Numerical Methods

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 June 2020) | Viewed by 24956

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Special Issue Editors

Department of Mathematics, Technical University of Cluj-Napoca, 400641 Cluj-Napoca, Romania
Interests: wavelet analysis
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Special Issue Information

Dear Colleagues,

This Special Issue, “Numerical Methods” is open for submissions and welcomes papers from a broad interdisciplinary area, since ‘numerical methods’ are a specific form of mathematics that involves creating and use of algorithms to map out the mathematical core of a practical problem.

Numerical methods naturally find application in all fields of engineering, physical sciences, life sciences, social sciences, medicine, business, and even arts. The common uses of numerical methods include approximation, simulation, and estimation, and there is almost no scientific field in which numerical methods do not find a use.

Some subjects included in ‘numerical methods’ are: IEEE arithmetic, root finding, systems of equations, least-squares estimation, maximum likelihood estimation, interpolation, numeric integration and differentiation—the list may go on and on. MSC 2010 subject classification for numerical methods includes: 30C30 (in conformal mapping theory), 31C20 (in connection with discrete potential theory), and 60H35 (computational methods for stochastic equations), and most of the subjects in 37Mxx (approximation methods and numerical treatment of dynamical systems), 49Mxx (numerical methods), 65XX (numerical analysis), 74Sxx (numerical methods for deformable solids), 76Mxx (basic methods in fluid mechanics), 78Mxx (basic methods for optics and electromagnetic theory), 80Mxx (basic methods for classical thermodynamics and heat transfer), 82Bxx (equilibrium statistical mechanics), 82Cxx (time-dependent statistical mechanics), and 91Gxx (mathematical finance). Topics of interest include (but are not limited to) the numerical methods for approximation, simulation, and estimation.

Prof. Lorentz Jäntschi
Prof. Daniela Daniela Roșca
Guest Editors

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Keywords

  • Asymptotic stability
  • Boundary element method
  • Diffusion
  • Elasticity
  • Errors
  • Finite element method
  • Flow of fluids
  • Hydrodynamics
  • Image processing
  • Integration
  • Markov processes
  • Monte–Carlo methods
  • Numerical algorithms
  • Numerical methods
  • Optimization
  • Partial differential equations
  • Robustness
  • Simulation
  • Stress analysis
  • System stability
  • Turbulence
  • Uncertainty analysis
  • Vibrations
  • Wavelets

Published Papers (10 papers)

