible irreversible [...] Read more.
Axial-Flux Permanent Magnet (AFPM) machines have gained popularity over the past few years due to their compact design. Their application can be found, for example, in the automotive and medical sectors. For typically considered materials, excessive heat can be generated, causing possible irreversible damage to the magnets, bonding, or other structural parts. In order to optimize cooling, knowledge of the flow and the consequent temperature distribution is required. This paper discusses the flow types and heat transfer present inside a typical AFPM machine. An Isogeometric Analysis (IGA) laminar-energy model is developed using the Nutils open-source Python package. The developed analysis tool is used to study the effects of various important design parameters, such as the air-inlet, the gap-length, and the rotation speed on the heat transfer in an AFPM machine. It is observed that the convective heat transfer at the stator core is negatively affected by adding an air-inlet. However, the heat dissipation of the entire stator improves as convective heat transfer occurs within the air-inlet. Full article
(This article belongs to the Special Issue Advances in Computational Fluid Dynamics and Heat & Mass Transfer)
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Article
A Non-Standard Finite Difference Scheme for Magneto-Hydro Dynamics Boundary Layer Flows of an Incompressible Fluid Past a Flat Plate
Math. Comput. Appl. 2021, 26(1), 22; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010022 - 09 Mar 2021
Viewed by 436
Abstract
This paper deals with a non-standard implicit finite difference scheme that is defined on a quasi-uniform mesh for approximate solutions of the Magneto-Hydro Dynamics (MHD) boundary layer flow of an incompressible fluid past a flat plate for a wide range of the magnetic [...] Read more.
This paper deals with a non-standard implicit finite difference scheme that is defined on a quasi-uniform mesh for approximate solutions of the Magneto-Hydro Dynamics (MHD) boundary layer flow of an incompressible fluid past a flat plate for a wide range of the magnetic parameter. The proposed approach allows imposing the given boundary conditions at infinity exactly. We show how to improve the obtained numerical results via a mesh refinement and a Richardson extrapolation. The obtained numerical results are favourably compared with those available in the literature. Full article
(This article belongs to the Special Issue Advances in Computational Fluid Dynamics and Heat & Mass Transfer)
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Article
Robust H Loop Shaping Controller Synthesis for SISO Systems by Complex Modular Functions
Math. Comput. Appl. 2021, 26(1), 21; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010021 - 03 Mar 2021
Viewed by 394
Abstract
In this paper, a loop shaping controller design methodology for single input and a single output (SISO) system is proposed. The theoretical background for this approach is based on complex elliptic functions which allow a flexible design of a SISO controller considering that [...] Read more.
In this paper, a loop shaping controller design methodology for single input and a single output (SISO) system is proposed. The theoretical background for this approach is based on complex elliptic functions which allow a flexible design of a SISO controller considering that elliptic functions have a double periodicity. The gain and phase margins of the closed-loop system can be selected appropriately with this new loop shaping design procedure. The loop shaping design methodology consists of implementing suitable filters to obtain a desired frequency response of the closed-loop system by selecting appropriate poles and zeros by the Abel theorem that are fundamental in the theory of the elliptic functions. The elliptic function properties are implemented to facilitate the loop shaping controller design along with their fundamental background and contributions from the complex analysis that are very useful in the automatic control field. Finally, apart from the filter design, a PID controller loop shaping synthesis is proposed implementing a similar design procedure as the first part of this study. Full article
(This article belongs to the Special Issue Advances of Modern Control Systems and Robotic Applications)
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Editorial
Advanced Mathematics and Computational Applications in Control Systems Engineering
Math. Comput. Appl. 2021, 26(1), 20; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010020 - 03 Mar 2021
Viewed by 439
Abstract
Control-systems engineering is a multidisciplinary subject that applies automatic-control theory to design systems with desired behaviors in control environments [...] Full article
Article
A Framework for Analysis and Prediction of Operational Risk Stress
Math. Comput. Appl. 2021, 26(1), 19; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010019 - 24 Feb 2021
Cited by 1 | Viewed by 489
Abstract
A model for financial stress testing and stability analysis is presented. Given operational risk loss data within a time window, short-term projections are made using Loess fits to sequences of lognormal parameters. The projections can be scaled by a sequence of risk factors, [...] Read more.
