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Numerical Analysis and Scientific Computing, 3rd Edition

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 17137

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Prof. Dr. Theodore E. Simos

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1. Laboratory of Applied Mathematics for Solving Interdisciplinary Problems of Energy Production, Ulyanovsk State Technical University, Severny Venetz Street 32, 432027 Ulyanovsk, Russia
2. Digital Industry REC, South Ural State University, 76, Lenin Avenue, 454080 Chelyabinsk, Russia
3. Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, 67100 Xanthi, Greece
Interests: numerical analysis; scientific computing; applied numerical analysis; computational chemistry; computational material sciences; computational physics; parallel algorithm and expert systems
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Prof. Dr. Charampos Tsitouras

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Guest Editor

Mathematics and Applied Mathematics, National and Kapodistrian University of Athens, 10679 Athens, Greece
Interests: numerical analysis; scientific computing
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the last few decades, the role of numerical analysis and scientific computing has increased, especially for the solution of real-world problems.

This Special Issue will present recent research results in the field of numerical analysis and scientific computing.

Papers on the production, analysis, and computational performance of new and original methods of all areas of numerical analysis and scientific computing are welcome. More specifically, we welcome papers on, but not limited to, the following topics:

Numerical analysis of ODEs; numerical analysis of PDEs (including BVPs); scientific computing and algorithms; stochastic differential equations; approximation theory; numerical linear algebra; numerical integral equations; error analysis and interval analysis; difference equations and recurrence relations; numerical problems in dynamical systems; scientific applications (computational physics, computational statistics, computational chemistry, computational engineering, etc.); differential algebraic equations, numerical methods in Fourier analysis; mathematical physics; mathematical chemistry; mathematical biology and mathematical medicine; optimization and operational research; theoretical mechanics; discrete applied mathematics; statistics; probability; dynamical systems; algorithms; experimental mathematics; theoretical computer science; applied analysis; mathematical modeling (including, but not limited to, mathematical modeling of engineering and environmental processes manufacturing, and industrial systems, heat transfer, fluid mechanics, CFD, and transport phenomena solid mechanics and mechanics of metals, electromagnets and MHD, reliability modeling and system optimization, decision sciences in an industrial and manufacturing context, civil engineering systems and structures, mineral and energy resources, relevant software engineering issues associated with CAD and CAE, materials and metallurgical engineering; mathematical modelling of social, behavioral and other sciences); decomposition and reconstruction algorithms, subdivision algorithms; continuous and discrete wavelet transform; time–frequency localization; phase space analysis; sub-band coding; image compression; real-time filtering; radar and sonar applications; transient analysis; medical imaging; multigrid methods; frames; bifurcation and singularity theory; deterministic chaos and fractals; soliton and coherent phenomena; formation of patterns; evolution; complexity theory and neural networks; analytical approaches and simulations for more accurate descriptions; predictions; experimental observations and applications of non-linear phenomena in science and engineering; theoretical and applied aspects of computational geometry; control theory and automation; fuzzy sets and systems and fuzzy logic; applied algebra; quality theory of differential equations; neural networks.

We also welcome papers that explore applications of numerical and mathematical methods to real-world problems in science, engineering, and technology.

Prof. Dr. Theodore E. Simos
Prof. Dr. Charampos Tsitouras
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

numerical analysis
computational mathematics
scientific computing
computational methods and algorithms
applied and industrial mathematics
mathematical methods
optimization
applications in science, engineering and technology

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Published Papers (10 papers)

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Research

12 pages, 509 KiB

Open AccessArticle

Multithreaded Reproducible Banded Matrix-Vector Multiplication

by Tao Tang, Haijun Qi, Qingfeng Lu and Hao Jiang

Mathematics 2024, 12(3), 422; https://doi.org/10.3390/math12030422 - 28 Jan 2024

Viewed by 951

Abstract

Reproducibility refers to getting bitwise identical floating point results from multiple runs of the same program, is an important basis for debugging or correctness checking in many codes. However the round-off error and non-associativity of floating point makes attaining reproducibility a challenge in [...] Read more.

