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Correction: A Grassmann manifold handbook: basic geometry and computational aspects 更正：格拉斯曼流形手册：基本几何与计算方面
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-03-22 DOI: 10.1007/s10444-024-10115-w
Thomas Bendokat, Ralf Zimmermann, P.-A. Absil
{"title":"Correction: A Grassmann manifold handbook: basic geometry and computational aspects","authors":"Thomas Bendokat, Ralf Zimmermann, P.-A. Absil","doi":"10.1007/s10444-024-10115-w","DOIUrl":"https://doi.org/10.1007/s10444-024-10115-w","url":null,"abstract":"","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140214636","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
Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equation 应用于非线性时间分数扩散方程的二维空间最佳二次样条配位法数值分析
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-03-22 DOI: 10.1007/s10444-024-10116-9
Xiao Ye, Xiangcheng Zheng, Jun Liu, Yue Liu
{"title":"Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equation","authors":"Xiao Ye, Xiangcheng Zheng, Jun Liu, Yue Liu","doi":"10.1007/s10444-024-10116-9","DOIUrl":"https://doi.org/10.1007/s10444-024-10116-9","url":null,"abstract":"<p>Optimal quadratic spline collocation (QSC) method has been widely used in various problems due to its high-order accuracy, while the corresponding numerical analysis is rarely investigated since, e.g., the perturbation terms result in the asymmetry of optimal QSC discretization. We present numerical analysis for the optimal QSC method in two space dimensions via discretizing a nonlinear time-fractional diffusion equation for demonstration. The <i>L</i>2-1<span>(_sigma )</span> formula on the graded mesh is used to account for the initial solution singularity, leading to an optimal QSC–<i>L</i>2-1<span>(_{sigma })</span> scheme where the nonlinear term is treated by the extrapolation. We provide the existence and uniqueness of the numerical solution, as well as the second-order temporal accuracy and fourth-order spatial accuracy with proper grading parameters. Furthermore, we consider the fast implementation based on the sum-of-exponentials technique to reduce the computational cost. Numerical experiments are performed to verify the theoretical analysis and the effectiveness of the proposed scheme.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140188703","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 probabilistic reduced basis method for parameter-dependent problems 参数相关问题的概率还原法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-03-13 DOI: 10.1007/s10444-024-10114-x
Marie Billaud-Friess, Arthur Macherey, Anthony Nouy, Clémentine Prieur
{"title":"A probabilistic reduced basis method for parameter-dependent problems","authors":"Marie Billaud-Friess, Arthur Macherey, Anthony Nouy, Clémentine Prieur","doi":"10.1007/s10444-024-10114-x","DOIUrl":"https://doi.org/10.1007/s10444-024-10114-x","url":null,"abstract":"<p>Probabilistic variants of model order reduction (MOR) methods have recently emerged for improving stability and computational performance of classical approaches. In this paper, we propose a probabilistic reduced basis method (RBM) for the approximation of a family of parameter-dependent functions. It relies on a probabilistic greedy algorithm with an error indicator that can be written as an expectation of some parameter-dependent random variable. Practical algorithms relying on Monte Carlo estimates of this error indicator are discussed. In particular, when using probably approximately correct (PAC) bandit algorithm, the resulting procedure is proven to be a weak-greedy algorithm with high probability. Intended applications concern the approximation of a parameter-dependent family of functions for which we only have access to (noisy) pointwise evaluations. As a particular application, we consider the approximation of solution manifolds of linear parameter-dependent partial differential equations with a probabilistic interpretation through the Feynman-Kac formula.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140123898","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
Structured interpolation for multivariate transfer functions of quadratic-bilinear systems 二次线性系统多元传递函数的结构插值
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-03-12 DOI: 10.1007/s10444-024-10109-8
Peter Benner, Serkan Gugercin, Steffen W. R. Werner
{"title":"Structured interpolation for multivariate transfer functions of quadratic-bilinear systems","authors":"Peter Benner, Serkan Gugercin, Steffen W. R. Werner","doi":"10.1007/s10444-024-10109-8","DOIUrl":"https://doi.org/10.1007/s10444-024-10109-8","url":null,"abstract":"<p>High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting large-scale nonlinear systems. The high dimensionality of the dynamics causes computational bottlenecks, especially when these large-scale systems need to be simulated for a variety of situations such as different forcing terms. This motivates model reduction where the goal is to replace the full-order dynamics with accurate reduced-order surrogates. Interpolation-based model reduction has been proven to be an effective tool for the construction of cheap-to-evaluate surrogate models that preserve the internal structure in the case of weak nonlinearities. In this paper, we consider the construction of multivariate interpolants in frequency domain for structured quadratic-bilinear systems. We propose definitions for structured variants of the symmetric subsystem and generalized transfer functions of quadratic-bilinear systems and provide conditions for structure-preserving interpolation by projection. The theoretical results are illustrated using two numerical examples including the simulation of molecular dynamics in crystal structures.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140123990","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
