Article
Mathematics, Applied
Hyeong Moon Yoon, Young Joon Ahn
Summary: In this paper, the approximation of a circular arc using hexic polynomial curves with 12 contacts is considered. Two methods are presented to obtain Gk approximation curves, where k=3, 4, and these curves interpolate at both endpoints and the midpoint of the circular arc. The approximation curves can be obtained by solving a degree six equation. It is shown that the approximation orders of the methods are 12. The optimal approximation is found for each method and numerical examples are provided to illustrate the approximation orders are 12.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Emil Zagar
Summary: This paper focuses on the interpolation problem of two points, two corresponding tangent directions, curvatures, and arc length sampled from a circular arc (circular arc data). A general approach using Planar Pythagorean-hodograph (PH) curves of degree seven is presented, and the strong dependence of the solution on the general data is demonstrated. For circular arc data, a complicated system of nonlinear equations is reduced to a numerical solution of only one algebraic equation of degree 6, and the existence of admissible solutions is analyzed. Criteria for selecting the most appropriate solution in the case of multiple solutions are described, and an asymptotic analysis is provided. Numerical examples are used to validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Irina Georgieva, Clemens Hofreither
Summary: The algorithm is based on a formulation of the problem as a nonlinear system of equations and barycentric interpolation for best uniform rational approximation of real continuous functions on real intervals, and can handle singularities and arbitrary degrees for numerator and denominator. Numerical experiments show that it typically converges globally and exhibits superlinear convergence in a neighborhood of the solution. Interesting auxiliary results include formulae for the derivatives of barycentric rational interpolants and for the derivative of the nullspace of a full-rank matrix.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Michael S. Floater
Summary: In this note, a solution is derived for the problem of finding a polynomial of degree at most $n$ that best approximates data at $n+2$ points in the $l_{p}$ norm. The solution can be expressed as a convex combination of Lagrange interpolants over subsets of $n+1$ points, and the error oscillates in sign.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Computer Science, Information Systems
Liqiang Lin, Pengdi Huang, Fuyou Xue, Kai Xu, Daniel Cohen-Or, Hui Huang
Summary: This study introduces a shape-aware point convolution method for irregular point clouds to accurately learn the geometric shapes and extract deep features. The results show that HPC-DNN outperforms traditional point convolution methods in semantic segmentation tasks, achieving performance improvements.
SCIENCE CHINA-INFORMATION SCIENCES
(2021)
Article
Mathematics, Applied
Clemens Hofreither
Summary: The algorithm, named BRASIL, iteratively adjusts interval lengths to achieve the best rational approximation of a scalar function by exploiting the equioscillation of local maximum errors. It utilizes the barycentric rational formula for stable computation and demonstrates improved convergence rate with the Anderson acceleration method. The algorithm shows excellent numerical stability and computational efficiency for high degree rational approximations.
NUMERICAL ALGORITHMS
(2021)
Article
Engineering, Electrical & Electronic
Kun Li, Shihong Yue, Yongguang Tan, Huaxiang Wang, Xinshan Zhu
Summary: Electric tomography (ET) is an advanced visualization technique that offers low-cost, rapid-response, nonradiative, and nonintrusive advantages over other tomography modalities. However, the imaging resolution of ET is significantly low, resulting in a shortage of required measurements compared to the number of pixels in a detection field. This study proposes a reproducing kernel-based best interpolation (RKBI) method, which effectively increases the number of numeric measurements in the ET process. RKBI outperforms existing interpolation methods in terms of approximation error for a set of available measurements. The optimality of RKBI is validated through both theoretical and experimental frameworks, highlighting its ability to enhance the spatial resolution and steadiness of ET images.
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT
(2023)
Article
Automation & Control Systems
Taweechai Nuntawisuttiwong, Natasha Dejdumrong
Summary: A new scheme is proposed to approximate an arbitrary degree Bezier curve by a sequence of circular arcs, which represents the shape of the curve in a more efficient way. The technique used for segmentation involves investigating inner angles and tangent vectors, allowing the curve to be divided into subcurves.
INFORMATION TECHNOLOGY AND CONTROL
(2021)
Article
Computer Science, Software Engineering
Young Joon Ahn
Summary: This paper presents a method for conic approximation using polynomial curves of odd degree, showing the relationship between the coefficients of the polynomial curve and Bernoulli's triangle. The paper also determines the scaling factor for equi-oscillation and Hausdorff distance between the half circle and the polynomial approximation curve.
