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Axioms
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Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition
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Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 4635

Special Issue Editor

Prof. Dr. Peixin Ye

E-Mail Website
Guest Editor
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
Interests: approximation theory and applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

A basic and important problem in numerical computation is the need to resolve complicated functions into simpler, easier-to-compute functions. Good numerical methods based on the theory of the approximation of functions have many applications in numerous branches of applied mathematics, such as computer-aided geometric design, machine learning, and signal processing. The primary purpose of this Special Issue is to highlight the recent progress made in the theory and application of function approximation. Topics may include, but are not limited to, the following: multivariate approximation, numerical integration, optimization, machine learning, signal processing, and computer-aided geometric design.

Prof. Dr. Peixin Ye
Guest Editor

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. Axioms is an international peer-reviewed open access monthly 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 2400 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

  • approximation of functions
  • numerical computation
  • computational complexity
  • optimization
  • machine learning
  • signal processing

Related Special Issue

Published Papers (7 papers)

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Research

14 pages, 3346 KiB  
Article
Numerical Computation of 2D Domain Integrals in Boundary Element Method by (α, β) Distance Transformation for Transient Heat Conduction Problems
by Yunqiao Dong, Zhengxu Tan and Hengbo Sun
Axioms 2024, 13(7), 490; https://doi.org/10.3390/axioms13070490 - 22 Jul 2024
Viewed by 246
Abstract
When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient [...] Read more.
When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient heat conduction. However, as the time step decreases progressively and approaches zero, the integrand of the domain integrals is close to singular, resulting in large errors when employing standard Gaussian quadrature directly. To solve the problem and further improve the calculation accuracy of the domain integrals, an (α, β) distance transformation is presented. Distance transformation is a simple and efficient method for eliminating near-singularity, typically applied to nearly singular integrals. Firstly, the (α, β) coordinate transformation is introduced. Then, a new distance transformation for the domain integrals is constructed by replacing the shortest distance with the time step. With the new method, the integrand of the domain integrals is substantially smoothed, and the singularity arising from small time steps in the domain integrals is effectively eliminated. Thus, more accurate results can be obtained by the (α, β) distance transformation. Different sizes of time steps, positions of source point, and shapes of integration elements are considered in numerical examples. Comparative studies of the numerical results for the domain integrals using various methods demonstrate that higher accuracy and efficiency are achieved by the proposed method. Full article
(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)
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11 pages, 253 KiB  
Article
Estimates of Eigenvalues and Approximation Numbers for a Class of Degenerate Third-Order Partial Differential Operators
by Mussakan Muratbekov, Ainash Suleimbekova and Mukhtar Baizhumanov
Axioms 2024, 13(7), 451; https://doi.org/10.3390/axioms13070451 - 3 Jul 2024
Viewed by 294
Abstract
In this paper, we study the spectral properties of a class of degenerate third-order partial differential operators with variable coefficients presented in a rectangle. Conditions are found to ensure the existence and compactness of the inverse operator. A theorem on estimates of approximation [...] Read more.
In this paper, we study the spectral properties of a class of degenerate third-order partial differential operators with variable coefficients presented in a rectangle. Conditions are found to ensure the existence and compactness of the inverse operator. A theorem on estimates of approximation numbers is proven. Here, we note that finding estimates of approximation numbers, as well as extremal subspaces, for a set of solutions to the equation is a task that is certainly important from both a theoretical and a practical point of view. The paper also obtained an upper bound for the eigenvalues. Note that, in this paper, estimates of eigenvalues and approximation numbers for the degenerate third-order partial differential operators are obtained for the first time. Full article
(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)
12 pages, 930 KiB  
Article
Constructing Approximations to Bivariate Piecewise-Smooth Functions
by David Levin
Axioms 2024, 13(7), 428; https://doi.org/10.3390/axioms13070428 - 26 Jun 2024
Viewed by 685
Abstract
This paper demonstrates that the space of piecewise-smooth bivariate functions can be well-approximated by the space of the functions defined by a set of simple (non-linear) operations on smooth uniform tensor product splines. The examples include bivariate functions with jump discontinuities or normal [...] Read more.
This paper demonstrates that the space of piecewise-smooth bivariate functions can be well-approximated by the space of the functions defined by a set of simple (non-linear) operations on smooth uniform tensor product splines. The examples include bivariate functions with jump discontinuities or normal discontinuities across curves, and even across more involved geometries such as a three-corner discontinuity. The provided data may be uniform or non-uniform, and noisy, and the approximation procedure involves non-linear least-squares minimization. Also included is a basic approximation theorem for functions with jump discontinuity across a smooth curve. Full article