A model for financial stress testing and stability analysis is presented. Given operational risk loss data within a time window, short-term projections are made using Loess fits to sequences of lognormal parameters. The projections can be scaled by a sequence of risk factors, derived from economic data in response to international regulatory requirements. Historic and projected loss data are combined using a lengthy nonlinear algorithm to calculate a capital reserve for the upcoming year. The model is embedded in a general framework, in which arrays of risk factors can be swapped in and out to assess their effect on the projected losses. Risk factor scaling is varied to assess the resilience and stability of financial institutions to economic shock. Symbolic analysis of projected losses shows that they are well-conditioned with respect to risk factors. Specific reference is made to the effect of the 2020 COVID-19 pandemic. For a 1-year projection, the framework indicates a requirement for an increase in regulatory capital of approximately 3% for mild stress, 8% for moderate stress, and 32% for extreme stress. The proposed framework is significant because it is the first formal methodology to link financial risk with economic factors in an objective way without recourse to correlations. Full article
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Article
Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method
Math. Comput. Appl. 2021, 26(1), 18; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010018 - 23 Feb 2021
Viewed by 329
Abstract
This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution [...] Read more.
This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context, the numerical test is illustrated by two examples where we find meaningful numerical results. Full article
(This article belongs to the Section Engineering)
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Article
Data Augmentation and Feature Selection for Automatic Model Recommendation in Computational Physics
Math. Comput. Appl. 2021, 26(1), 17; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010017 - 16 Feb 2021
Viewed by 528
Abstract
Classification algorithms have recently found applications in computational physics for the selection of numerical methods or models adapted to the environment and the state of the physical system. For such classification tasks, labeled training data come from numerical simulations and generally correspond to [...] Read more.
Classification algorithms have recently found applications in computational physics for the selection of numerical methods or models adapted to the environment and the state of the physical system. For such classification tasks, labeled training data come from numerical simulations and generally correspond to physical fields discretized on a mesh. Three challenging difficulties arise: the lack of training data, their high dimensionality, and the non-applicability of common data augmentation techniques to physics data. This article introduces two algorithms to address these issues: one for dimensionality reduction via feature selection, and one for data augmentation. These algorithms are combined with a wide variety of classifiers for their evaluation. When combined with a stacking ensemble made of six multilayer perceptrons and a ridge logistic regression, they enable reaching an accuracy of 90% on our classification problem for nonlinear structural mechanics. Full article
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Article
Fizzle Testing: An Equation Utilizing Random Surveillance to Help Reduce COVID-19 Risks
Math. Comput. Appl. 2021, 26(1), 16; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010016 - 13 Feb 2021
Cited by 1 | Viewed by 828
Abstract
A closed-form equation, the Fizzle Equation, was derived from a mathematical model predicting Severe Acute Respiratory Virus-2 dynamics, optimized for a 4000-student university cohort. This equation sought to determine the frequency and percentage of random surveillance testing required to prevent an outbreak, [...] Read more.