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Research

15 pages, 959 KiB  
Article
dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS
by Monica Dessole, Fabio Marcuzzi and Marco Vianello
Mathematics 2020, 8(7), 1122; https://doi.org/10.3390/math8071122 - 09 Jul 2020
Cited by 3 | Viewed by 1922
Abstract
We provide a numerical package for the computation of a d-variate near G-optimal polynomial regression design of degree m on a finite design space X R d , by few iterations of a basic multiplicative algorithm followed by Tchakaloff-like compression of [...] Read more.
We provide a numerical package for the computation of a d-variate near G-optimal polynomial regression design of degree m on a finite design space X R d , by few iterations of a basic multiplicative algorithm followed by Tchakaloff-like compression of the discrete measure keeping the reached G-efficiency, via an accelerated version of the Lawson-Hanson algorithm for Non-Negative Least Squares (NNLS) problems. This package can solve on a personal computer large-scale problems where c a r d ( X ) × dim ( P 2 m d ) is up to 10 8 10 9 , being dim ( P 2 m d ) = 2 m + d d = 2 m + d 2 m . Several numerical tests are presented on complex shapes in d = 3 and on hypercubes in d > 3 . Full article
(This article belongs to the Special Issue Numerical Methods)
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15 pages, 782 KiB  
Article
On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence
by Janak Raj Sharma, Sunil Kumar and Lorentz Jäntschi
Mathematics 2020, 8(7), 1091; https://doi.org/10.3390/math8071091 - 03 Jul 2020
Cited by 22 | Viewed by 1774
Abstract
A number of optimal order multiple root techniques that require derivative evaluations in the formulas have been proposed in literature. However, derivative-free optimal techniques for multiple roots are seldom obtained. By considering this factor as motivational, here we present a class of optimal [...] Read more.
A number of optimal order multiple root techniques that require derivative evaluations in the formulas have been proposed in literature. However, derivative-free optimal techniques for multiple roots are seldom obtained. By considering this factor as motivational, here we present a class of optimal fourth order methods for computing multiple roots without using derivatives in the iteration. The iterative formula consists of two steps in which the first step is a well-known Traub–Steffensen scheme whereas second step is a Traub–Steffensen-like scheme. The Methodology is based on two steps of which the first is Traub–Steffensen iteration and the second is Traub–Steffensen-like iteration. Effectiveness is validated on different problems that shows the robust convergent behavior of the proposed methods. It has been proven that the new derivative-free methods are good competitors to their existing counterparts that need derivative information. Full article
(This article belongs to the Special Issue Numerical Methods)
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13 pages, 754 KiB  
Article
A Simple Method for Network Visualization
by Jintae Park, Sungha Yoon, Chaeyoung Lee and Junseok Kim
Mathematics 2020, 8(6), 1020; https://doi.org/10.3390/math8061020 - 22 Jun 2020
Cited by 2 | Viewed by 2297
Abstract
In this article, we present a simple method for network visualization. The proposed method is based on distmesh [P.O. Persson and G. Strang, A simple mesh generator in MATLAB, SIAM Review 46 (2004) pp. 329–345], which is a simple unstructured triangular mesh generator [...] Read more.
In this article, we present a simple method for network visualization. The proposed method is based on distmesh [P.O. Persson and G. Strang, A simple mesh generator in MATLAB, SIAM Review 46 (2004) pp. 329–345], which is a simple unstructured triangular mesh generator for geometries represented by a signed distance function. We demonstrate a good performance of the proposed algorithm through several network visualization examples. Full article
(This article belongs to the Special Issue Numerical Methods)
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15 pages, 514 KiB  
Article
Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron
by Adrian Holhoş and Daniela Roşca
Mathematics 2020, 8(6), 994; https://doi.org/10.3390/math8060994 - 17 Jun 2020
Cited by 1 | Viewed by 1543
Abstract
We construct a new volume preserving map from the unit ball B 3 to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded [...] Read more.
We construct a new volume preserving map from the unit ball B 3 to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of L 2 ( B 3 ) , consisting in piecewise constant functions with small local support. Full article
(This article belongs to the Special Issue Numerical Methods)
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14 pages, 817 KiB  
Article
On q-Quasi-Newton’s Method for Unconstrained Multiobjective Optimization Problems
by Kin Keung Lai, Shashi Kant Mishra and Bhagwat Ram
Mathematics 2020, 8(4), 616; https://doi.org/10.3390/math8040616 - 17 Apr 2020
Cited by 12 | Viewed by 2692
Abstract
A parameter-free optimization technique is applied in Quasi-Newton’s method for solving unconstrained multiobjective optimization problems. The components of the Hessian matrix are constructed using q-derivative, which is positive definite at every iteration. The step-length is computed by an Armijo-like rule which is [...] Read more.
A parameter-free optimization technique is applied in Quasi-Newton’s method for solving unconstrained multiobjective optimization problems. The components of the Hessian matrix are constructed using q-derivative, which is positive definite at every iteration. The step-length is computed by an Armijo-like rule which is responsible to escape the point from local minimum to global minimum at every iteration due to q-derivative. Further, the rate of convergence is proved as a superlinear in a local neighborhood of a minimum point based on q-derivative. Finally, the numerical experiments show better performance. Full article
(This article belongs to the Special Issue Numerical Methods)
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21 pages, 3507 KiB  
Article
Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions
by Lorentz Jäntschi
Mathematics 2020, 8(2), 216; https://doi.org/10.3390/math8020216 - 08 Feb 2020
Cited by 26 | Viewed by 2714
Abstract
In the subject of statistics for engineering, physics, computer science, chemistry, and earth sciences, one of the sampling challenges is the accuracy, or, in other words, how representative the sample is of the population from which it was drawn. A series of statistics [...] Read more.
