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An optimal control framework for adaptive neural ODEs 自适应神经 ODE 的优化控制框架
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-23 DOI: 10.1007/s10444-024-10149-0
Joubine Aghili, Olga Mula
{"title":"An optimal control framework for adaptive neural ODEs","authors":"Joubine Aghili,&nbsp;Olga Mula","doi":"10.1007/s10444-024-10149-0","DOIUrl":"10.1007/s10444-024-10149-0","url":null,"abstract":"<div><p>In recent years, the notion of neural ODEs has connected deep learning with the field of ODEs and optimal control. In this setting, neural networks are defined as the mapping induced by the corresponding time-discretization scheme of a given ODE. The learning task consists in finding the ODE parameters as the optimal values of a sampled loss minimization problem. In the limit of infinite time steps, and data samples, we obtain a notion of continuous formulation of the problem. The practical implementation involves two discretization errors: a sampling error and a time-discretization error. In this work, we develop a general optimal control framework to analyze the interplay between the above two errors. We prove that to approximate the solution of the fully continuous problem at a certain accuracy, we not only need a minimal number of training samples, but also need to solve the control problem on the sampled loss function with some minimal accuracy. The theoretical analysis allows us to develop rigorous adaptive schemes in time and sampling, and gives rise to a notion of adaptive neural ODEs. The performance of the approach is illustrated in several numerical examples.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141085343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
An unfitted finite element method with direct extension stabilization for time-harmonic Maxwell problems on smooth domains 用于平滑域上时谐麦克斯韦问题的直接扩展稳定的非拟合有限元方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-22 DOI: 10.1007/s10444-024-10148-1
Fanyi Yang, Xiaoping Xie
{"title":"An unfitted finite element method with direct extension stabilization for time-harmonic Maxwell problems on smooth domains","authors":"Fanyi Yang,&nbsp;Xiaoping Xie","doi":"10.1007/s10444-024-10148-1","DOIUrl":"10.1007/s10444-024-10148-1","url":null,"abstract":"<div><p>We propose an unfitted finite element method for numerically solving the time-harmonic Maxwell equations on a smooth domain. The embedded boundary of the domain is allowed to cut through the background mesh arbitrarily. The unfitted scheme is based on a mixed interior penalty formulation, where the Nitsche penalty method is applied to enforce the boundary condition in a weak sense, and a penalty stabilization technique is adopted based on a local direct extension operator to ensure the stability for cut elements. We prove the inf-sup stability and obtain optimal convergence rates under the energy norm and the <span>(L^2)</span> norm for both variables. Numerical examples in both two and three dimensions are presented to illustrate the accuracy of the method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141085335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 sparse approximation for fractional Fourier transform 分数傅里叶变换的稀疏近似值
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-20 DOI: 10.1007/s10444-024-10127-6
Fang Yang, Jiecheng Chen, Tao Qian, Jiman Zhao
{"title":"A sparse approximation for fractional Fourier transform","authors":"Fang Yang,&nbsp;Jiecheng Chen,&nbsp;Tao Qian,&nbsp;Jiman Zhao","doi":"10.1007/s10444-024-10127-6","DOIUrl":"10.1007/s10444-024-10127-6","url":null,"abstract":"<div><p>The paper promotes a new sparse approximation for fractional Fourier transform, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the upper half-plane. Under this methodology, the local polynomial Fourier transform characterization of Hardy space is established, which is an analog of the Paley-Wiener theorem. Meanwhile, a sparse fractional Fourier series for chirp <span>( L^2 )</span> function is proposed, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the unit disk. Besides the establishment of the theoretical foundation, the proposed approximation provides a sparse solution for a forced Schr<span>(ddot{textrm{o}})</span>dinger equations with a harmonic oscillator.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141069489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Error analysis of a collocation method on graded meshes for a fractional Laplacian problem 针对分数拉普拉斯问题的梯度网格上的拼合方法的误差分析
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-20 DOI: 10.1007/s10444-024-10146-3
Minghua Chen, Weihua Deng, Chao Min, Jiankang Shi, Martin Stynes