COMPUTER AIDED GEOMETRIC DESIGN
(2021)
Article
Engineering, Mechanical
Christian Adams, Joachim Boes, Tobias Melz
Summary: This paper introduces similitude analysis methods for vibration analysis of rectangular plates and a performance measure, the Mahalanobis distance, to evaluate the accuracy of scaling laws in approximating the actual vibration responses. The study validates that the Mahalanobis distance is effective in assessing the performance of similitude analyses. It is found that scaling laws can sufficiently approximate the vibration responses of rectangular plates up to a maximum permissible amount of geometrical distortion.
MECHANICAL SYSTEMS AND SIGNAL PROCESSING
(2022)
Article
Mathematics, Applied
Saisai Shi, Bo Tan, Qinglong Zhou
Summary: This passage discusses the attractor and points with multiple codings in an iterated function system on a compact metric space. It introduces the concepts of digit sequences and shortest distance functions to describe the relationships between points. The paper focuses on the asymptotic behavior of the shortest distance function as n approaches infinity, calculates the Hausdorff dimensions of exceptional sets, and studies exceptional sets in specific systems.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Xiaolong Zhang
Summary: Both the best polynomial approximation and the Chebyshev approximation play important roles in numerical analysis. While the best approximation was traditionally considered superior to the Chebyshev approximation in the uniform norm, recent studies have shown that this is not always the case, especially for functions with singularities. This paper presents findings regarding functions with logarithmic endpoint singularities, showing that the pointwise errors of the Chebyshev approximation are smaller than those of the best approximation of the same degree, except at the narrow boundary layer. The paper also provides theorems to explain this phenomenon.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Arpita Mal, Kallol Paul
Summary: In this paper, several distance formulae in the space of compact operators are presented in terms of extreme points and semi-inner-products. The best approximation to an element out of a subspace is characterized, and a sufficient condition for unique best approximation is obtained. The usefulness of the results is demonstrated with examples. Finally, using the distance formulae, the approximate Birkhoff-James orthogonality of an element to a subspace in the space of compact operators is characterized.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Feifei Qu, Xin Wei
Summary: This paper investigates Jordan curves and their constant distance boundaries in the complex plane, proving the convergence of Gamma(lambda) to Gamma under certain conditions. This has important implications for understanding the properties of Jordan curves.
Article
Mathematics, Applied
Marjeta Knez, Francesca Pelosi, Maria Lucia Sampoli
Summary: In this paper, we focus on the construction of G(2) planar Pythagorean-hodograph (PH) spline curves that interpolate points, tangent directions, and curvatures, while also having a prescribed arc-length. The interpolation method used is completely local, and each spline segment is defined as a PH biarc curve of degree 7. By fixing two free parameters to zero, it is shown that the length constraint can be satisfied for any data and any chosen ratio between boundary tangents. The bending energy is then used to select the best solution, and numerical examples are provided to illustrate the theoretical results and confirm the approximation order of 5.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Software Engineering
Grega Cigler, Emil Zagar
Summary: This paper discusses the interpolation problem of planar parametric cubic curves with prescribed arc lengths for two points and two tangent directions. By using Pythagorean-hodograph (PH) curves, the problem can be simplified and closed form solutions are provided.
COMPUTER AIDED GEOMETRIC DESIGN
(2022)
Article
Mathematics, Applied
M. S. Bruzon, T. M. Garrido, R. de la Rosa
Summary: We study a family of generalized Zakharov-Kuznetsov modified equal width equations in (2+1)-dimensions involving an arbitrary function and three parameters. By using the Lie group theory, we classify the Lie point symmetries of these equations and obtain exact solutions. We also show that this family of equations admits local low-order multipliers and derive all local low-order conservation laws through the multiplier approach.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Dohee Jung, Changbum Chun
Summary: The paper presents a general approach to enhance the Pade iterations for computing the matrix sign function by selecting an arbitrary three-point family of methods based on weight functions. The approach leads to a multi-parameter family of iterations and allows for the discovery of new methods. Convergence and stability analysis as well as numerical experiments confirm the improved performance of the new methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Abhishek Yadav, Amit Setia, M. Thamban Nair
Summary: In this paper, we propose a Galerkin's residual-based numerical scheme for solving a system of Cauchy-type singular integral equations using Chebyshev polynomials. We prove the well-posedness of the system and derive a theoretical error bound and convergence order. The numerical examples validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fernando Chacon-Gomez, M. Eugenia Cornejo, Jesus Medina, Eloisa Ramirez-Poussa
Summary: The use of decision rules allows for reliable extraction of information and inference of conclusions from relational databases, but the concepts of decision algorithms need to be extended in fuzzy environments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Ilhame Amirali, Gabil M. Amiraliyev