A closed-form equation, the Fizzle Equation, was derived from a mathematical model predicting Severe Acute Respiratory Virus-2 dynamics, optimized for a 4000-student university cohort. This equation sought to determine the frequency and percentage of random surveillance testing required to prevent an outbreak, enabling an institution to develop scientifically sound public health policies to bring the effective reproduction number of the virus below one, halting virus progression. Model permutations evaluated the potential spread of the virus based on the level of random surveillance testing, increased viral infectivity and implementing additional safety measures. The model outcomes included: required level of surveillance testing, the number of infected individuals, and the number of quarantined individuals. Using the derived equations, this study illustrates expected infection load and how testing policy can prevent outbreaks in an institution. Furthermore, this process is iterative, making it possible to develop responsive policies scaling the amount of surveillance testing based on prior testing results, further conserving resources. Full article
(This article belongs to the Collection Mathematical Modelling of COVID-19)
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Article
Approximation of Gaussian Curvature by the Angular Defect: An Error Analysis
Math. Comput. Appl. 2021, 26(1), 15; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010015 - 09 Feb 2021
Viewed by 479
Abstract
It is common practice in science and engineering to approximate smooth surfaces and their geometric properties by using triangle meshes with vertices on the surface. Here, we study the approximation of the Gaussian curvature through the Gauss–Bonnet scheme. In this scheme, the Gaussian [...] Read more.
It is common practice in science and engineering to approximate smooth surfaces and their geometric properties by using triangle meshes with vertices on the surface. Here, we study the approximation of the Gaussian curvature through the Gauss–Bonnet scheme. In this scheme, the Gaussian curvature at a vertex on the surface is approximated by the quotient of the angular defect and the area of the Voronoi region. The Voronoi region is the subset of the mesh that contains all points that are closer to the vertex than to any other vertex. Numerical error analyses suggest that the Gauss–Bonnet scheme always converges with quadratic convergence speed. However, the general validity of this conclusion remains uncertain. We perform an analytical error analysis on the Gauss–Bonnet scheme. Under certain conditions on the mesh, we derive the convergence speed of the Gauss–Bonnet scheme as a function of the maximal distance between the vertices. We show that the conditions are sufficient and necessary for a linear convergence speed. For the special case of locally spherical surfaces, we find a better convergence speed under weaker conditions. Furthermore, our analysis shows that the Gauss–Bonnet scheme, while generally efficient and effective, can give erroneous results in some specific cases. Full article
(This article belongs to the Section Natural Sciences)
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Article
An Epidemic Grid Model to Address the Spread of Covid-19: A Comparison between Italy, Germany and France
Math. Comput. Appl. 2021, 26(1), 14; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010014 - 08 Feb 2021
Viewed by 478
Abstract
This paper presents a discrete compartmental Susceptible–Exposed–Infected–Recovered/Dead (SEIR/D) model to address the expansion of Covid-19. This model is based on a grid. As time passes, the status of the cells updates by means of binary rules following a neighborhood and a delay pattern. [...] Read more.
This paper presents a discrete compartmental Susceptible–Exposed–Infected–Recovered/Dead (SEIR/D) model to address the expansion of Covid-19. This model is based on a grid. As time passes, the status of the cells updates by means of binary rules following a neighborhood and a delay pattern. This model has already been analyzed in previous works and successfully compared with the corresponding continuous models solved by ordinary differential equations (ODE), with the intention of finding the homologous parameters between both approaches. Thus, it has been possible to prove that the combination neighborhood-update rule is responsible for the rate of expansion and recovering/death of the disease. The delays (between Susceptible and Asymptomatic, Asymptomatic and Infected, Infected and Recovered/Dead) may have a crucial impact on both height and timing of the peak of Infected and the Recovery/Death rate. This theoretical model has been successfully tested in the case of the dissemination of information through mobile social networks and in the case of plant pests. Full article
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2019)
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Article
Differential Evolution under Fixed Point Arithmetic and FP16 Numbers
Math. Comput. Appl. 2021, 26(1), 13; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010013 - 04 Feb 2021
Viewed by 390
Abstract
In this work, the differential evolution algorithm behavior under a fixed point arithmetic is analyzed also using half-precision floating point (FP) numbers of 16 bits, and these last numbers are known as FP16. In this paper, it is considered that it is important [...] Read more.