{"title":"Error analysis of a collocation method on graded meshes for a fractional Laplacian problem","authors":"Minghua Chen,&nbsp;Weihua Deng,&nbsp;Chao Min,&nbsp;Jiankang Shi,&nbsp;Martin Stynes","doi":"10.1007/s10444-024-10146-3","DOIUrl":"10.1007/s10444-024-10146-3","url":null,"abstract":"<div><p>The numerical solution of a 1D fractional Laplacian boundary value problem is studied. Although the fractional Laplacian is one of the most important and prominent nonlocal operators, its numerical analysis is challenging, partly because the problem’s solution has in general a weak singularity at the boundary of the domain. To solve the problem numerically, we use piecewise linear collocation on a mesh that is graded to handle the boundary singularity. A rigorous analysis yields a bound on the maximum nodal error which shows how the order of convergence of the method depends on the grading of the mesh; hence, one can determine the optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141069476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
An adaptive certified space-time reduced basis method for nonsmooth parabolic partial differential equations 非光滑抛物型偏微分方程的自适应认证时空还原基方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-15 DOI: 10.1007/s10444-024-10137-4
Marco Bernreuther, Stefan Volkwein
{"title":"An adaptive certified space-time reduced basis method for nonsmooth parabolic partial differential equations","authors":"Marco Bernreuther,&nbsp;Stefan Volkwein","doi":"10.1007/s10444-024-10137-4","DOIUrl":"10.1007/s10444-024-10137-4","url":null,"abstract":"<div><p>In this paper, a nonsmooth semilinear parabolic partial differential equation (PDE) is considered. For a reduced basis (RB) approach, a space-time formulation is used to develop a certified a-posteriori error estimator. This error estimator is adopted to the presence of the discrete empirical interpolation method (DEIM) as approximation technique for the nonsmoothness. The separability of the estimated error into an RB and a DEIM part then guides the development of an adaptive RB-DEIM algorithm, combining both offline phases into one. Numerical experiments show the capabilities of this novel approach in comparison with classical RB and RB-DEIM approaches.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10137-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Local behaviors of Fourier expansions for functions of limited regularities 有限正则函数傅里叶展开的局部行为
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-09 DOI: 10.1007/s10444-024-10136-5
Shunfeng Yang, Shuhuang Xiang
{"title":"Local behaviors of Fourier expansions for functions of limited regularities","authors":"Shunfeng Yang,&nbsp;Shuhuang Xiang","doi":"10.1007/s10444-024-10136-5","DOIUrl":"10.1007/s10444-024-10136-5","url":null,"abstract":"<div><p>Based on the explicit formula of the pointwise error of Fourier projection approximation and by applying van der Corput-type Lemma, optimal convergence rates for periodic functions with different degrees of smoothness are established. It shows that the convergence rate enjoys a decay rate one order higher in the smooth parts than that at the singularities. In addition, it also depends on the distance from the singularities. Ample numerical experiments illustrate the perfect coincidence with the estimates.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140895409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Optimally convergent mixed finite element methods for the time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise 具有乘法噪声的时变二维/三维随机闭环地热系统的最佳收敛混合有限元方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-08 DOI: 10.1007/s10444-024-10122-x
Xinyue Gao, Yi Qin, Jian Li
{"title":"Optimally convergent mixed finite element methods for the time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise","authors":"Xinyue Gao,&nbsp;Yi Qin,&nbsp;Jian Li","doi":"10.1007/s10444-024-10122-x","DOIUrl":"10.1007/s10444-024-10122-x","url":null,"abstract":"<div><p>In this paper, a new time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise is developed and studied. This model considers heat transfer between the free flow in the pipe region and the porous media flow in the porous media region. Darcy’s law and stochastic Navier-Stokes equations are used to control the flows in the pipe and porous media regions, respectively. The heat equation is coupled with the flow equation to describe the heat transfer in these both regions. In order to avoid sub-optimal convergence, a new mixed finite element method is proposed by using the Helmholtz decomposition that drives the multiplicative noise. Then, the stability of the proposed method is proved, and we obtain the optimal convergence order <span>(o(Delta t^{frac{1}{2}}+h))</span> of global error estimation. Finally, numerical results indicate the efficiency of the proposed model and the accuracy of the numerical method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140895378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 Lagrangian approach for solving an axisymmetric thermo-electromagnetic problem. Application to time-varying geometry processes 解决轴对称热电磁问题的拉格朗日方法。时变几何过程的应用
