Pub Date : 2023-09-01DOI: 10.4208/aam.oa-2023-0024
Jianguo Huang, Sen Lin and Yue Yu
. This paper devises a new lowest-order conforming virtual element method (VEM) for planar linear elasticity with the pure displacement/traction boundary condition. The main trick is to view a generic polygon K as a new one (cid:101) K with additional vertices consisting of interior points on edges of K , so that the discrete admissible space is taken as the V 1 type virtual element space related to the partition { (cid:101) K } instead of { K } . The method is proved to converge with optimal convergence order both in H 1 and L 2 norms and uniformly with respect to the Lam´e constant λ . Numerical tests are presented to illustrate the good performance of the proposed VEM and confirm the theoretical results.
{"title":"A New Locking-Free Virtual Element Method for Linear Elasticity Problems","authors":"Jianguo Huang, Sen Lin and Yue Yu","doi":"10.4208/aam.oa-2023-0024","DOIUrl":"https://doi.org/10.4208/aam.oa-2023-0024","url":null,"abstract":". This paper devises a new lowest-order conforming virtual element method (VEM) for planar linear elasticity with the pure displacement/traction boundary condition. The main trick is to view a generic polygon K as a new one (cid:101) K with additional vertices consisting of interior points on edges of K , so that the discrete admissible space is taken as the V 1 type virtual element space related to the partition { (cid:101) K } instead of { K } . The method is proved to converge with optimal convergence order both in H 1 and L 2 norms and uniformly with respect to the Lam´e constant λ . Numerical tests are presented to illustrate the good performance of the proposed VEM and confirm the theoretical results.","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44305830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.4208/aam.oa-2023-0017
Luyu Cen and Xiaoping Wang
. In this paper, we propose a simple energy decaying iterative thresh-olding algorithm to solve the heat transfer problem. The material domain is implicitly represented by its characteristic function, and the problem is formulated into a minimum-minimum problem. We prove that the energy is decreasing in each iteration. Numerical experiments for two types the heat transfer problems (volume to point and volume to sides) are performed to demonstrate the effectiveness of the proposed methods.
{"title":"An Iterative Thresholding Method for the Heat Transfer Problem","authors":"Luyu Cen and Xiaoping Wang","doi":"10.4208/aam.oa-2023-0017","DOIUrl":"https://doi.org/10.4208/aam.oa-2023-0017","url":null,"abstract":". In this paper, we propose a simple energy decaying iterative thresh-olding algorithm to solve the heat transfer problem. The material domain is implicitly represented by its characteristic function, and the problem is formulated into a minimum-minimum problem. We prove that the energy is decreasing in each iteration. Numerical experiments for two types the heat transfer problems (volume to point and volume to sides) are performed to demonstrate the effectiveness of the proposed methods.","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47606132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.4208/aam.oa-2023-0021
Yuling Jiao, Xiliang Lu, Jerry Zhijian Yang, Cheng Yuan and Pingwen Zhang
. In this paper, we present an improved analysis of the Physics In-formed Neural Networks (PINNs) method for solving second-order elliptic equations. By assuming an intrinsic sparse structure in the underlying solution, we provide a convergence rate analysis that can overcome the curse of dimensionality (CoD). Specifically, using some approximation theory in Sobolev space together with the multivariate Faa di Bruno formula, we first derive the approximation error for composition functions with a small degree of freedom in each compositional layer. Furthermore, by integrating several results on the statistical error of neural networks, we obtain a refined convergence rate analysis for PINNs in solving elliptic equations with compositional solutions. We also demonstrate the benefits of the intrinsic sparse structure with two simple numerical examples.
。本文提出了求解二阶椭圆方程的物理信息神经网络(PINNs)方法的改进分析。通过在底层解中假设一个固有的稀疏结构，我们提供了一个收敛速度分析，可以克服维数诅咒(CoD)。具体而言，利用Sobolev空间中的近似理论，结合多元的Faa di Bruno公式，首先推导出了小自由度组合函数在各组合层中的近似误差。此外，通过对神经网络统计误差的几个结果的综合，我们得到了pinn在求解具有组合解的椭圆型方程时收敛速度的精细分析。我们还通过两个简单的数值例子证明了本征稀疏结构的优点。
{"title":"Improved Analysis of PINNs: Alleviate the CoD for Compositional Solutions","authors":"Yuling Jiao, Xiliang Lu, Jerry Zhijian Yang, Cheng Yuan and Pingwen Zhang","doi":"10.4208/aam.oa-2023-0021","DOIUrl":"https://doi.org/10.4208/aam.oa-2023-0021","url":null,"abstract":". In this paper, we present an improved analysis of the Physics In-formed Neural Networks (PINNs) method for solving second-order elliptic equations. By assuming an intrinsic sparse structure in the underlying solution, we provide a convergence rate analysis that can overcome the curse of dimensionality (CoD). Specifically, using some approximation theory in Sobolev space together with the multivariate Faa di Bruno formula, we first derive the approximation error for composition functions with a small degree of freedom in each compositional layer. Furthermore, by integrating several results on the statistical error of neural networks, we obtain a refined convergence rate analysis for PINNs in solving elliptic equations with compositional solutions. We also demonstrate the benefits of the intrinsic sparse structure with two simple numerical examples.","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47061352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.4208/aam.oa-2023-0007
Dianming Hou, Zhonghua Qiaoand Tao Tang
