{"title":"HRW: Hybrid Residual and Weak Form Loss for Solving Elliptic Interface Problems with Neural Network","authors":"Muzhou Hou, Yinghao Chen, Shen Cao, Yuntian Chen and Jinyong Ying","doi":"10.4208/nmtma.oa-2023-0097","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2023-0097","url":null,"abstract":"","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135194756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-01DOI: 10.4208/jcm.2206-m2021-0240
T. Sun, Chengjian Sun
This paper deals with numerical methods for solving one-dimensional (1D) and two-dimensional (2D) initial-boundary value problems (IBVPs) of space-fractional sine-Gordon equations (SGEs) with distributed delay. For 1D problems, we construct a kind of one-parameter finite difference (OPFD) method. It is shown that, under a suitable condition, the proposed method is convergent with second order accuracy both in time and space. In implementation, the preconditioned conjugate gradient (PCG) method with the Strang circulant preconditioner is carried out to improve the computational efficiency of the OPFD method. For 2D problems, we develop another kind of OPFD method. For such a method, two classes of accelerated schemes are suggested, one is alternative direction implicit (ADI) scheme and the other is ADI-PCG scheme. In particular, we prove that ADI scheme can arrive at second-order accuracy in time and space. With some numerical experiments, the computational effectiveness and accuracy of the methods are further verified. Moreover, for the suggested methods, a numerical comparison in computational efficiency is presented.
{"title":"One-Parameter Finite Difference Methods and Their Accelerated Schemes for Space-Fractional Sine-Gordon Equations with Distributed Delay","authors":"T. Sun, Chengjian Sun","doi":"10.4208/jcm.2206-m2021-0240","DOIUrl":"https://doi.org/10.4208/jcm.2206-m2021-0240","url":null,"abstract":"This paper deals with numerical methods for solving one-dimensional (1D) and two-dimensional (2D) initial-boundary value problems (IBVPs) of space-fractional sine-Gordon equations (SGEs) with distributed delay. For 1D problems, we construct a kind of one-parameter finite difference (OPFD) method. It is shown that, under a suitable condition, the proposed method is convergent with second order accuracy both in time and space. In implementation, the preconditioned conjugate gradient (PCG) method with the Strang circulant preconditioner is carried out to improve the computational efficiency of the OPFD method. For 2D problems, we develop another kind of OPFD method. For such a method, two classes of accelerated schemes are suggested, one is alternative direction implicit (ADI) scheme and the other is ADI-PCG scheme. In particular, we prove that ADI scheme can arrive at second-order accuracy in time and space. With some numerical experiments, the computational effectiveness and accuracy of the methods are further verified. Moreover, for the suggested methods, a numerical comparison in computational efficiency is presented.","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46489530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-01DOI: 10.4208/jcm.2212-m2021-0231
Yuping Zeng, M. Zhong
A mixed finite element method is presented for the Biot consolidation problem in poroe-lasticity. More precisely, the displacement is approximated by using the Crouzeix-Raviart nonconforming finite elements, while the fluid pressure is approximated by using the node conforming finite elements. The well-posedness of the fully discrete scheme is established, and a corresponding priori error estimate with optimal order in the energy norm is also derived. Numerical experiments are provided to validate the theoretical results. Mathematics
{"title":"A Coupled Method Combining Crouzeix-Raviart Nonconforming and Node Conforming Finite Element Spaces for Boit Consolidation Model","authors":"Yuping Zeng, M. Zhong","doi":"10.4208/jcm.2212-m2021-0231","DOIUrl":"https://doi.org/10.4208/jcm.2212-m2021-0231","url":null,"abstract":"A mixed finite element method is presented for the Biot consolidation problem in poroe-lasticity. More precisely, the displacement is approximated by using the Crouzeix-Raviart nonconforming finite elements, while the fluid pressure is approximated by using the node conforming finite elements. The well-posedness of the fully discrete scheme is established, and a corresponding priori error estimate with optimal order in the energy norm is also derived. Numerical experiments are provided to validate the theoretical results. Mathematics","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42854561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-01DOI: 10.4208/jcm.2301-m2022-0099
Yuhong Dai, Jiani Zhang
Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas. While there have been many numerical algorithms for solving smooth convex-concave minimax problems, numerical algorithms for nonsmooth convex-concave minimax problems are rare. This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem. A semi-proximal point method (SPP) is proposed, in which a quadratic convex-concave function is adopted for approximating the smooth part of the objective function and semi-proximal terms are added in each subproblem. This construction enables the subproblems at each iteration are solvable and even easily solved when the semiproximal terms are cleverly chosen. We prove the global convergence of our algorithm under mild assumptions, without requiring strong convexity-concavity condition. Under the locally metrical subregularity of the solution mapping, we prove that our algorithm has the linear rate of convergence. Preliminary numerical results are reported to verify the efficiency of our algorithm.
{"title":"Semi-Proximal Point Method for Nonsmooth Convex-Concave Minimax Optimization","authors":"Yuhong Dai, Jiani Zhang","doi":"10.4208/jcm.2301-m2022-0099","DOIUrl":"https://doi.org/10.4208/jcm.2301-m2022-0099","url":null,"abstract":"Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas. While there have been many numerical algorithms for solving smooth convex-concave minimax problems, numerical algorithms for nonsmooth convex-concave minimax problems are rare. This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem. A semi-proximal point method (SPP) is proposed, in which a quadratic convex-concave function is adopted for approximating the smooth part of the objective function and semi-proximal terms are added in each subproblem. This construction enables the subproblems at each iteration are solvable and even easily solved when the semiproximal terms are cleverly chosen. We prove the global convergence of our algorithm under mild assumptions, without requiring strong convexity-concavity condition. Under the locally metrical subregularity of the solution mapping, we prove that our algorithm has the linear rate of convergence. Preliminary numerical results are reported to verify the efficiency of our algorithm.","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43697412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-01DOI: 10.4208/jcm.2210-m2021-0106
Jonathan W. Siegel and Jinchao Xu
{"title":"Extended Regularized Dual Averaging Methods for Stochastic Optimization","authors":"Jonathan W. Siegel and Jinchao Xu","doi":"10.4208/jcm.2210-m2021-0106","DOIUrl":"https://doi.org/10.4208/jcm.2210-m2021-0106","url":null,"abstract":"","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45716207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-01DOI: 10.4208/jcm.2210-m2021-0257
Baoli Yin, Yang Liu, Hong Zhang
{"title":"On Discrete Energy Dissipation of Maxwell’s Equations in a Cole-Cole Dispersive Medium","authors":"Baoli Yin, Yang Liu, Hong Zhang","doi":"10.4208/jcm.2210-m2021-0257","DOIUrl":"https://doi.org/10.4208/jcm.2210-m2021-0257","url":null,"abstract":"","PeriodicalId":50225,"journal":{"name":"Journal of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44048569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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