dgbmv

performs general banded matrix-vector multiplication for double precision, is the most basic Level-2 operation in BLAS. First, we designed a reproducible algorithm for banded matrix-vector multiplication

repro_dgbmv

based on the technique of error-free transformation. Then the error of the algorithm is analyzed. Second, the algorithm is parallelized into

repro_dgbmv_thread

on ARM and x86 platforms. The numerical test results verify that

repro_dgbmv_thread

is reproducible and has higher accuracy than ordinary

dgbmv

. In numerical experiments on ARM platform, as the number of threads increases from 1 to 8, the run time of this algorithm is reduced by 5.2–7 times, while the run time of multithreaded

dgbmv

is only reduced by 2.2–3.8 times. In numerical experiments on x86 platform, as the number of threads increases from 1 to 15, the run time of this algorithm is reduced by 7.7–10.6 times, while the run time of multithreaded

dgbmv

is only reduced by 4.2–6.8 times. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing, 3rd Edition)

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14 pages, 393 KiB

Open AccessArticle

Nine-Stage Runge–Kutta–Nyström Pairs Sharing Orders Eight and Six

by Hadeel Alharbi, Kusum Yadav, Rabie A. Ramadan, Houssem Jerbi, Theodore E. Simos and Charalampos Tsitouras

Mathematics 2024, 12(2), 316; https://doi.org/10.3390/math12020316 - 18 Jan 2024

Viewed by 974

Abstract

We explore second-order systems of non-stiff initial-value problems (IVPs), particularly those cases where the first derivatives are absent. These types of problems are of significant interest and have applications in various domains, such as astronomy and physics. Runge–Kutta–Nyström (RKN) pairs stand out as highly effective methods of addressing these IVPs. In order to create a pair with eighth and sixth orders, we need to address a certain known set of equations concerning the coefficients. When constructing such pairs for use in double-precision arithmetic, we often need to meet various conditions. Primarily, we aim to maintain small coefficient magnitudes to prevent a loss of accuracy. Nevertheless, in the context of quadruple precision, we can tolerate larger coefficients. This flexibility enables us to establish pairs with eighth and sixth orders that exhibit significantly reduced truncation errors. Traditionally, these pairs are constructed to go through eight stages per step. Here, we propose using nine stages per step. Then we have available more coefficients in order to further reduce truncation errors. As a result, we construct a novel pair that, as anticipated, achieves superior performance compared to equivalent-order pairs in various significant problem scenarios. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing, 3rd Edition)

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14 pages, 533 KiB

Open AccessArticle

The Shape Parameter in the Shifted Surface Spline—A Sharp and Friendly Approach

by Lin-Tian Luh

Mathematics 2024, 12(2), 229; https://doi.org/10.3390/math12020229 - 10 Jan 2024

Viewed by 700

Abstract

This is a continuation of our previous study on the shape parameter contained in the shifted surface spline. We insist that the data points be purely scattered without meshes and the domain can be of any shape when conducting function interpolation by shifted surface splines. We also endeavor to make our approach easily accessible for scientists, not only mathematicians. However, the space of interpolated functions is smaller than that used before, leading to sharper function approximation. This function space has particular significance in numerical partial differential equations, especially for equations whose solutions lie in Sobolev space. Although the Fourier transform is deeply involved, scientists without a background in Fourier analysis can easily understand and use our approach. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing, 3rd Edition)

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21 pages, 3057 KiB

Open AccessArticle

Towards Higher-Order Zeroing Neural Networks for Calculating Quaternion Matrix Inverse with Application to Robotic Motion Tracking

by Rabeh Abbassi, Houssem Jerbi, Mourad Kchaou, Theodore E. Simos, Spyridon D. Mourtas and Vasilios N. Katsikis

Mathematics 2023, 11(12), 2756; https://doi.org/10.3390/math11122756 - 18 Jun 2023

Cited by 7 | Viewed by 1445

Abstract

The efficient solution of the time-varying quaternion matrix inverse (TVQ-INV) is a challenging but crucial topic due to the significance of quaternions in many disciplines, including physics, engineering, and computer science. The main goal of this research is to employ the higher-order zeroing neural network (HZNN) strategy to address the TVQ-INV problem. HZNN is a family of zeroing neural network models that correlates to the hyperpower family of iterative methods with adjustable convergence order. Particularly, three novel HZNN models are created in order to solve the TVQ-INV both directly in the quaternion domain and indirectly in the complex and real domains. The noise-handling version of these models is also presented, and the performance of these models under various types of noises is theoretically and numerically tested. The effectiveness and practicality of these models are further supported by their use in robotic motion tracking. According to the principal results, each of these six models can solve the TVQ-INV effectively, and the HZNN strategy offers a faster convergence rate than the conventional zeroing neural network strategy. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing, 3rd Edition)