In this work, the differential evolution algorithm behavior under a fixed point arithmetic is analyzed also using half-precision floating point (FP) numbers of 16 bits, and these last numbers are known as FP16. In this paper, it is considered that it is important to analyze differential evolution (DE) in these circumstances with the goal of reducing its consumption power, storage size of the variables, and improve its speed behavior. All these aspects become important if one needs to design a dedicated hardware, as an embedded DE within a circuit chip, that performs optimization. With these conditions DE is tested using three common multimodal benchmark functions: Rosenbrock, Rastrigin, and Ackley, in 10 dimensions. Results are obtained in software by simulating all numbers using C programming language. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)
Article
Market Behavior and Evolution of Wealth Distribution: A Simulation Model Based on Artificial Agents
Math. Comput. Appl. 2021, 26(1), 12; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010012 - 03 Feb 2021
Viewed by 862
Abstract
The aim of this work is to simulate a market behavior in order to study the evolution of wealth distribution. The numerical simulations are carried out on a simple economical model with a finite number of economic agents, which are able to exchange [...] Read more.
The aim of this work is to simulate a market behavior in order to study the evolution of wealth distribution. The numerical simulations are carried out on a simple economical model with a finite number of economic agents, which are able to exchange goods/services and money; the various agents interact each other by means of random exchanges. The model is micro founded, self-consistent, and predictive. Despite the simplicity of the model, the simulations show a complex and non-trivial behavior. First of all, we are able to recognize two solution classes, namely two phases, separated by a threshold region. The analysis of the wealth distribution of the model agents, in the threshold region, shows functional forms resembling empirical quantitative studies of the probability distributions of wealth and income in the United Kingdom and the United States. Furthermore, the decile distribution of the population wealth of the simulated model, in the threshold region, overlaps in a suggestive way with the real data of the Italian population wealth in the last few years. Finally, the results of the simulated model allow us to draw important considerations for designing effective policies for economic and human development. Full article
(This article belongs to the Section Social Sciences)
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Article
Curve and Surface Construction Using Hermite Trigonometric Interpolant
Math. Comput. Appl. 2021, 26(1), 11; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010011 - 29 Jan 2021
Viewed by 426
Abstract
In this paper, the constructions of both open and closed trigonometric Hermite interpolation curves while using the derivatives are presented. The combination of tension, continuity, and bias control is used as a very powerful type of interpolation; they are applied to open and [...] Read more.
In this paper, the constructions of both open and closed trigonometric Hermite interpolation curves while using the derivatives are presented. The combination of tension, continuity, and bias control is used as a very powerful type of interpolation; they are applied to open and closed Hermite interpolation curves. Surface construction utilizing the studied trigonometric Hermite interpolation is explored and several examples obtained by the C1 trigonometric Hermite interpolation surface are given to show the usefulness of this method. Full article
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Editorial
Acknowledgment to Reviewers of MCA in 2020
Math. Comput. Appl. 2021, 26(1), 10; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010010 - 27 Jan 2021
Viewed by 397
Abstract
Peer review is the driving force of journal development, and reviewers are gatekeepers who ensure that MCA maintains its standards for the high quality of its published papers [...] Full article
Article
Univariate Theory of Functional Connections Applied to Component Constraints
Math. Comput. Appl. 2021, 26(1), 9; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010009 - 14 Jan 2021
Viewed by 474
Abstract
This work presents a methodology to derive analytical functionals, with embedded linear constraints among the components of a vector (e.g., coordinates) that is a function a single variable (e.g., time). This work prepares the background necessary for the indirect solution of optimal control [...] Read more.
