Research

16 pages, 5703 KiB

Open AccessArticle

An Efficient Method for Solving Problems of Acoustic Scattering on Three-Dimensional Transparent Structures

by Alexander B. Samokhin and Ivan A. Yurchenkov

Mathematics 2024, 12(6), 789; https://doi.org/10.3390/math12060789 - 07 Mar 2024

Viewed by 418

The article contains a study of methods for solving integral equations in the context of acoustic problems. The methodology considered is applied to describe acoustic wave propagation and scattering. Efficient discretization methods are used together with iterative methods to solve the operator equations, [...] Read more.

The article contains a study of methods for solving integral equations in the context of acoustic problems. The methodology considered is applied to describe acoustic wave propagation and scattering. Efficient discretization methods are used together with iterative methods to solve the operator equations, including an apparatus for fast multiplication of the resulting post-discretization Toeplitz matrices by a vector using the fast Fourier transform. The theoretical analysis of the proposed numerical algorithm demonstrates its efficiency in terms of the required number of arithmetic operations and the memory footprint of the computing system. The presented numerical simulation demonstrates the possibility of solving the problem of acoustic wave propagation in transparent media using the proposed methods. A visualization of the obtained solutions for a practical problem with a high level of discretization of the solution volume domain is also presented. Full article

(This article belongs to the Special Issue Applied and Computational Mathematics for Digital Environments, 2nd Edition)

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14 pages, 559 KiB

Open AccessArticle

A Stabilisation System Synthesis for Motion along a Preset Trajectory and Its Solution by Symbolic Regression

by Askhat Diveev, Elena Sofronova and Nurbek Konyrbaev

Mathematics 2024, 12(5), 706; https://doi.org/10.3390/math12050706 - 28 Feb 2024

Viewed by 415

Abstract

The problem of a stabilisation system synthesis for the motion of a control object along a given spatial trajectory is considered. The complexity of the problem is that the preset trajectory is defined in the state subspace and not in time. This paper [...] Read more.

The problem of a stabilisation system synthesis for the motion of a control object along a given spatial trajectory is considered. The complexity of the problem is that the preset trajectory is defined in the state subspace and not in time. This paper describes a stabilisation system synthesis for motion along a trajectory specified in time and along a trajectory specified in the form of a manifold in a state space. In order to construct a stabilisation system, it is necessary to determine a distance between an object and the given trajectory at each moment in time. For trajectories that are not given in time, the determination of this distance can be ambiguous. An object may be exactly on a trajectory but at a different time. This paper proposes some approaches to solve the problem. One of the approaches is to transform a given trajectory in a state subspace into a trajectory given in time. A description of a universal method to perform this transformation is presented. In order to solve the synthesis problem automatically, without having to analyse the mathematical model of the control object, it is suggested that machine learning control by symbolic regression is used. In computational experiments, examples of stabilisation system syntheses for quadcopter motion along a given spatial trajectory are presented. Full article

(This article belongs to the Special Issue Applied and Computational Mathematics for Digital Environments, 2nd Edition)

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22 pages, 368 KiB

Open AccessArticle

Allocation of Starting Points in Global Optimization Problems

by Oleg Khamisov, Eugene Semenkin and Vladimir Nelyub

Mathematics 2024, 12(4), 606; https://doi.org/10.3390/math12040606 - 18 Feb 2024

Viewed by 432

Abstract

We propose new multistart techniques for finding good local solutions in global optimization problems. The objective function is assumed to be differentiable, and the feasible set is a convex compact set. The techniques are based on finding maximum distant points on the feasible [...] Read more.

We propose new multistart techniques for finding good local solutions in global optimization problems. The objective function is assumed to be differentiable, and the feasible set is a convex compact set. The techniques are based on finding maximum distant points on the feasible set. A special global optimization problem is used to determine the maximum distant points. Preliminary computational results are given. Full article

(This article belongs to the Special Issue Applied and Computational Mathematics for Digital Environments, 2nd Edition)

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40 pages, 1869 KiB

Open AccessArticle

Decision-Making on the Diagnosis of Oncological Diseases Using Cost-Sensitive SVM Classifiers Based on Datasets with a Variety of Features of Different Natures

by Liliya A. Demidova

Mathematics 2024, 12(4), 538; https://doi.org/10.3390/math12040538 - 08 Feb 2024

Viewed by 569

Abstract

This paper discusses the problem of detecting cancer using such biomarkers as blood protein markers. The purpose of this research is to propose an approach for making decisions in the diagnosis of cancer through the creation of cost-sensitive SVM classifiers on the basis [...] Read more.