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13 pages, 447 KiB

Open AccessArticle

New Family of Multi-Step Iterative Methods Based on Homotopy Perturbation Technique for Solving Nonlinear Equations

by Huda J. Saeed, Ali Hasan Ali, Rayene Menzer, Ana Danca Poțclean and Himani Arora

Mathematics 2023, 11(12), 2603; https://doi.org/10.3390/math11122603 - 7 Jun 2023

Cited by 4 | Viewed by 1466

Abstract

This research aims to propose a new family of one-parameter multi-step iterative methods that combine the homotopy perturbation method with a quadrature formula for solving nonlinear equations. The proposed methods are based on a higher-order convergence scheme that allows for faster and more efficient convergence compared to existing methods. It aims also to demonstrate that the efficiency index of the proposed iterative methods can reach up to

\sqrt[3]{4} \approx 1.587

and

\sqrt[4]{8} \approx 1.681

, respectively, indicating a high degree of accuracy and efficiency in solving nonlinear equations. To evaluate the effectiveness of the suggested methods, several numerical examples including their performance are provided and compared with existing methods. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing, 3rd Edition)

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18 pages, 505 KiB

Open AccessArticle

Novel Parametric Families of with and without Memory Iterative Methods for Multiple Roots of Nonlinear Equations

by G Thangkhenpau, Sunil Panday, Shubham Kumar Mittal and Lorentz Jäntschi

Mathematics 2023, 11(9), 2036; https://doi.org/10.3390/math11092036 - 25 Apr 2023

Cited by 7 | Viewed by 1394

Abstract

The methods that use memory using accelerating parameters for computing multiple roots are almost non-existent in the literature. Furthermore, the only paper available in this direction showed an increase in the order of convergence of 0.5 from the without memory to the with memory extension. In this paper, we introduce a new fifth-order without memory method, which we subsequently extend to two higher-order with memory methods using a self-accelerating parameter. The proposed with memory methods extension demonstrate a significant improvement in the order of convergence from 5 to 7, making this the first paper to achieve at least a 2-order improvement. In addition to this improvement, our paper is also the first to use Hermite interpolating polynomials to approximate the accelerating parameter in the proposed with memory methods for multiple roots. We also provide rigorous theoretical proofs of convergence theorems to establish the order of the proposed methods. Finally, we demonstrate the potential impact of the proposed methods through numerical experimentation on a diverse range of problems. Overall, we believe that our proposed methods have significant potential for various applications in science and engineering. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing, 3rd Edition)

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16 pages, 4592 KiB

Open AccessArticle

Unsteady Hydromagnetic Flow over an Inclined Rotating Disk through Neural Networking Approach

by Ishtiaq Ali, Taza Gul and Arshad Khan

Mathematics 2023, 11(8), 1893; https://doi.org/10.3390/math11081893 - 17 Apr 2023

Cited by 19 | Viewed by 1591

Abstract

The goal of this research is to examine how a magnetic field affects the unsteady flow of a hybrid nanofluid over a spinning disk that is inclined and stretched while the flow is surrounded by a non-Darcy porous medium. Furthermore, for heat transmission mechanisms, Joule heating and viscous dissipation are considered. The current article is made more realistic by imposing thermal radiation to enhance the heat transmission system under the effects of convection. Moreover, thermal and velocity slip conditions have also been incorporated into the current study. The equations that administer the flow problem along with constraints at the boundaries are converted to dimension-free form by employing a set of appropriate similarity transformations, which are then solved by the numerical technique Runge-Kutta method of order four (RK-4). The new and advanced trend for the convergence of the obtained results is validated through a neural networking approach. The temperature of hybrid nanofluid is augmented by an increase in the porosity parameter, the unsteadiness factor, the Eckert number, the magnetic field, and the Forchheimmer number, while for the values of the radiation factor, the thermal heat is decreasing near the disk and increasing away from the disk. The precision of the obtained results has been ensured by comparing them with established results, with good agreement among these results. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing, 3rd Edition)