This work presents a methodology to derive analytical functionals, with embedded linear constraints among the components of a vector (e.g., coordinates) that is a function a single variable (e.g., time). This work prepares the background necessary for the indirect solution of optimal control problems via the application of the Pontryagin Maximum Principle. The methodology presented is part of the univariate Theory of Functional Connections that has been developed to solve constrained optimization problems. To increase the clarity and practical aspects of the proposed method, the work is mostly presented via examples of applications rather than via rigorous mathematical definitions and proofs. Full article
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Article
Chaotic Multi-Objective Simulated Annealing and Threshold Accepting for Job Shop Scheduling Problem
Math. Comput. Appl. 2021, 26(1), 8; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010008 - 12 Jan 2021
Viewed by 801
Abstract
The Job Shop Scheduling Problem (JSSP) has enormous industrial applicability. This problem refers to a set of jobs that should be processed in a specific order using a set of machines. For the single-objective optimization JSSP problem, Simulated Annealing is among the best [...] Read more.
The Job Shop Scheduling Problem (JSSP) has enormous industrial applicability. This problem refers to a set of jobs that should be processed in a specific order using a set of machines. For the single-objective optimization JSSP problem, Simulated Annealing is among the best algorithms. However, in Multi-Objective JSSP (MOJSSP), these algorithms have barely been analyzed, and the Threshold Accepting Algorithm has not been published for this problem. It is worth mentioning that the researchers in this area have not reported studies with more than three objectives, and the number of metrics they used to measure their performance is less than two or three. In this paper, we present two MOJSSP metaheuristics based on Simulated Annealing: Chaotic Multi-Objective Simulated Annealing (CMOSA) and Chaotic Multi-Objective Threshold Accepting (CMOTA). We developed these algorithms to minimize three objective functions and compared them using the HV metric with the recently published algorithms, MOMARLA, MOPSO, CMOEA, and SPEA. The best algorithm is CMOSA (HV of 0.76), followed by MOMARLA and CMOTA (with HV of 0.68), and MOPSO (with HV of 0.54). In addition, we show a complexity comparison of these algorithms, showing that CMOSA, CMOTA, and MOMARLA have a similar complexity class, followed by MOPSO. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)
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Article
Estimating the Market Share for New Products with a Split Questionnaire Survey
Math. Comput. Appl. 2021, 26(1), 7; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010007 - 09 Jan 2021
Viewed by 552
Abstract
When designing a new product, conjoint analysis is a powerful tool to estimate the perceived value of the prospects. However, it has a drawback: when the product has too many attributes and levels, it may be difficult to administrate the survey to respondents [...] Read more.
When designing a new product, conjoint analysis is a powerful tool to estimate the perceived value of the prospects. However, it has a drawback: when the product has too many attributes and levels, it may be difficult to administrate the survey to respondents because they will be overwhelmed by the too numerous questions. In this paper, we propose an alternative approach that permits us to bypass this problem. Contrary to conjoint analysis, which estimates respondents’ utility functions, our approach directly estimates market shares. This enables us to split the questionnaire among respondents and, therefore, to reduce the burden on each respondent as much as desired. However, this new method has two weaknesses that conjoint analysis does not have: first, inferences on a single respondent cannot be made; second, the competition’s product profiles have to be known before administrating the survey. Therefore, our method has to be used when traditional methods are less easily implementable, i.e., when the number of attributes and levels is large. Full article
(This article belongs to the Section Social Sciences)
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Article
Prediction of Maximum Pressure at the Roofs of Rectangular Water Tanks Subjected to Harmonic Base Excitation Using the Multi-Gene Genetic Programming Method
Math. Comput. Appl. 2021, 26(1), 6; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010006 - 02 Jan 2021
Viewed by 549
Abstract
Liquid storage tanks subjected to base excitation can cause large impact forces on the tank roof, which can lead to structural damage as well as economic and environmental losses. The use of artificial intelligence in solving engineering problems is becoming popular in various [...] Read more.