This paper discusses the problem of detecting cancer using such biomarkers as blood protein markers. The purpose of this research is to propose an approach for making decisions in the diagnosis of cancer through the creation of cost-sensitive SVM classifiers on the basis of datasets with a variety of features of different nature. Such datasets may include compositions of known features corresponding to blood protein markers and new features constructed using methods for calculating entropy and fractal dimensions, as well as using the UMAP algorithm. Based on these datasets, multiclass SVM classifiers were developed. They use cost-sensitive learning principles to overcome the class imbalance problem, which is typical for medical datasets. When implementing the UMAP algorithm, various variants of the loss function were considered. This was performed in order to select those that provide the formation of such new features that ultimately allow us to develop the best cost-sensitive SVM classifiers in terms of maximizing the mean value of the metric

M a c r o F_{1} - s c o r e

. The experimental results proved the possibility of applying the UMAP algorithm, approximate entropy and, in addition, Higuchi and Katz fractal dimensions to construct new features using blood protein markers. It turned out that when working with the UMAP algorithm, the most promising is the application of a loss function on the basis of fuzzy cross-entropy, and the least promising is the application of a loss function on the basis of intuitionistic fuzzy cross-entropy. Augmentation of the original dataset with either features on the basis of the UMAP algorithm, features on the basis of the UMAP algorithm and approximate entropy, or features on the basis of approximate entropy provided the creation of the three best cost-sensitive SVM classifiers with mean values of the metric

M a c r o F_{1} - s c o r e

increased by 5.359%, 5.245% and 4.675%, respectively, compared to the mean values of this metric in the case when only the original dataset was utilized for creating the base SVM classifier (without performing any manipulations to overcome the class imbalance problem, and also without introducing new features). Full article

(This article belongs to the Special Issue Applied and Computational Mathematics for Digital Environments, 2nd Edition)

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22 pages, 1474 KiB

Open AccessArticle

Adaptation of the Scaling Factor Based on the Success Rate in Differential Evolution

by Vladimir Stanovov and Eugene Semenkin

Mathematics 2024, 12(4), 516; https://doi.org/10.3390/math12040516 - 07 Feb 2024

Viewed by 493

Abstract

Differential evolution is a popular heuristic black-box numerical optimization algorithm which is often used due to its simplicity and efficiency. Parameter adaptation is one of the main directions of study regarding the differential evolution algorithm. The main reason for this is that differential [...] Read more.

Differential evolution is a popular heuristic black-box numerical optimization algorithm which is often used due to its simplicity and efficiency. Parameter adaptation is one of the main directions of study regarding the differential evolution algorithm. The main reason for this is that differential evolution is highly sensitive to the scaling factor and crossover rate parameters. In this study, a novel adaptation technique is proposed which uses the success rate to replace the popular success history-based adaptation for scaling factor tuning. In particular, the scaling factor is sampled with a Cauchy distribution, whose location parameter is set as an nth order root of the current success rate, i.e., the ratio of improved solutions to the current population size. The proposed technique is universal and can be applied to any differential evolution variant. Here it is tested with several state-of-the-art variants of differential evolution, and on two benchmark sets, CEC 2017 and CEC 2022. The performed experiments, which include modifications of algorithms developed by other authors, show that in many cases using the success rate to determine the scaling factor can be beneficial, especially with relatively small computational resource. Full article

(This article belongs to the Special Issue Applied and Computational Mathematics for Digital Environments, 2nd Edition)

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19 pages, 591 KiB

Open AccessArticle

Fractional-Differential Models of the Time Series Evolution of Socio-Dynamic Processes with Possible Self-Organization and Memory

by Dmitry Zhukov, Konstantin Otradnov and Vladimir Kalinin

Mathematics 2024, 12(3), 484; https://doi.org/10.3390/math12030484 - 02 Feb 2024

Viewed by 606

Abstract

This article describes the solution of two problems. First, based on the fractional diffusion equation, a boundary problem with arbitrary values of derivative indicators was formulated and solved, describing more general cases than existing solutions. Secondly, from the consideration of the probability schemes [...] Read more.

This article describes the solution of two problems. First, based on the fractional diffusion equation, a boundary problem with arbitrary values of derivative indicators was formulated and solved, describing more general cases than existing solutions. Secondly, from the consideration of the probability schemes of transitions between states of the process, which can be observed in complex systems, a fractional-differential equation of the telegraph type with multiples is obtained (in time: β, 2β, 3β, … and state: α, 2α, 3α, …) using orders of fractional derivatives and its analytical solution for one particular boundary problem is considered. In solving edge problems, the Fourier method was used. This makes it possible to represent the solution in the form of a nested time series (one in time t, the second in state x), each of which is a function of the Mittag-Leffler type. The eigenvalues of the Mittag-Leffler function for describing states can be found using boundary conditions and the Fourier coefficient based on the initial condition and orthogonality conditions of the eigenfunctions. An analysis of the characteristics of time series of changes in the emotional color of users’ comments on published news in online mass media and the electoral campaigns of the US presidential elections showed that for the mathematical expectation of amplitudes of deviations of series levels from the size of the amplitude calculation interval (“sliding window”), a root dependence of fractional degree was observed; for dispersion, a power law with a fractional index greater than 1.5 was observed; and the behavior of the excess showed the presence of so-called “heavy tails”. The obtained results indicate that time series have unsteady non-locality, both in time and state. This provides the rationale for using differential equations with partial fractional derivatives to describe time series dynamics. Full article

(This article belongs to the Special Issue Applied and Computational Mathematics for Digital Environments, 2nd Edition)

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21 pages, 8966 KiB

Open AccessArticle

Exploratory Landscape Validation for Bayesian Optimization Algorithms

by Taleh Agasiev and Anatoly Karpenko

Mathematics 2024, 12(3), 426; https://doi.org/10.3390/math12030426 - 28 Jan 2024

Viewed by 602

Abstract

Bayesian optimization algorithms are widely used for solving problems with a high computational complexity in terms of objective function evaluation. The efficiency of Bayesian optimization is strongly dependent on the quality of the surrogate models of an objective function, which are built and [...] Read more.