{"title":"Fast High Order and Energy Dissipative Schemes with Variable Time Steps for Time-Fractional Molecular Beam Epitaxial Growth Model","authors":"Dianming Hou, Zhonghua Qiaoand Tao Tang","doi":"10.4208/aam.oa-2023-0007","DOIUrl":"https://doi.org/10.4208/aam.oa-2023-0007","url":null,"abstract":"","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43436056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.4208/aam.oa-2023-0016
Yanming Lai, Kewei Liang, Ping Lin, Xiliang Lu and Qimeng Quan
{"title":"Error Analysis of the Nonconforming $P_1$ Finite Element Method to the Sequential Regularization Formulation for Unsteady Navier-Stokes Equations","authors":"Yanming Lai, Kewei Liang, Ping Lin, Xiliang Lu and Qimeng Quan","doi":"10.4208/aam.oa-2023-0016","DOIUrl":"https://doi.org/10.4208/aam.oa-2023-0016","url":null,"abstract":"","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42301682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.4208/aam.oa-2023-0018
Shaohong Du, Qianqian Hou and Xiaoping Xie
. In this paper, we consider a modified nonlinear dynamic diffusion (DD) method for convection-diffusion-reaction equations. This method is free of stabilization parameters and capable of precluding spurious oscillations. We develop a reliable and efficient residual-type a posteriori error estimator
{"title":"A Linearized Adaptive Dynamic Diffusion Finite Element Method for Convection-Diffusion-Reaction Equations","authors":"Shaohong Du, Qianqian Hou and Xiaoping Xie","doi":"10.4208/aam.oa-2023-0018","DOIUrl":"https://doi.org/10.4208/aam.oa-2023-0018","url":null,"abstract":". In this paper, we consider a modified nonlinear dynamic diffusion (DD) method for convection-diffusion-reaction equations. This method is free of stabilization parameters and capable of precluding spurious oscillations. We develop a reliable and efficient residual-type a posteriori error estimator","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45816750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/aam.oa-2023-0014
Xiaohui Hu
. In this paper, we derive a generalized nonisospectral semi-infinite Lotka–Volterra equation, which possesses a determinant solution. We also give its a Lax pair expressed in terms of symmetric orthogonal polynomials. In addition, if the simplified case of the moment evolution relation is considered, that is, without the convolution term, we also give a generalized nonisospectral finite Lotka–Volterra equation with an explicit determinant solution. Finally, an application of the generalized nonisospectral continuous-time Lotka–Volterra equation in the food chain is investigated by numerical simulation. Our approach is mainly based on Hirota’s bilinear method and determinant techniques.
{"title":"Nonisospectral Lotka--Volterra Systems as a Candidate Model for Food Chain","authors":"Xiaohui Hu","doi":"10.4208/aam.oa-2023-0014","DOIUrl":"https://doi.org/10.4208/aam.oa-2023-0014","url":null,"abstract":". In this paper, we derive a generalized nonisospectral semi-infinite Lotka–Volterra equation, which possesses a determinant solution. We also give its a Lax pair expressed in terms of symmetric orthogonal polynomials. In addition, if the simplified case of the moment evolution relation is considered, that is, without the convolution term, we also give a generalized nonisospectral finite Lotka–Volterra equation with an explicit determinant solution. Finally, an application of the generalized nonisospectral continuous-time Lotka–Volterra equation in the food chain is investigated by numerical simulation. Our approach is mainly based on Hirota’s bilinear method and determinant techniques.","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42017586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/aam.oa-2023-0010
Guang-Jing Song null, Michael K. Ng
{"title":"Nonnegative Low Rank Matrix Completion by Riemannian Optimalization Methods","authors":"Guang-Jing Song null, Michael K. Ng","doi":"10.4208/aam.oa-2023-0010","DOIUrl":"https://doi.org/10.4208/aam.oa-2023-0010","url":null,"abstract":"","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43304932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/aam.oa-2023-0009
Hui Liang, Jingtang Ma null, Zhengguang Shi
. In this paper, a rough Heston model with variable volatility of volatility (vol-of-vol) is derived by modifying the generalized nonlinear Hawkes process and extending the scaling techniques. Then the nonlinear fractional Riccati equation for the characteristic function of the asset log-price is derived. The existence, uniqueness and regularity of the solution to the nonlinear fractional Riccati equation are proved and the equation is solved by the Adams methods. Finally the Fourier-cosine methods are combined with the Adams methods to price the options.
在本文中，通过修改广义非线性Hawkes过程和扩展标度技术，导出了一个具有可变波动率（vol of vol）的粗糙Heston模型。然后导出了资产日志价格特征函数的非线性分式Riccati方程。证明了非线性分式Riccati方程解的存在性、唯一性和正则性，并用Adams方法求解了该方程。最后将傅立叶余弦方法与亚当斯方法相结合，对期权进行定价。
{"title":"Rough Heston Models with Variable Vol-of-Vol and Option Pricing","authors":"Hui Liang, Jingtang Ma null, Zhengguang Shi","doi":"10.4208/aam.oa-2023-0009","DOIUrl":"https://doi.org/10.4208/aam.oa-2023-0009","url":null,"abstract":". In this paper, a rough Heston model with variable volatility of volatility (vol-of-vol) is derived by modifying the generalized nonlinear Hawkes process and extending the scaling techniques. Then the nonlinear fractional Riccati equation for the characteristic function of the asset log-price is derived. The existence, uniqueness and regularity of the solution to the nonlinear fractional Riccati equation are proved and the equation is solved by the Adams methods. Finally the Fourier-cosine methods are combined with the Adams methods to price the options.","PeriodicalId":58853,"journal":{"name":"应用数学年刊:英文版","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44622738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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