Applied Numerical Mathematics最新文献

{"title":"A surface mesh DG-VEM for elliptic membrane shell model","authors":"Qian Yang ,&nbsp;Xiaoqin Shen ,&nbsp;Jikun Zhao ,&nbsp;Zhiming Gao","doi":"10.1016/j.apnum.2025.06.014","DOIUrl":"10.1016/j.apnum.2025.06.014","url":null,"abstract":"<div><div>Elliptic membrane shell (EMS), characterized by a system with complex variable coefficients on a surface, poses significant challenges for numerical discretization. In this paper, leveraging the differing regularity of displacement components, we propose a discontinuous Galerkin virtual element method (DG-VEM) for the EMS model. Specifically, we construct <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span>-continuous virtual element spaces for the first two components, whereas the third component is discretized on each element using a polynomial of degree <em>l</em>, with no continuity enforced across element boundaries. This method offers high mesh flexibility, eliminates the need for explicit basis function expressions, and improves accuracy to achieve convergence of any desired order. Furthermore, we establish the existence, uniqueness, stability, and convergence of the numerical solution, along with rigorous error estimates. Several numerical examples are presented to test the convergence and stability of the DG-VEM. Additionally, we demonstrate the method's adaptability to diverse grid subdivisions and show that, for comparable error levels, the DG-VEM for the EMS model requires significantly fewer degrees of freedom than traditional finite element methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 298-318"},"PeriodicalIF":2.2,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exponentially accurate spectral Monte Carlo method for linear PDEs and their error estimates","authors":"Jiaying Feng,&nbsp;Changtao Sheng,&nbsp;Chenglong Xu","doi":"10.1016/j.apnum.2025.06.010","DOIUrl":"10.1016/j.apnum.2025.06.010","url":null,"abstract":"<div><div>This paper introduces a spectral Monte Carlo iterative method (SMC) for solving linear Poisson and parabolic equations driven by <em>α</em>-stable Lévy processes with <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, which was initially proposed and developed by Gobet and Maire in their pioneering works ((2004) <span><span>[24]</span></span>, and (2005) <span><span>[25]</span></span>) for the case <span><math><mi>α</mi><mo>=</mo><mn>2</mn></math></span>. The novel method effectively integrates multiple computational techniques, including the interpolation based on generalized Jacobi functions (GJFs), space-time spectral methods, control variates techniques, and a novel walk-on-spheres method (WOS). The exponential convergence of the error bounds is rigorously established through finite iterations for both Poisson and parabolic equations involving the integral fractional Laplacian operator. Remarkably, the proposed space-time spectral Monte Carlo method (ST-SMC) for the parabolic equation is unified for both <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> and <span><math><mi>α</mi><mo>=</mo><mn>2</mn></math></span>. Extensive numerical results are provided to demonstrate the spectral accuracy and efficiency of the proposed method, thereby validating the theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 278-297"},"PeriodicalIF":2.2,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalization error analysis of deep backward dynamic programming for solving nonlinear PDEs","authors":"Du Ouyang,&nbsp;Jichang Xiao,&nbsp;Xiaoqun Wang","doi":"10.1016/j.apnum.2025.06.013","DOIUrl":"10.1016/j.apnum.2025.06.013","url":null,"abstract":"<div><div>We explore the application of the quasi-Monte Carlo (QMC) method in deep backward dynamic programming (DBDP) <span><span>[1]</span></span> for numerically solving high-dimensional nonlinear partial differential equations (PDEs). Our study focuses on examining the generalization error as a component of the total error in the DBDP framework, discovering that the rate of convergence for the generalization error is influenced by the choice of sampling methods. Specifically, for a given batch size <em>m</em>, the generalization error under QMC methods exhibits a convergence rate of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>)</mo></math></span>, where <span><math><mi>ε</mi><mo>&gt;</mo><mn>0</mn></math></span> can be made arbitrarily small. This rate is notably more favorable than that of the traditional Monte Carlo (MC) methods, which is <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>)</mo></math></span>. Our theoretical analysis shows that the generalization error under QMC methods achieves a higher order of convergence than their MC counterparts. Numerical experiments demonstrate that QMC indeed surpasses MC in delivering solutions that are both more precise and stable.