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Symmetry
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Computational Mathematics and Its Applications in Numerical Analysis
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Computational Mathematics and Its Applications in Numerical Analysis

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 December 2025 | Viewed by 7746

Special Issue Editors

Prof. Dr. Shufei Wu

E-Mail Website
Guest Editor
Department of Statistics, Tamkang University, Taipei 251301, Taiwan
Interests: statistics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Shou-Fu Tian

E-Mail Website
Guest Editor
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
Interests: partial differential equations

Special Issue Information

Dear Colleagues,

Computational mathematics has became increasingly significant across diverse fields in science and engineering. This Special Issue aims to showcase recent research findings in computational mathematics and advanced numerical methods. The publication will encompass original and survey papers delving into analysis, modeling, control, and optimization. Contributions employing advanced computational mathematics and numerical methods, with applications spanning various realms of science, technology, engineering, and related disciplines, are encouraged and warmly welcomed.

Potential topics include, but are not limited to, the following:

  • Computational mathematics for data science;
  • Applied mathematics in optimization problems;
  • Signal processing and mathematical images;
  • Scientific computing and algorithms;
  • Numerical analysis for problems arising in various fields.

Prof. Dr. Shufei Wu
Prof. Dr. Shou-Fu Tian
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. Symmetry 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

  • computational mathematics for data science
  • applied mathematics in optimization problems
  • signal processing and mathematical images
  • scientific computing and algorithms
  • numerical analysis for problems arising in various fields 

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (7 papers)