Liquid storage tanks subjected to base excitation can cause large impact forces on the tank roof, which can lead to structural damage as well as economic and environmental losses. The use of artificial intelligence in solving engineering problems is becoming popular in various research fields, and the Genetic Programming (GP) method is receiving more attention in recent years as a regression tool and also as an approach for finding empirical expressions between the data. In this study, an OpenFOAM numerical model that was validated by the authors in a previous study is used to simulate various tank sizes with different liquid heights. The tanks are excited in three different orientations with harmonic sinusoidal loadings. The excitation frequencies are chosen as equal to the tanks’ natural frequencies so that they would be subject to a resonance condition. The maximum pressure in each case is recorded and made dimensionless; then, using Multi-Gene Genetic Programming (MGGP) methods, a relationship between the dimensionless maximum pressure and dimensionless liquid height is acquired. Finally, some error measurements are calculated, and the sensitivity and uncertainty of the proposed equation are analyzed. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)
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Review
Surrogate Modeling Approaches for Multiobjective Optimization: Methods, Taxonomy, and Results
Math. Comput. Appl. 2021, 26(1), 5; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010005 - 31 Dec 2020
Cited by 1 | Viewed by 890
Abstract
Most practical optimization problems are comprised of multiple conflicting objectives and constraints which involve time-consuming simulations. Construction of metamodels of objectives and constraints from a few high-fidelity solutions and a subsequent optimization of metamodels to find in-fill solutions in an iterative manner remain [...] Read more.
Most practical optimization problems are comprised of multiple conflicting objectives and constraints which involve time-consuming simulations. Construction of metamodels of objectives and constraints from a few high-fidelity solutions and a subsequent optimization of metamodels to find in-fill solutions in an iterative manner remain a common metamodeling based optimization strategy. The authors have previously proposed a taxonomy of 10 different metamodeling frameworks for multiobjective optimization problems, each of which constructs metamodels of objectives and constraints independently or in an aggregated manner. Of the 10 frameworks, five follow a generative approach in which a single Pareto-optimal solution is found at a time and other five frameworks were proposed to find multiple Pareto-optimal solutions simultaneously. Of the 10 frameworks, two frameworks (M3-2 and M4-2) are detailed here for the first time involving multimodal optimization methods. In this paper, we also propose an adaptive switching based metamodeling (ASM) approach by switching among all 10 frameworks in successive epochs using a statistical comparison of metamodeling accuracy of all 10 frameworks. On 18 problems from three to five objectives, the ASM approach performs better than the individual frameworks alone. Finally, the ASM approach is compared with three other recently proposed multiobjective metamodeling methods and superior performance of the ASM approach is observed. With growing interest in metamodeling approaches for multiobjective optimization, this paper evaluates existing strategies and proposes a viable adaptive strategy by portraying importance of using an ensemble of metamodeling frameworks for a more reliable multiobjective optimization for a limited budget of solution evaluations. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)
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Article
Mathematical Model and Numerical Simulation for Electric Field Induced Cancer Cell Migration
Math. Comput. Appl. 2021, 26(1), 4; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010004 - 31 Dec 2020
Viewed by 579
Abstract
A mathematical model describing the interaction of cancer cells with the urokinase plasminogen activation system is represented by a system of partial differential equations, in which cancer cell dynamics accounts for diffusion, chemotaxis, and haptotaxis contributions. The mutual relations between nerve fibers and [...] Read more.
A mathematical model describing the interaction of cancer cells with the urokinase plasminogen activation system is represented by a system of partial differential equations, in which cancer cell dynamics accounts for diffusion, chemotaxis, and haptotaxis contributions. The mutual relations between nerve fibers and tumors have been recently investigated, in particular, the role of nerves in the development of tumors, as well neurogenesis induced by cancer cells. Such mechanisms are mediated by neurotransmitters released by neurons as a consequence of electrical stimuli flowing along the nerves, and therefore electric fields can be present inside biological tissues, in particular, inside tumors. Considering cancer cells as negatively charged particles immersed in the correct biological environment and subjected to an external electric field, the effect of the latter on cancer cell dynamics is still unknown. Here, we implement a mathematical model that accounts for the interaction of cancer cells with the urokinase plasminogen activation system subjected to a uniform applied electric field, simulating the first stage of cancer cell dynamics in a three-dimensional axial symmetric domain. The obtained numerical results predict that cancer cells can be moved along a preferred direction by an applied electric field, suggesting new and interesting strategies in cancer therapy. Full article
(This article belongs to the Section Natural Sciences)
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Article
A Continuous Model of Marital Relations with Stochastic Differential Equations
Math. Comput. Appl. 2021, 26(1), 3; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010003 - 31 Dec 2020
Viewed by 580
Abstract
Marital relations depend on many factors which can increase the amount of satisfaction or unhappiness in the relation. A large percentage of marriages end up in divorce. While there are many studies about the causes of divorce and how to prevent it, there [...] Read more.