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 319-339"},"PeriodicalIF":2.2,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144522669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical method and analysis for fluid-structure model on unbounded domains","authors":"Hongwei Li,&nbsp;Xinyue Chen","doi":"10.1016/j.apnum.2025.06.009","DOIUrl":"10.1016/j.apnum.2025.06.009","url":null,"abstract":"<div><div>Numerically solving the fluid-structure model on unbounded domains poses a challenge, due to the unbounded nature of the physical domain. To overcome this challenge, the artificial boundary method is specifically applied to numerically solve the fluid-structure model on unbounded domains, which can be used to analyze fluid-structure interactions in various scientific and engineering fields. Drawing inspiration from the artificial boundary method, we employ artificial boundaries to truncate the unbounded domain, subsequently designing the high order local artificial boundary conditions thereon based on the Padé approximation. Then, the initial value problem on the unbounded domain is reduced into an initial boundary value problem on the computational domain, which can be efficiently solved by adopting the finite difference method. Furthermore, a series of auxiliary variables is introduced specifically to address the issue of mixed derivatives arising in the artificial boundary conditions, and the stability, convergence and solvability of the reduced problem are rigorously analyzed. Numerical experiments are reported to demonstrate the effectiveness of artificial boundary conditions and theoretical analysis.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 255-277"},"PeriodicalIF":2.2,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical scheme for the solution of the “bad” Boussinesq equation","authors":"Christophe Charlier ,&nbsp;Daniel Eriksson ,&nbsp;Jonatan Lenells","doi":"10.1016/j.apnum.2025.06.011","DOIUrl":"10.1016/j.apnum.2025.06.011","url":null,"abstract":"<div><div>We present a numerical scheme for the solution of the initial-value problem for the “bad” Boussinesq equation. The accuracy of the scheme is tested by comparison with exact soliton solutions as well as with recently obtained asymptotic formulas for the solution.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 216-233"},"PeriodicalIF":2.2,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical simulation of a dynamic human capital model with demographic delays via the local discrete Galerkin method","authors":"Yadollah Ordokhani ,&nbsp;Alireza Hosseinian ,&nbsp;Pouria Assari","doi":"10.1016/j.apnum.2025.06.007","DOIUrl":"10.1016/j.apnum.2025.06.007","url":null,"abstract":"<div><div>A strong and dynamic economy depends on various factors, with human capital playing a crucial role in fostering resilience and adaptability in an ever-changing world. Human capital, which depends on the past behavior of the system, requires strategic investments in education, health, and skill development. This study presents a numerical approach for solving the human capital model with age-structured delays, formulated as integro-differential equations with double delays and difference kernels. The proposed method employs a local meshless discrete Galerkin approach based on the moving least squares (MLS) technique, which can work with irregular or non-uniform data. The localized nature of the MLS scheme enhances computational efficiency by focusing on small neighborhoods. Moreover, the stabilized MLS framework, achieved by using shifted and scaled polynomial basis functions, enhances numerical stability and reduces sensitivity to the distribution of nodes, thereby transferring these advantageous properties to the method. The simplicity of the proposed algorithm makes it easy to implement on standard personal computers and extend to a wider class of delay integro-differential equations. To assess its reliability, we analyzed its error and determined the convergence order of the presented method. We have applied it to solve several numerical examples, and the obtained results confirm the method's accuracy, stability, and alignment with theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 234-254"},"PeriodicalIF":2.2,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Note on an inertial projection-based approach for solving extended general quasi-variational inequalities and its convergence analysis","authors":"Saudia Jabeen ,&nbsp;Siegfried Macías ,&nbsp;Saleem Ullah ,&nbsp;Muhammad Aslam Noor ,&nbsp;Jorge E. Macías-Díaz","doi":"10.1016/j.apnum.2025.06.012","DOIUrl":"10.1016/j.apnum.2025.06.012","url":null,"abstract":"<div><div>In this work, an inertial projection-based method is proposed to find approximate solutions to a new class of quasi-variational inequalities. The approach utilizes projection techniques to develop an iterative scheme, and its convergence properties are rigorously examined under appropriate conditions. Various specific cases are derived from the general framework, highlighting the adaptability of the proposed method. The comparative performance of this approach with existing techniques remains an open area for exploration. It is expected that the theoretical framework and techniques proposed in this study will stimulate further research in this domain.