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Research

19 pages, 1819 KiB  
Article
Adaptive Optics Retinal Image Restoration Using Total Variation with Overlapping Group Sparsity
by Xiaotong Chen, Yurong Shi and Hongsun Fu
Symmetry 2025, 17(5), 660; https://doi.org/10.3390/sym17050660 (registering DOI) - 26 Apr 2025
Viewed by 89
Abstract
Adaptive optics (AO)-corrected retina flood illumination imaging technology is widely used for investigating both structural and functional aspects of the retina. Given the inherent low-contrast nature of original retinal images, it is necessary to perform image restoration. Total variation (TV) regularization is an [...] Read more.
Adaptive optics (AO)-corrected retina flood illumination imaging technology is widely used for investigating both structural and functional aspects of the retina. Given the inherent low-contrast nature of original retinal images, it is necessary to perform image restoration. Total variation (TV) regularization is an efficient regularization technique for AO retinal image restoration. However, a main shortcoming of TV regularization is its potential to experience the staircase effects, particularly in smooth regions of the image. To overcome the drawback, a new image restoration model is proposed for AO retinal images. This model utilizes the overlapping group sparse total variation (OGSTV) as a regularization term. Due to the structural characteristics of AO retinal images, only partial information regarding the PSF is known. Consequently, we have to solve a more complicated myopic deconvolution problem. To address this computational challenge, we propose an ADMM-MM-LAP method to solve the proposed model. First, we apply the alternating direction method of multiplier (ADMM) as the outer-layer optimization method. Then, appropriate algorithms are employed to solve the ADMM subproblems based on their inherent structures. Specifically, the majorization–minimization (MM) method is applied to handle the asymmetry OGSTV regularization component, while a modified version of the linearize and project (LAP) method is adopted to address the tightly coupled subproblem. Theoretically, we establish the complexity analysis of the proposed method. Numerical results demonstrate that the proposed model outperforms the existing state-of-the-art TV model across several metrics. Full article
(This article belongs to the Special Issue Computational Mathematics and Its Applications in Numerical Analysis)
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22 pages, 2491 KiB  
Article
Computational Analysis of the Comprehensive Lifetime Performance Index for Exponentiated Fréchet Lifetime Distribution Products with Multi-Components
by Shu-Fei Wu and Hsueh-Chien Yeh
Symmetry 2024, 16(8), 1060; https://doi.org/10.3390/sym16081060 - 16 Aug 2024
Viewed by 1335
Abstract
The lifetime performance index is commonly used in the manufacturing industry to evaluate the performance of the capabilities of the production process. For products with multiple components, the comprehensive lifetime performance index, which is a monotonically increasing function of the overall process yield, [...] Read more.
The lifetime performance index is commonly used in the manufacturing industry to evaluate the performance of the capabilities of the production process. For products with multiple components, the comprehensive lifetime performance index, which is a monotonically increasing function of the overall process yield, is used to relate to each individual lifetime performance index. For products where the lifetime of the ith component follows an exponentiated Fréchet lifetime distribution, we examine the maximum likelihood estimators for both the comprehensive and individual lifetime performance indices based on the progressive type I interval-censored samples, deriving their asymptotic distributions. By specifying the target level for the comprehensive lifetime performance index, we can set the desired level for individual indices. A testing procedure, using the maximum likelihood estimator as the test statistic, was developed to determine if the comprehensive lifetime performance index meets the target. Given that the lifetime distribution is asymmetric, this study pertains to asymmetrical probability distributions and their applications across diverse fields. We illustrate the power analysis of this testing procedure with figures and summarize key findings. Finally, we demonstrate the application of this testing algorithm with a practical example involving two components to verify if the overall production process achieves the assigned target level. Full article
(This article belongs to the Special Issue Computational Mathematics and Its Applications in Numerical Analysis)
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14 pages, 802 KiB  
Article
A Generalized Iterated Tikhonov Method in the Fourier Domain for Determining the Unknown Source of the Time-Fractional Diffusion Equation
by Bin Zheng, Junfeng Liu, Zhenyu Zhao, Zhihong Dou and Benxue Gong
Symmetry 2024, 16(7), 864; https://doi.org/10.3390/sym16070864 - 8 Jul 2024
Viewed by 1210
Abstract
In this paper, an inverse problem of determining a source in a time-fractional diffusion equation is investigated. A Fourier extension scheme is used to approximate the solution to avoid the impact on smoothness caused by directly using singular system eigenfunctions for approximation. A [...] Read more.
In this paper, an inverse problem of determining a source in a time-fractional diffusion equation is investigated. A Fourier extension scheme is used to approximate the solution to avoid the impact on smoothness caused by directly using singular system eigenfunctions for approximation. A modified implicit iteration method is proposed as a regularization technique to stabilize the solution process. The convergence rates are derived when a discrepancy principle serves as the principle for choosing the regularization parameters. Numerical tests are provided to further verify the efficacy of the proposed method. Full article
(This article belongs to the Special Issue Computational Mathematics and Its Applications in Numerical Analysis)
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17 pages, 325 KiB  
Article
Solutions for the Nonlinear Mixed Variational Inequality Problem in the System
by Husain Gissy, Abdullah Ali H. Ahmadini and Salahuddin
Symmetry 2024, 16(7), 796; https://doi.org/10.3390/sym16070796 - 25 Jun 2024