Summary: This paper considers the one-dimensional initial-boundary problem for a pseudoparabolic equation with a time delay. To solve this problem numerically, a higher-order difference method is constructed and the error estimate for its solution is obtained. Based on the method of energy estimates, the fully discrete scheme is shown to be convergent of order four in space and of order two in time. The given numerical results illustrate the convergence and effectiveness of the numerical method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Tong-tong Shang, Guo-ji Tang, Wen-sheng Jia
Summary: The goal of this paper is to investigate a class of linear complementarity problems over tensor-spaces, denoted by TLCP, which is an extension of the classical linear complementarity problem. First, two classes of structured tensors over tensor-spaces (i.e., T-R tensor and T-RO tensor) are introduced and some equivalent characterizations are discussed. Then, the lower bound and upper bound of the solutions in the sense of the infinity norm of the TLCP are obtained when the problem has a solution.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Fabio Difonzo, Pawel Przybylowicz, Yue Wu
Summary: This paper focuses on the existence, uniqueness, and approximation of solutions of delay differential equations (DDEs) with Caratheodory type right-hand side functions. It presents the construction of the randomized Euler scheme for DDEs and investigates its error. Furthermore, the paper reports the results of numerical experiments.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Priyanka Roy, Geetanjali Panda, Dong Qiu
Summary: In this article, a gradient based descent line search scheme is proposed for solving interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational perspectives. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by the proposed scheme.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Zhongqian Wang, Changqing Ye, Eric T. Chung
Summary: In this paper, the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions for elasticity equations in high contrast media is developed. The method offers advantages such as independence of target region's contrast from precision and significant impact of oversampling domain sizes on numerical accuracy. Furthermore, this is the first proof of convergence of CEM-GMsFEM with mixed boundary conditions for elasticity equations. Numerical experiments demonstrate the method's performance.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Samaneh Soradi-Zeid, Maryam Alipour
Summary: The Laguerre polynomials are a new set of basic functions used to solve a specific class of optimal control problems specified by integro-differential equations, namely IOCP. The corresponding operational matrices of derivatives are calculated to extend the solution of the problem in terms of Laguerre polynomials. By considering the basis functions and using the collocation method, the IOCP is simplified into solving a system of nonlinear algebraic equations. The proposed method has been proven to have an error bound and convergence analysis for the approximate optimal value of the performance index. Finally, examples are provided to demonstrate the validity and applicability of this technique.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Almudena P. Marquez, Maria L. Gandarias, Stephen C. Anco
Summary: A generalization of the KP equation involving higher-order dispersion is studied. The Lie point symmetries and conservation laws of the equation are obtained using Noether's theorem and the introduction of a potential. Sech-type line wave solutions are found and their features, including dark solitary waves on varying backgrounds, are discussed.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Susanne Saminger-Platz, Anna Kolesarova, Adam Seliga, Radko Mesiar, Erich Peter Klement
Summary: In this article, we study real functions defined on the unit square satisfying basic properties and explore the conditions for generating bivariate copulas using parameterized transformations and other constructions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Lulu Tian, Nattaporn Chuenjarern, Hui Guo, Yang Yang
Summary: In this paper, a new local discontinuous Galerkin (LDG) algorithm is proposed to solve the incompressible Euler equation in two dimensions on overlapping meshes. The algorithm solves the vorticity, velocity field, and potential function on different meshes. The method employs overlapping meshes to ensure continuity of velocity along the interfaces of the primitive meshes, allowing for the application of upwind fluxes. The article introduces two sufficient conditions to maintain the maximum principle of vorticity.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Cheng Wang, Jilu Wang, Steven M. Wise, Zeyu Xia, Liwei Xu
Summary: In this paper, a temporally second-order accurate numerical scheme for the Cahn-Hilliard-Magnetohydrodynamics system of equations is proposed and analyzed. The scheme utilizes a modified Crank-Nicolson-type approximation for time discretization and a mixed finite element method for spatial discretization. The modified Crank-Nicolson approximation allows for mass conservation and energy stability analysis. Error estimates are derived for the phase field, velocity, and magnetic fields, and numerical examples are presented to validate the proposed scheme's theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Mingyu He, Wenyuan Liao
Summary: This paper presents a numerical method for solving reaction-diffusion equations in spatially heterogeneous domains, which are commonly used to model biological applications. The method utilizes a fourth-order compact alternative directional implicit scheme based on Pade approximation-based operator splitting techniques. Stability analysis shows that the method is unconditionally stable, and numerical examples demonstrate its high efficiency and high order accuracy in both space and time.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)