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 190-198"},"PeriodicalIF":2.2,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of Tseng algorithm with inertial extrapolation step for stochastic variational inequality problem","authors":"Olawale K. Oyewole ,&nbsp;Seithuti P. Moshokoa ,&nbsp;Sani Salisu ,&nbsp;Yekini Shehu","doi":"10.1016/j.apnum.2025.06.008","DOIUrl":"10.1016/j.apnum.2025.06.008","url":null,"abstract":"<div><div>In this paper, we design an inertial version of the Tseng extragradient algorithm (also called the Forward-Backward-Forward Algorithm) with self-adaptive step sizes to solve the stochastic variational inequality problem. We prove that the sequence of iterates generated by our proposed algorithm converges to a solution of the stochastic variational inequality problem under mild conditions. Furthermore, we obtain some convergence rates and numerical simulations of our proposed algorithm with comparisons with related algorithms to show the superiority of our algorithm.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 199-215"},"PeriodicalIF":2.2,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An efficient multi projection methods for systems of Fredholm integral equations with mixed weakly singular kernels: A superconvergence approach","authors":"Krishna Murari Malav ,&nbsp;Kapil Kant ,&nbsp;Joydip Dhar ,&nbsp;Samiran Chakraborty","doi":"10.1016/j.apnum.2025.06.006","DOIUrl":"10.1016/j.apnum.2025.06.006","url":null,"abstract":"<div><div>In this article, we develop the multi-Galerkin and iterated multi-Galerkin methods to solve systems of second-kind linear Fredholm integral equations (FIEs) with smooth and mixed weakly singular kernels. First, we develop the mathematical formulation of the multi-Galerkin and iterated multi-Galerkin methods using piecewise polynomial approximations to solve such systems and obtain superconvergence results. These methods transform the linear system of FIEs into corresponding matrix equations. We derive error estimates and obtain the convergence analysis. We prove that the convergence rates for the multi-Galerkin method are <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>3</mn><mi>r</mi></mrow></msup><mo>)</mo></math></span> for smooth kernels and <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>r</mi><mo>−</mo><mi>α</mi></mrow></msup><mi>log</mi><mo>⁡</mo><mi>h</mi><mo>)</mo></math></span> for mixed weakly singular kernels, where <em>r</em> denote the degree of the piecewise polynomials, <em>h</em> is the norm of partitions and <span><math><mi>α</mi><mo>=</mo><munder><mi>max</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mo>⁡</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span>. Moreover, we establish that the iterated multi-Galerkin method achieves improved convergence rates of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>4</mn><mi>r</mi></mrow></msup><mo>)</mo></math></span> for smooth kernels and <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>2</mn><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></msup><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>h</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> for mixed weakly singular kernels. Hence, the results show that the iterated multi-Galerkin method improves the multi-Galerkin method. Finally, the theoretical results are validated through the numerical examples.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 172-189"},"PeriodicalIF":2.2,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Localization of tumor through a non-conventional numerical shape optimization technique","authors":"Julius Fergy Tiongson Rabago","doi":"10.1016/j.apnum.2025.06.005","DOIUrl":"10.1016/j.apnum.2025.06.005","url":null,"abstract":"<div><div>This paper presents a method for estimating the shape and location of an embedded tumor using shape optimization techniques, specifically through the coupled complex boundary method. The inverse problem—characterized by a measured temperature profile and corresponding heat flux (e.g., from infrared thermography)—is reformulated as a complex boundary value problem with a complex Robin boundary condition, thereby simplifying its over-specified nature. The geometry of the tumor is identified by optimizing an objective functional that depends on the imaginary part of the solution throughout the domain. The shape derivative of the functional is derived through shape sensitivity analysis. An iterative algorithm is developed to numerically recover the tumor shape, based on the Riesz representative of the gradient and implemented using the finite element method. In addition, the mesh sensitivity of the finite element solution to the state problem is analyzed, and bounds are established for its variation with respect to mesh deformation and its gradient. Numerical examples are presented to validate the theoretical results and to demonstrate the accuracy and effectiveness of the proposed method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"217 ","pages":"Pages 135-171"},"PeriodicalIF":2.2,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}