Viewed by 888
Abstract
Our paper proposes a system of nonlinear mixed variational inequality problems (SNMVIPs) on Banach spaces. Under suitable assumptions, using the K-Fan fixed point theorem and Minty techniques, we demonstrate that the solution set to the SNMVIP is nonempty, weakly compact, and unique. Additionally, [...] Read more.
Our paper proposes a system of nonlinear mixed variational inequality problems (SNMVIPs) on Banach spaces. Under suitable assumptions, using the K-Fan fixed point theorem and Minty techniques, we demonstrate that the solution set to the SNMVIP is nonempty, weakly compact, and unique. Additionally, we suggest a stability result for the SNMVIPs by perturbing the duality mappings. Furthermore, we present an optimal control problem that is governed by the SNMVIPs and show that it can be solved. Full article
(This article belongs to the Special Issue Computational Mathematics and Its Applications in Numerical Analysis)
23 pages, 1001 KiB  
Article
A Fast Method for the Off-Boundary Evaluation of Laplace Layer Potentials by Convolution Sums
by Wenchao Guan, Zhicheng Wang, Leqi Xue and Yueen Hou
Symmetry 2024, 16(6), 764; https://doi.org/10.3390/sym16060764 - 18 Jun 2024
Viewed by 1260
Abstract
In off-boundary computations of layer potentials, the near-singularities in integrals near the boundary presents challenges for conventional quadrature methods in achieving high precision. Additionally, the significant complexity of O(n2) interactions between n targets and n sources reduces the efficiency [...] Read more.
In off-boundary computations of layer potentials, the near-singularities in integrals near the boundary presents challenges for conventional quadrature methods in achieving high precision. Additionally, the significant complexity of O(n2) interactions between n targets and n sources reduces the efficiency of these methods. A fast and accurate numerical algorithm is presented for computing the Laplace layer potentials on a circle with a boundary described by a polar curve. This method can maintain high precision even when evaluating targets located at a close distance from the boundary. The radial symmetry of the integral kernels simplifies their description. By exploiting the polar form of the boundary and applying a one-dimensional exponential sum approximation along the radial direction, an approximation of layer potentials by the convolution sum is obtained. The algorithm uses FFT convolution to accelerate computation and employs a local quadrature to maintain accuracy for nearly singular terms. Consequently, it achieves spectral accuracy in regions outside of a sufficiently small neighborhood of the boundary and requires O(nlogn) arithmetic operations. With the help of this algorithm, layer potentials can be efficiently evaluated on a computational domain. Full article
(This article belongs to the Special Issue Computational Mathematics and Its Applications in Numerical Analysis)
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13 pages, 237 KiB  
Article
An Efficient Solution of Multiplicative Differential Equations through Laguerre Polynomials
by Hatice Yalman Kosunalp, Selcuk Bas and Selahattin Kosunalp
Symmetry 2024, 16(6), 748; https://doi.org/10.3390/sym16060748 - 15 Jun 2024
Cited by 1 | Viewed by 935
Abstract
The field of multiplicative analysis has recently garnered significant attention, particularly in the context of solving multiplicative differential equations (MDEs). The symmetry concept in MDEs facilitates the determination of invariant solutions and the reduction of these equations by leveraging their intrinsic symmetrical properties. [...] Read more.
The field of multiplicative analysis has recently garnered significant attention, particularly in the context of solving multiplicative differential equations (MDEs). The symmetry concept in MDEs facilitates the determination of invariant solutions and the reduction of these equations by leveraging their intrinsic symmetrical properties. This study is motivated by the need for efficient methods to address MDEs, which are critical in various applications. Our novel contribution involves leveraging the fundamental properties of orthogonal polynomials, specifically Laguerre polynomials, to derive new solutions for MDEs. We introduce the definitions of Laguerre multiplicative differential equations and multiplicative Laguerre polynomials. By applying the power series method, we construct these multiplicative Laguerre polynomials and rigorously prove their basic properties. The effectiveness of our proposed solution is validated through illustrative examples, demonstrating its practical applicability and potential for advancing the field of multiplicative analysis. Full article
(This article belongs to the Special Issue Computational Mathematics and Its Applications in Numerical Analysis)
19 pages, 366 KiB  
Article
A New Adaptive Levenberg–Marquardt Method for Nonlinear Equations and Its Convergence Rate under the Hölderian Local Error Bound Condition
by Yang Han and Shaoping Rui
Symmetry 2024, 16(6), 674; https://doi.org/10.3390/sym16060674 - 30 May 2024
Cited by 2 | Viewed by 1047
Abstract
The Levenberg–Marquardt (LM) method is one of the most significant methods for solving nonlinear equations as well as symmetric and asymmetric linear equations. To improve the method, this paper proposes a new adaptive LM algorithm by modifying the LM parameter, combining the trust [...] Read more.
The Levenberg–Marquardt (LM) method is one of the most significant methods for solving nonlinear equations as well as symmetric and asymmetric linear equations. To improve the method, this paper proposes a new adaptive LM algorithm by modifying the LM parameter, combining the trust region technique and the non-monotone technique. It is interesting that the new algorithm is constantly optimized by adaptively choosing the LM parameter. To evaluate the effectiveness of the new algorithm, we conduct tests using various examples. To extend the convergence results, we prove the convergence of the new algorithm under the Hölderian local error bound condition rather than the commonly used local error bound condition. Theoretical analysis and numerical results show that the new algorithm is stable and effective. Full article
(This article belongs to the Special Issue Computational Mathematics and Its Applications in Numerical Analysis)
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