(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)
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20 pages, 280 KiB  
Article
Tractability of Multivariate Approximation Problem on Euler and Wiener Integrated Processes
by Jie Zhang
Axioms 2024, 13(5), 326; https://doi.org/10.3390/axioms13050326 - 15 May 2024
Viewed by 472
Abstract
This paper examines the tractability of multivariate approximation problems under the normalized error criterion for a zero-mean Gaussian measure in an average-case setting. The Gaussian measure is associated with a covariance kernel, which is represented by the tensor product of one-dimensional kernels corresponding [...] Read more.
This paper examines the tractability of multivariate approximation problems under the normalized error criterion for a zero-mean Gaussian measure in an average-case setting. The Gaussian measure is associated with a covariance kernel, which is represented by the tensor product of one-dimensional kernels corresponding to Euler and Wiener integrated processes with non-negative and nondecreasing smoothness parameters {rd}d∈N. We give matching sufficient and necessary conditions for various concepts of tractability in terms of the asymptotic properties of the regularity parameters, except for (s, 0)-WT. Full article
(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)
17 pages, 916 KiB  
Article
Sparse Signal Recovery via Rescaled Matching Pursuit
by Wan Li and Peixin Ye
Axioms 2024, 13(5), 288; https://doi.org/10.3390/axioms13050288 - 24 Apr 2024
Viewed by 683
Abstract
We propose the Rescaled Matching Pursuit (RMP) algorithm to recover sparse signals in high-dimensional Euclidean spaces. The RMP algorithm has less computational complexity than other greedy-type algorithms, such as Orthogonal Matching Pursuit (OMP). We show that if the restricted isometry property is satisfied, [...] Read more.
We propose the Rescaled Matching Pursuit (RMP) algorithm to recover sparse signals in high-dimensional Euclidean spaces. The RMP algorithm has less computational complexity than other greedy-type algorithms, such as Orthogonal Matching Pursuit (OMP). We show that if the restricted isometry property is satisfied, then the upper bound of the error between the original signal and its approximation can be derived. Furthermore, we prove that the RMP algorithm can find the correct support of sparse signals from random measurements with a high probability. Our numerical experiments also verify this conclusion and show that RMP is stable with the noise. So, the RMP algorithm is a suitable method for recovering sparse signals. Full article
(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)
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18 pages, 382 KiB  
Article
On the Convergence of an Approximation Scheme of Fractional-Step Type, Associated to a Nonlinear Second-Order System with Coupled In-Homogeneous Dynamic Boundary Conditions
by Constantin Fetecău, Costică Moroşanu and Silviu-Dumitru Pavăl
Axioms 2024, 13(5), 286; https://doi.org/10.3390/axioms13050286 - 23 Apr 2024
Viewed by 595
Abstract
The paper concerns a nonlinear second-order system of coupled PDEs, having the principal part in divergence form and subject to in-homogeneous dynamic boundary conditions, for both θ(t,x) and φ(t,x). Two main topics [...] Read more.
The paper concerns a nonlinear second-order system of coupled PDEs, having the principal part in divergence form and subject to in-homogeneous dynamic boundary conditions, for both θ(t,x) and φ(t,x). Two main topics are addressed here, as follows. First, under a certain hypothesis on the input data, f1, f2, w1, w2, α, ξ, θ0, α0, φ0, and ξ0, we prove the well-posedness of a solution θ,α,φ,ξ, which is θ(t,x),α(t,x)∈Wp1,2(Q)×Wp1,2(Σ), φ(t,x),ξ(t,x)∈Wν1,2(Q)×Wp1,2(Σ), ν=min{q,μ}. According to the new formulation of the problem, we extend the previous results, allowing the new mathematical model to be even more complete to describe the diversity of physical phenomena to which it can be applied: interface problems, image analysis, epidemics, etc. The main goal of the present paper is to develop an iterative scheme of fractional-step type in order to approximate the unique solution to the nonlinear second-order system. The convergence result is established for the new numerical method, and on the basis of this approach, a conceptual algorithm, alg-frac_sec-ord_u+varphi_dbc, is elaborated. The benefit brought by such a method consists of simplifying the computations so that the time required to approximate the solutions decreases significantly. Some conclusions are given as well as new research topics for the future. Full article
(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)
16 pages, 896 KiB  
Article
Vector-Valued Shepard Processes: Approximation with Summability
by Oktay Duman and Biancamaria Della Vecchia
Axioms 2023, 12(12), 1124; https://doi.org/10.3390/axioms12121124 - 15 Dec 2023
Cited by 1 | Viewed by 960
Abstract
In this work, vector-valued continuous functions are approximated uniformly on the unit hypercube by Shepard operators. If λ denotes the usual parameter of the Shepard operators and m is the dimension of the hypercube, then our results show that it is possible to [...] Read more.
In this work, vector-valued continuous functions are approximated uniformly on the unit hypercube by Shepard operators. If λ denotes the usual parameter of the Shepard operators and m is the dimension of the hypercube, then our results show that it is possible to obtain a uniform approximation of a continuous vector-valued function by these operators when λ≥m+1. By using three-dimensional parametric plots, we illustrate this uniform approximation for some vector-valued functions. Finally, the influence in approximation by regular summability processes is studied, and their motivation is shown. Full article
(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)
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