Marital relations depend on many factors which can increase the amount of satisfaction or unhappiness in the relation. A large percentage of marriages end up in divorce. While there are many studies about the causes of divorce and how to prevent it, there are very few mathematical models dealing with marital relations. In this paper, we present a continuous model based on the ideas presented by Gottman and coauthors. We show that the type of influence functions that describe the interaction between husband and wife is critical in determining the outcome of a marriage. We also introduce stochasticity into the model to account for the many factors that affect the marriage and that are not easily quantified, such as economic climate, work stress, and family relations. We show that these factors are able to change the equilibrium state of the couple. Full article
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2019)
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Article
A Localized Collocation Solver Based on T-Complete Functions for Anti-Plane Transverse Elastic Wave Propagation Analysis in 2D Phononic Crystals
Math. Comput. Appl. 2021, 26(1), 2; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010002 - 30 Dec 2020
Cited by 1 | Viewed by 464
Abstract
In this paper, we introduce a novel localized collocation solver for two-dimensional (2D) phononic crystal analysis. In the proposed collocation solver, the displacement at each node is expressed as a linear combination of T-complete functions in each stencil support and the sparse linear [...] Read more.
In this paper, we introduce a novel localized collocation solver for two-dimensional (2D) phononic crystal analysis. In the proposed collocation solver, the displacement at each node is expressed as a linear combination of T-complete functions in each stencil support and the sparse linear system is obtained by satisfying the considered governing equation at interior nodes and boundary conditions at boundary nodes. As compared with finite element method (FEM) results and the analytical solutions, the efficiency and accuracy of the proposed localized collocation solver are verified under a benchmark example. Then, the proposed method is applied to 2D phononic crystals with various lattice forms and scatterer shapes, where the related band structures, transmission spectra, and displacement amplitude distributions are calculated as compared with the FEM. Full article
(This article belongs to the Section Engineering)
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Article
A Saint-Venant Model for Overland Flows with Precipitation and Recharge
Math. Comput. Appl. 2021, 26(1), 1; https://0-doi-org.brum.beds.ac.uk/10.3390/mca26010001 - 29 Dec 2020
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Abstract
We propose a one-dimensional Saint-Venant (open-channel) model for overland flows, including a water input–output source term modeling recharge via rainfall and infiltration (or exfiltration). We derive the model via asymptotic reduction from the two-dimensional Navier–Stokes equations under the shallow water assumption, with boundary [...] Read more.
We propose a one-dimensional Saint-Venant (open-channel) model for overland flows, including a water input–output source term modeling recharge via rainfall and infiltration (or exfiltration). We derive the model via asymptotic reduction from the two-dimensional Navier–Stokes equations under the shallow water assumption, with boundary conditions including recharge via ground infiltration and runoff. This new model recovers existing models as special cases, and adds more scope by adding water-mixing friction terms that depend on the rate of water recharge. We propose a novel entropy function and its flux, which are useful in validating the model’s conservation or dissipation properties. Based on this entropy function, we propose a finite volume scheme extending a class of kinetic schemes and provide numerical comparisons with respect to the newly introduced mixing friction coefficient. We also provide a comparison with experimental data. Full article
(This article belongs to the Section Natural Sciences)
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