Bayesian optimization algorithms are widely used for solving problems with a high computational complexity in terms of objective function evaluation. The efficiency of Bayesian optimization is strongly dependent on the quality of the surrogate models of an objective function, which are built and refined at each iteration. The quality of surrogate models, and hence the performance of an optimization algorithm, can be greatly improved by selecting the appropriate hyperparameter values of the approximation algorithm. The common approach to finding good hyperparameter values for each iteration of Bayesian optimization is to build surrogate models with different hyperparameter values and choose the best one based on some estimation of the approximation error, for example, a cross-validation score. Building multiple surrogate models for each iteration of Bayesian optimization is computationally demanding and significantly increases the time required to solve an optimization problem. This paper suggests a new approach, called exploratory landscape validation, to find good hyperparameter values with less computational effort. Exploratory landscape validation metrics can be used to predict the best hyperparameter values, which can improve both the quality of the solutions found by Bayesian optimization and the time needed to solve problems. Full article

(This article belongs to the Special Issue Applied and Computational Mathematics for Digital Environments, 2nd Edition)

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18 pages, 1035 KiB

Open AccessArticle

Adaptive Synthesized Control for Solving the Optimal Control Problem

by Askhat Diveev and Elizaveta Shmalko

Mathematics 2023, 11(19), 4035; https://doi.org/10.3390/math11194035 - 22 Sep 2023

Viewed by 560

Abstract

The development of artificial intelligence systems assumes that a machine can independently generate an algorithm of actions or a control system to solve the tasks. To do this, the machine must have a formal description of the problem and possess computational methods for [...] Read more.

The development of artificial intelligence systems assumes that a machine can independently generate an algorithm of actions or a control system to solve the tasks. To do this, the machine must have a formal description of the problem and possess computational methods for solving it. This article deals with the problem of optimal control, which is the main task in the development of control systems, insofar as all systems being developed must be optimal from the point of view of a certain criterion. However, there are certain difficulties in implementing the resulting optimal control modes. This paper considers an extended formulation of the optimal control problem, which implies the creation of such systems that would have the necessary properties for its practical implementation. To solve it, an adaptive synthesized optimal control approach based on the use of numerical methods of machine learning is proposed. Such control moves the control object, optimally changing the position of the stable equilibrium point in the presence of some initial position uncertainty. As a result, from all possible synthesized controls, one is chosen that is less sensitive to changes in the initial state. As an example, the optimal control problem of a quadcopter with complex phase constraints is considered. To solve this problem, according to the proposed approach, the control synthesis problem is firstly solved to obtain a stable equilibrium point in the state space using a machine learning method of symbolic regression. After that, optimal positions of the stable equilibrium point are searched using a particle swarm optimization algorithm using the source functional from the initial optimal control problem statement. It is shown that such an approach allows for generating the control system automatically by computer, basing this on the formal statement of the problem and then directly implementing it onboard as far as the stabilization system has already been introduced. Full article

(This article belongs to the Special Issue Applied and Computational Mathematics for Digital Environments, 2nd Edition)

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20 pages, 896 KiB

Open AccessArticle

Universal Stabilisation System for Control Object Motion along the Optimal Trajectory

by Askhat Diveev and Elena Sofronova

Mathematics 2023, 11(16), 3556; https://doi.org/10.3390/math11163556 - 17 Aug 2023

Cited by 1 | Viewed by 646

Abstract

An attempt to construct a universal stabilisation system that ensures the object motion along specified trajectory from certain class is presented. If such a stabilisation system is constructed, then only the problem of optimal control is solved, but for a model of the [...] Read more.

An attempt to construct a universal stabilisation system that ensures the object motion along specified trajectory from certain class is presented. If such a stabilisation system is constructed, then only the problem of optimal control is solved, but for a model of the object, which includes a stabilisation system and a subsystem with a reference model for generating a specified trajectory. In this case, the desired control is the control in the reference model. Statement of complete optimal control problem includes two problems, optimal control problem and stabilisation system synthesis problem for motion along given trajectory in the state space. Numerical methods for solving these problems based on evolutionary computation and symbolic regression are described. It is shown that when solving the stabilisation system synthesis problem, it is possible to obtain a universal system that provides stabilisation of the object motion relative to any trajectory from a certain class. Therefore, it is advisable to formulate an optimal control problem for an object with a motion stabilisation system. A computational example of solving the problem for the spatial motion of a quadrocopter is given. Full article

(This article belongs to the Special Issue Applied and Computational Mathematics for Digital Environments, 2nd Edition)

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