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54 pages, 8310 KiB

Open AccessArticle

Evolutionary Multiobjective Design Approach for Robust Balancing of the Shaking Force, Shaking Moment, and Torque under Uncertainties: Application to Robotic Manipulators

by Ricardo Mejia-Rodriguez , Miguel Gabriel Villarreal-Cervantes , Alejandro Rodríguez-Molina , José Humberto Pérez-Cruz and Víctor Manuel Silva-García

Mathematics 2023, 11(8), 1776; https://doi.org/10.3390/math11081776 - 7 Apr 2023

Cited by 1 | Viewed by 4155

Abstract

In this paper, the environmental uncertainties are taken into account when designing a robotic manipulator to balance the shaking force, shaking moment, and torque. The proposed robust balancing design approach does not consider the probability distributions of the uncertainties and is addressed without dependence on specific trajectories. This is expressed as a nonlinear constrained multiobjective optimization problem in which the nominal performance in the time-independent terms of the shaking force balancing, the shaking moment balancing, and the torque delivery, as well as their three sensitivities to uncertainties, are simultaneously optimized to provide a set of link shapes that match link mass distributions in a single stage. The proposal is applied to a three-degree-of-freedom serial-parallel manipulator, and the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) is used to solve the associated problem. Comparative results with other design approaches reveal that the selected design achieves a suitable tradeoff in balancing the shaking force balancing, the shaking moment balancing, and the torque delivery and their sensitivities, leading to a reduction in their values and variations under mass changes in the manipulator end-effector with different operating conditions (tasks). Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing, 3rd Edition)

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17 pages, 683 KiB

Open AccessArticle

A Fresnel Cosine Integral WASD Neural Network for the Classification of Employee Attrition

by Hadeel Alharbi, Obaid Alshammari, Houssem Jerbi, Theodore E. Simos, Vasilios N. Katsikis, Spyridon D. Mourtas and Romanos D. Sahas

Mathematics 2023, 11(6), 1506; https://doi.org/10.3390/math11061506 - 20 Mar 2023

Cited by 3 | Viewed by 1807

Abstract

Employee attrition, defined as the voluntary resignation of a subset of a company’s workforce, represents a direct threat to the financial health and overall prosperity of a firm. From lost reputation and sales to the undermining of the company’s long-term strategy and corporate secrets, the effects of employee attrition are multidimensional and, in the absence of thorough planning, may endanger the very existence of the firm. It is thus impeccable in today’s competitive environment that a company acquires tools that enable timely prediction of employee attrition and thus leave room either for retention campaigns or for the formulation of strategical maneuvers that will allow the firm to undergo their replacement process with its economic activity left unscathed. To this end, a weights and structure determination (WASD) neural network utilizing Fresnel cosine integrals in the determination of its activation functions, termed FCI-WASD, is developed through a process of three discrete stages. Those consist of populating the hidden layer with a sufficient number of neurons, fine-tuning the obtained structure through a neuron trimming process, and finally, storing the necessary portions of the network that will allow for its successful future recreation and application. Upon testing the FCI-WASD on two publicly available employee attrition datasets and comparing its performance to that of five popular and well-established classifiers, the vast majority of them coming from MATLAB’s classification learner app, the FCI-WASD demonstrated superior performance with the overall results suggesting that it is a competitive as well as reliable model that may be used with confidence in the task of employee attrition classification. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing, 3rd Edition)

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30 pages, 1806 KiB

Open AccessArticle

A Hybrid CM-BEM Formulation for Solving Large-Scale 3D Eddy-Current Problems Based on ℋ-Matrices and Randomized Singular Value Decomposition for BEM Matrix Compression

by Federico Moro and Lorenzo Codecasa

Mathematics 2023, 11(6), 1324; https://doi.org/10.3390/math11061324 - 9 Mar 2023

Viewed by 1380

Abstract

We present a novel

a, v

We present a novel

a, v

-q hybrid method for solving large-scale time-harmonic eddy-current problems. This method combines a hybrid unsymmetric formulation based on the cell method and the boundary element method with a hierarchical matrix-compression technique based on randomized singular value decomposition. The main advantage is that the memory requirements are strongly reduced compared to the corresponding hybrid method without matrix compression while retaining the same robust solution strategy consisting of a simple construction of the preconditioner. In addition, the matrix-compression accuracy and efficiency are enhanced compared to traditional compression methods, such as adaptive cross approximation. The numerical results show that the proposed hybrid approach can also be effectively used to analyze large-scale eddy-current problems of engineering interest. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing, 3rd Edition)

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