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Numerische Mathematik最新文献

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On the accuracy of the finite volume approximations to nonlocal conservation laws 关于非局部守恒定律的有限体积近似的准确性
IF 2.1 2区 数学
Numerische Mathematik Pub Date : 2023-12-13 DOI: 10.1007/s00211-023-01388-2
Aekta Aggarwal, Helge Holden, Ganesh Vaidya
{"title":"On the accuracy of the finite volume approximations to nonlocal conservation laws","authors":"Aekta Aggarwal, Helge Holden, Ganesh Vaidya","doi":"10.1007/s00211-023-01388-2","DOIUrl":"https://doi.org/10.1007/s00211-023-01388-2","url":null,"abstract":"<p>In this article, we discuss the error analysis for a certain class of monotone finite volume schemes approximating nonlocal scalar conservation laws, modeling traffic flow and crowd dynamics, without any additional assumptions on monotonicity or linearity of the kernel <span>(mu )</span> or the flux <i>f</i>. We first prove a novel Kuznetsov-type lemma for this class of PDEs and thereby show that the finite volume approximations converge to the entropy solution at the rate of <span>(sqrt{Delta t})</span> in <span>(L^1(mathbb {R}))</span>. To the best of our knowledge, this is the first proof of any type of convergence rate for this class of conservation laws. We also present numerical experiments to illustrate this result.</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"145 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138683261","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}
引用次数: 0
Lattice enumeration via linear programming 通过线性规划进行网格枚举
IF 2.1 2区 数学
Numerische Mathematik Pub Date : 2023-12-11 DOI: 10.1007/s00211-023-01376-6
Moulay Abdellah Chkifa
{"title":"Lattice enumeration via linear programming","authors":"Moulay Abdellah Chkifa","doi":"10.1007/s00211-023-01376-6","DOIUrl":"https://doi.org/10.1007/s00211-023-01376-6","url":null,"abstract":"<p>Given a positive integer <i>d</i> and <span>({{varvec{a}}}_{1},dots ,{{varvec{a}}}_{r})</span> a family of vectors in <span>({{mathbb {R}}}^d)</span>, <span>({k_1{{varvec{a}}}_{1}+dots +k_r{{varvec{a}}}_{r}: k_1,dots ,k_r in {{mathbb {Z}}}}subset {{mathbb {R}}}^d)</span> is the so-called lattice generated by the family. In high dimensional integration, prescribed lattices are used for constructing reliable quadrature schemes. The quadrature points are the lattice points lying on the integration domain, typically the unit hypercube <span>([0,1)^d)</span> or a rescaled shifted hypercube. It is crucial to have a cost-effective method for enumerating lattice points within such domains. Undeniably, the lack of such fast enumeration procedures hinders the applicability of lattice rules. Existing enumeration procedures exploit intrinsic properties of the lattice at hand, such as periodicity, orthogonality, recurrences, etc. In this paper, we unveil a general-purpose fast lattice enumeration algorithm based on linear programming (named <b>FLE-LP</b>).</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"2 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138572065","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}
引用次数: 0
Stability and guaranteed error control of approximations to the Monge–Ampère equation 蒙日-安培方程近似值的稳定性和保证误差控制
IF 2.1 2区 数学
Numerische Mathematik Pub Date : 2023-12-07 DOI: 10.1007/s00211-023-01385-5
Dietmar Gallistl, Ngoc Tien Tran
{"title":"Stability and guaranteed error control of approximations to the Monge–Ampère equation","authors":"Dietmar Gallistl, Ngoc Tien Tran","doi":"10.1007/s00211-023-01385-5","DOIUrl":"https://doi.org/10.1007/s00211-023-01385-5","url":null,"abstract":"<p>This paper analyzes a regularization scheme of the Monge–Ampère equation by uniformly elliptic Hamilton–Jacobi–Bellman equations. The main tools are stability estimates in the <span>(L^infty )</span> norm from the theory of viscosity solutions which are independent of the regularization parameter <span>(varepsilon )</span>. They allow for the uniform convergence of the solution <span>(u_varepsilon )</span> to the regularized problem towards the Alexandrov solution <i>u</i> to the Monge–Ampère equation for any nonnegative <span>(L^n)</span> right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the <span>(L^infty )</span> norm for continuously differentiable finite element approximations of <i>u</i> or <span>(u_varepsilon )</span>.\u0000</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"163 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138546841","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}
引用次数: 0
A discrete elasticity complex on three-dimensional Alfeld splits 三维Alfeld分裂上的离散弹性复合体
IF 2.1 2区 数学
Numerische Mathematik Pub Date : 2023-11-30 DOI: 10.1007/s00211-023-01381-9
Snorre H. Christiansen, Jay Gopalakrishnan, Johnny Guzmán, Kaibo Hu
{"title":"A discrete elasticity complex on three-dimensional Alfeld splits","authors":"Snorre H. Christiansen, Jay Gopalakrishnan, Johnny Guzmán, Kaibo Hu","doi":"10.1007/s00211-023-01381-9","DOIUrl":"https://doi.org/10.1007/s00211-023-01381-9","url":null,"abstract":"<p>We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via the linearized deformation operator, the linearized curvature operator, and the divergence operator, respectively. The construction is based on an algebraic machinery that derives the elasticity complex from de Rham complexes, and smoother finite element differential forms.\u0000</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138531609","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}