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-08 DOI: 10.1007/s10444-024-10121-y
Marta Benítez, Alfredo Bermúdez, Pedro Fontán, Iván Martínez, Pilar Salgado
{"title":"A Lagrangian approach for solving an axisymmetric thermo-electromagnetic problem. Application to time-varying geometry processes","authors":"Marta Benítez,&nbsp;Alfredo Bermúdez,&nbsp;Pedro Fontán,&nbsp;Iván Martínez,&nbsp;Pilar Salgado","doi":"10.1007/s10444-024-10121-y","DOIUrl":"10.1007/s10444-024-10121-y","url":null,"abstract":"<div><p>The aim of this work is to introduce a thermo-electromagnetic model for calculating the temperature and the power dissipated in cylindrical pieces whose geometry varies with time and undergoes large deformations; the motion will be a known data. The work will be a first step towards building a complete thermo-electromagnetic-mechanical model suitable for simulating electrically assisted forming processes, which is the main motivation of the work. The electromagnetic model will be obtained from the time-harmonic eddy current problem with an in-plane current; the source will be given in terms of currents or voltages defined at some parts of the boundary. Finite element methods based on a Lagrangian weak formulation will be used for the numerical solution. This approach will avoid the need to compute and remesh the thermo-electromagnetic domain along the time. The numerical tools will be implemented in FEniCS and validated by using a suitable test also solved in Eulerian coordinates.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10121-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140895489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stray field computation by inverted finite elements: a new method in micromagnetic simulations 用倒置有限元计算杂散场：微磁模拟中的一种新方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-07 DOI: 10.1007/s10444-024-10139-2
Tahar Z. Boulmezaoud, Keltoum Kaliche
{"title":"Stray field computation by inverted finite elements: a new method in micromagnetic simulations","authors":"Tahar Z. Boulmezaoud,&nbsp;Keltoum Kaliche","doi":"10.1007/s10444-024-10139-2","DOIUrl":"10.1007/s10444-024-10139-2","url":null,"abstract":"<div><p>In this paper, we propose a new method for computing the stray-field and the corresponding energy for a given magnetization configuration. Our approach is based on the use of inverted finite elements and does not need any truncation. After analyzing the problem in an appropriate functional framework, we describe the method and we prove its convergence. We then display some computational results which demonstrate its efficiency and confirm its full potential.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140845840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Unconditional superconvergence analysis of a structure-preserving finite element method for the Poisson-Nernst-Planck equations 针对泊松-纳斯特-普朗克方程的保结构有限元法的无条件超收敛分析
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-06 DOI: 10.1007/s10444-024-10145-4
Huaijun Yang, Meng Li
{"title":"Unconditional superconvergence analysis of a structure-preserving finite element method for the Poisson-Nernst-Planck equations","authors":"Huaijun Yang,&nbsp;Meng Li","doi":"10.1007/s10444-024-10145-4","DOIUrl":"10.1007/s10444-024-10145-4","url":null,"abstract":"<div><p>In this paper, a linearized structure-preserving Galerkin finite element method is investigated for Poisson-Nernst-Planck (PNP) equations. By making full use of the high accuracy estimation of the bilinear element, the mean value technique and rigorously dealing with the coupled nonlinear term, not only the unconditionally optimal error estimate in <span>(L^2)</span>-norm but also the unconditionally superclose error estimate in <span>(H^1)</span>-norm for the related variables are obtained. Then, the unconditionally global superconvergence error estimate in <span>(H^1)</span>-norm is derived by a simple and efficient interpolation post-processing approach, without any coupling restriction condition between the time step size and the space mesh width. Finally, numerical results are provided to confirm the theoretical findings. The numerical scheme preserves the global mass conservation and the electric energy decay, and this work has a great improvement of the error estimate results given in Prohl and Schmuck (Numer. Math. <b>111</b>, 591–630 2009) and Gao and He (J. Sci. Comput. <b>72</b>, 1269–1289 2017).</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140845221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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