In the subject of statistics for engineering, physics, computer science, chemistry, and earth sciences, one of the sampling challenges is the accuracy, or, in other words, how representative the sample is of the population from which it was drawn. A series of statistics were developed to measure the departure between the population (theoretical) and the sample (observed) distributions. Another connected issue is the presence of extreme values—possible observations that may have been wrongly collected—which do not belong to the population selected for study. By subjecting those two issues to study, we hereby propose a new statistic for assessing the quality of sampling intended to be used for any continuous distribution. Depending on the sample size, the proposed statistic is operational for known distributions (with a known probability density function) and provides the risk of being in error while assuming that a certain sample has been drawn from a population. A strategy for sample analysis, by analyzing the information about quality of the sampling provided by the order statistics in use, is proposed. A case study was conducted assessing the quality of sampling for ten cases, the latter being used to provide a pattern analysis of the statistics. Full article
(This article belongs to the Special Issue Numerical Methods)
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25 pages, 2778 KiB  
Article
Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1
by Soledad Moreno-Pulido, Francisco Javier Garcia-Pacheco, Clemente Cobos-Sanchez and Alberto Sanchez-Alzola
Mathematics 2020, 8(1), 85; https://doi.org/10.3390/math8010085 - 04 Jan 2020
Cited by 10 | Viewed by 2543
Abstract
In this manuscript we provide an exact solution to the maxmin problem max A x subject to B x 1 , where A and B are real matrices. This problem comes from a remodeling of [...] Read more.
In this manuscript we provide an exact solution to the maxmin problem max A x subject to B x 1 , where A and B are real matrices. This problem comes from a remodeling of max A x subject to min B x , because the latter problem has no solution. Our mathematical method comes from the Abstract Operator Theory, whose strong machinery allows us to reduce the first problem to max C x subject to x 1 , which can be solved exactly by relying on supporting vectors. Finally, as appendices, we provide two applications of our solution: first, we construct a truly optimal minimum stored-energy Transcranian Magnetic Stimulation (TMS) coil, and second, we find an optimal geolocation involving statistical variables. Full article
(This article belongs to the Special Issue Numerical Methods)
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24 pages, 3074 KiB  
Article
Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations
by Ampol Duangpan, Ratinan Boonklurb and Tawikan Treeyaprasert
Mathematics 2019, 7(12), 1201; https://doi.org/10.3390/math7121201 - 07 Dec 2019
Cited by 14 | Viewed by 2525
Abstract
The Burgers’ equation is one of the nonlinear partial differential equations that has been studied by many researchers, especially, in terms of the fractional derivatives. In this article, the numerical algorithms are invented to obtain the approximate solutions of time-fractional Burgers’ equations both [...] Read more.
The Burgers’ equation is one of the nonlinear partial differential equations that has been studied by many researchers, especially, in terms of the fractional derivatives. In this article, the numerical algorithms are invented to obtain the approximate solutions of time-fractional Burgers’ equations both in one and two dimensions as well as time-fractional coupled Burgers’ equations which their fractional derivatives are described in the Caputo sense. These proposed algorithms are constructed by applying the finite integration method combined with the shifted Chebyshev polynomials to deal the spatial discretizations and further using the forward difference quotient to handle the temporal discretizations. Moreover, numerical examples demonstrate the ability of the proposed method to produce the decent approximate solutions in terms of accuracy. The rate of convergence and computational cost for each example are also presented. Full article
(This article belongs to the Special Issue Numerical Methods)
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11 pages, 877 KiB  
Article
Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method
by Deepak Kumar, Janak Raj Sharma and Lorentz Jäntschi
Mathematics 2019, 7(10), 919; https://doi.org/10.3390/math7100919 - 02 Oct 2019
Cited by 11 | Viewed by 1895
Abstract
To locate a locally-unique solution of a nonlinear equation, the local convergence analysis of a derivative-free fifth order method is studied in Banach space. This approach provides radius of convergence and error bounds under the hypotheses based on the first Fréchet-derivative only. Such [...] Read more.
To locate a locally-unique solution of a nonlinear equation, the local convergence analysis of a derivative-free fifth order method is studied in Banach space. This approach provides radius of convergence and error bounds under the hypotheses based on the first Fréchet-derivative only. Such estimates are not introduced in the earlier procedures employing Taylor’s expansion of higher derivatives that may not exist or may be expensive to compute. The convergence domain of the method is also shown by a visual approach, namely basins of attraction. Theoretical results are endorsed via numerical experiments that show the cases where earlier results cannot be applicable. Full article
(This article belongs to the Special Issue Numerical Methods)
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11 pages, 877 KiB  
Article
Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function
by SAIRA, Shuhuang Xiang and Guidong Liu
Mathematics 2019, 7(10), 872; https://doi.org/10.3390/math7100872 - 20 Sep 2019
Cited by 5 | Viewed by 3740
Abstract
This paper aims to present a Clenshaw–Curtis–Filon quadrature to approximate the
solution of various cases of Cauchy-type singular integral equations (CSIEs) of the second kind with
a highly oscillatory kernel function. We adduce that the zero case oscillation (k = 0) proposed [...] Read more.
This paper aims to present a Clenshaw–Curtis–Filon quadrature to approximate the
solution of various cases of Cauchy-type singular integral equations (CSIEs) of the second kind with
a highly oscillatory kernel function. We adduce that the zero case oscillation (k = 0) proposed method
gives more accurate results than the scheme introduced in Dezhbord at el. (2016) and Eshkuvatov
at el. (2009) for small values of N. Finally, this paper illustrates some error analyses and numerical
results for CSIEs. Full article
(This article belongs to the Special Issue Numerical Methods)
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