引用次数: 15
Two-layer networks with the $$text {ReLU}^k$$ activation function: Barron spaces and derivative approximation 具有$$text {ReLU}^k$$激活函数的两层网络:巴伦空间和导数逼近
IF 2.1 2区 数学
Numerische Mathematik Pub Date : 2023-11-23 DOI: 10.1007/s00211-023-01384-6
Yuanyuan Li, Shuai Lu, Peter Mathé, Sergei V. Pereverzev
{"title":"Two-layer networks with the $$text {ReLU}^k$$ activation function: Barron spaces and derivative approximation","authors":"Yuanyuan Li, Shuai Lu, Peter Mathé, Sergei V. Pereverzev","doi":"10.1007/s00211-023-01384-6","DOIUrl":"https://doi.org/10.1007/s00211-023-01384-6","url":null,"abstract":"<p>We investigate the use of two-layer networks with the rectified power unit, which is called the <span>(text {ReLU}^k)</span> activation function, for function and derivative approximation. By extending and calibrating the corresponding Barron space, we show that two-layer networks with the <span>(text {ReLU}^k)</span> activation function are well-designed to simultaneously approximate an unknown function and its derivatives. When the measurement is noisy, we propose a Tikhonov type regularization method, and provide error bounds when the regularization parameter is chosen appropriately. Several numerical examples support the efficiency of the proposed approach.</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"18 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138531594","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}
引用次数: 0
Numerical stability and tensor nuclear norm 数值稳定性和张量核范数
2区 数学
Numerische Mathematik Pub Date : 2023-11-03 DOI: 10.1007/s00211-023-01377-5
Zhen Dai, Lek-Heng Lim
{"title":"Numerical stability and tensor nuclear norm","authors":"Zhen Dai, Lek-Heng Lim","doi":"10.1007/s00211-023-01377-5","DOIUrl":"https://doi.org/10.1007/s00211-023-01377-5","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"29 22","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818313","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}
引用次数: 2
A unifying framework for tangential interpolation of structured bilinear control systems 结构双线性控制系统切向插补的统一框架
2区 数学
Numerische Mathematik Pub Date : 2023-10-31 DOI: 10.1007/s00211-023-01380-w
Peter Benner, Serkan Gugercin, Steffen W. R. Werner
{"title":"A unifying framework for tangential interpolation of structured bilinear control systems","authors":"Peter Benner, Serkan Gugercin, Steffen W. R. Werner","doi":"10.1007/s00211-023-01380-w","DOIUrl":"https://doi.org/10.1007/s00211-023-01380-w","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135870870","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}
引用次数: 2
On generating Sobolev orthogonal polynomials 关于索博列夫正交多项式的生成
2区 数学
Numerische Mathematik Pub Date : 2023-10-31 DOI: 10.1007/s00211-023-01379-3
Niel Van Buggenhout
{"title":"On generating Sobolev orthogonal polynomials","authors":"Niel Van Buggenhout","doi":"10.1007/s00211-023-01379-3","DOIUrl":"https://doi.org/10.1007/s00211-023-01379-3","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"2 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135872727","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}
引用次数: 0
Grid-free weighted particle method applied to the Vlasov–Poisson equation 无网格加权粒子法在Vlasov-Poisson方程中的应用
2区 数学
Numerische Mathematik Pub Date : 2023-10-26 DOI: 10.1007/s00211-023-01378-4
Yoann Le Henaff
{"title":"Grid-free weighted particle method applied to the Vlasov–Poisson equation","authors":"Yoann Le Henaff","doi":"10.1007/s00211-023-01378-4","DOIUrl":"https://doi.org/10.1007/s00211-023-01378-4","url":null,"abstract":"We study a grid-free particle method based on following the evolution of the characteristics of the Vlasov–Poisson system, and we show that it converges for smooth enough initial data. This method is built as a combination of well-studied building blocks—mainly time integration and integral quadratures—and allows to obtain arbitrarily high orders. By making use of the Non-Uniform Fast Fourier Transform, the overall computational complexity is $$ {mathcal {O}}(P log P + K^d log K^d) $$ , where $$ P $$ is the total number of particles and where we only keep the Fourier modes $$ k in ({mathbb {Z}}^d)^* $$ such that $$ k_1^2 + dots + k_d^2 le K^2 $$ . Some numerical results are given for the Vlasov–Poisson system in the one-dimensional case.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"60 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136377031","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}
引用次数: 0
Statistical properties of BayesCG under the Krylov prior Krylov先验下BayesCG的统计性质
2区 数学
Numerische Mathematik Pub Date : 2023-10-12 DOI: 10.1007/s00211-023-01375-7
Tim W. Reid, Ilse C. F. Ipsen, Jon Cockayne, Chris J. Oates
{"title":"Statistical properties of BayesCG under the Krylov prior","authors":"Tim W. Reid, Ilse C. F. Ipsen, Jon Cockayne, Chris J. Oates","doi":"10.1007/s00211-023-01375-7","DOIUrl":"https://doi.org/10.1007/s00211-023-01375-7","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"169 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135969090","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}
引用次数: 0
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