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Semi-active damping optimization of vibrational systems using the reduced basis method 使用还原法优化振动系统的半主动阻尼
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-31 DOI: 10.1007/s10444-024-10141-8
Jennifer Przybilla, Igor Pontes Duff, Peter Benner
{"title":"Semi-active damping optimization of vibrational systems using the reduced basis method","authors":"Jennifer Przybilla,&nbsp;Igor Pontes Duff,&nbsp;Peter Benner","doi":"10.1007/s10444-024-10141-8","DOIUrl":"10.1007/s10444-024-10141-8","url":null,"abstract":"<div><p>In this article, we consider vibrational systems with semi-active damping that are described by a second-order model. In order to minimize the influence of external inputs to the system response, we are optimizing some damping values. As minimization criterion, we evaluate the energy response, that is the <span>(mathcal {H}_2)</span>-norm of the corresponding transfer function of the system. Computing the energy response includes solving Lyapunov equations for different damping parameters. Hence, the minimization process leads to high computational costs if the system is of large dimension. We present two techniques that reduce the optimization problem by applying the reduced basis method to the corresponding parametric Lyapunov equations. In the first method, we determine a reduced solution space on which the Lyapunov equations and hence the resulting energy response values are computed approximately in a reasonable time. The second method includes the reduced basis method in the minimization process. To evaluate the quality of the approximations, we introduce error estimators that evaluate the error in the controllability Gramians and the energy response. Finally, we illustrate the advantages of our methods by applying them to two different examples.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10141-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141185288","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
A rotational pressure-correction discontinuous Galerkin scheme for the Cahn-Hilliard-Darcy-Stokes system 卡恩-希利亚德-达西-斯托克斯系统的旋转压力校正非连续伽勒金方案
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-30 DOI: 10.1007/s10444-024-10151-6
Meiting Wang, Guang-an Zou, Jian Li
{"title":"A rotational pressure-correction discontinuous Galerkin scheme for the Cahn-Hilliard-Darcy-Stokes system","authors":"Meiting Wang,&nbsp;Guang-an Zou,&nbsp;Jian Li","doi":"10.1007/s10444-024-10151-6","DOIUrl":"10.1007/s10444-024-10151-6","url":null,"abstract":"<div><p>This paper is devoted to the numerical approximations of the Cahn-Hilliard-Darcy-Stokes system, which is a combination of the modified Cahn-Hilliard equation with the Darcy-Stokes equation. A novel discontinuous Galerkin pressure-correction scheme is proposed for solving the coupled system, which can achieve the desired level of linear, fully decoupled, and unconditionally energy stable. The developed scheme here is implemented by combining several effective techniques, including by adding an additional stabilization term artificially in Cahn-Hilliard equation for balancing the explicit treatment of the coupling term, the stabilizing strategy for the nonlinear energy potential, and a rotational pressure-correction scheme for the Darcy-Stokes equation. We rigorously prove the unique solvability, unconditional energy stability, and optimal error estimates of the proposed scheme. Finally, a number of numerical examples are provided to demonstrate numerically the efficiency of the present formulation.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141177501","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
Approximation in the extended functional tensor train format 扩展功能张量列车格式的近似值
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-28 DOI: 10.1007/s10444-024-10140-9
Christoph Strössner, Bonan Sun, Daniel Kressner
{"title":"Approximation in the extended functional tensor train format","authors":"Christoph Strössner,&nbsp;Bonan Sun,&nbsp;Daniel Kressner","doi":"10.1007/s10444-024-10140-9","DOIUrl":"10.1007/s10444-024-10140-9","url":null,"abstract":"<div><p>This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over <span>(96%)</span> compared to the algorithm from Gorodetsky et al. (Comput. Methods Appl. Mech. Eng. <b>347</b>, 59–84 2019).</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10140-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141159590","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
Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation 将变量变换与小波和方差分析结合起来进行高维逼近
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-05-23 DOI: 10.1007/s10444-024-10147-2
Daniel Potts, Laura Weidensager
{"title":"Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation","authors":"Daniel Potts,&nbsp;Laura Weidensager","doi":"10.1007/s10444-024-10147-2","DOIUrl":"10.1007/s10444-024-10147-2","url":null,"abstract":"<div><p>We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low-dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis on the torus. With a variable transformation, we are able to transform the approximation rates and fast algorithms from the torus to other domains. We perform and analyze scattered data approximation for smooth but arbitrary density functions by using a least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. For non-periodic functions, we propose a new extension method. A proper choice of the extension parameter together with the piecewise polynomial Chui-Wang wavelets extends the functions appropriately. In every case, we are able to bound the approximation error with high probability. Additionally, if the function has a low effective dimension (i.e., only interactions of a few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to decrease the approximation error. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.</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":"https://link.springer.com/content/pdf/10.1007/s10444-024-10147-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141085204","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
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
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