New low-order mixed finite element methods for linear elasticity 线性弹性的新低阶混合有限元方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-03-06 DOI: 10.1007/s10444-024-10112-z
Xuehai Huang, Chao Zhang, Yaqian Zhou, Yangxing Zhu
{"title":"New low-order mixed finite element methods for linear elasticity","authors":"Xuehai Huang, Chao Zhang, Yaqian Zhou, Yangxing Zhu","doi":"10.1007/s10444-024-10112-z","DOIUrl":"https://doi.org/10.1007/s10444-024-10112-z","url":null,"abstract":"<p>New low-order <span>({H}({{text {div}}}))</span>-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the <span>({(d+1)})</span>-order normal-normal face bubble space. The reduced counterpart has only <span>({d(d+1)}^{{2}})</span> degrees of freedom. Basis functions are explicitly given in terms of barycentric coordinates. Low-order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lamé coefficient <span>({lambda })</span>, and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140043445","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
Conditioning and spectral properties of isogeometric collocation matrices for acoustic wave problems 声波问题等几何配位矩阵的条件和频谱特性
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-03-04 DOI: 10.1007/s10444-024-10113-y
Elena Zampieri, Luca F. Pavarino
{"title":"Conditioning and spectral properties of isogeometric collocation matrices for acoustic wave problems","authors":"Elena Zampieri, Luca F. Pavarino","doi":"10.1007/s10444-024-10113-y","DOIUrl":"https://doi.org/10.1007/s10444-024-10113-y","url":null,"abstract":"<p>The conditioning and spectral properties of the mass and stiffness matrices for acoustic wave problems are here investigated when isogeometric analysis (IGA) collocation methods in space and Newmark methods in time are employed. Theoretical estimates and extensive numerical results are reported for the eigenvalues and condition numbers of the acoustic mass and stiffness matrices in the reference square domain with Dirichlet, Neumann, and absorbing boundary conditions. This study focuses in particular on the spectral dependence on the polynomial degree <i>p</i>, mesh size <i>h</i>, regularity <i>k</i>, of the IGA discretization and on the time step size <span>(Delta t)</span> and parameter <span>(beta )</span> of the Newmark method. Results on the sparsity of the matrices and the eigenvalue distribution with respect to the number of degrees of freedom <span> d.o.f.</span> and the number of nonzero entries <span>nz</span> are also reported. The results show that the spectral properties of the IGA collocation matrices are comparable with the available spectral estimates for IGA Galerkin matrices associated with the Poisson problem with Dirichlet boundary conditions, and in some cases, the IGA collocation results are better than the corresponding IGA Galerkin estimates, in particular for increasing <i>p</i> and maximal regularity <span>(k=p-1)</span>.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140026260","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 space–time DG method for the Schrödinger equation with variable potential 具有可变势能的薛定谔方程的时空 DG 方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-03-01 DOI: 10.1007/s10444-024-10108-9
Sergio Gómez, Andrea Moiola
{"title":"A space–time DG method for the Schrödinger equation with variable potential","authors":"Sergio Gómez, Andrea Moiola","doi":"10.1007/s10444-024-10108-9","DOIUrl":"https://doi.org/10.1007/s10444-024-10108-9","url":null,"abstract":"<p>We present a space–time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal <i>h</i>-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001071","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
Limitations of neural network training due to numerical instability of backpropagation 反向传播的数值不稳定性导致神经网络训练的局限性
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-02-11 DOI: 10.1007/s10444-024-10106-x
Clemens Karner, Vladimir Kazeev, Philipp Christian Petersen
{"title":"Limitations of neural network training due to numerical instability of backpropagation","authors":"Clemens Karner, Vladimir Kazeev, Philipp Christian Petersen","doi":"10.1007/s10444-024-10106-x","DOIUrl":"https://doi.org/10.1007/s10444-024-10106-x","url":null,"abstract":"<p>We study the training of deep neural networks by gradient descent where floating-point arithmetic is used to compute the gradients. In this framework and under realistic assumptions, we demonstrate that it is <i>highly unlikely</i> to find ReLU neural networks that maintain, in the course of training with gradient descent, <i>superlinearly</i> many affine pieces with respect to their number of layers. In virtually all approximation theoretical arguments which yield high order polynomial rates of approximation, sequences of ReLU neural networks with <i>exponentially</i> many affine pieces compared to their numbers of layers are used. As a consequence, we conclude that approximating sequences of ReLU neural networks resulting from gradient descent in practice differ substantially from theoretically constructed sequences. The assumptions and the theoretical results are compared to a numerical study, which yields concurring results.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139720270","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
Analysis of a $$varvec{P}_1oplus varvec{RT}_0$$ finite element method for linear elasticity with Dirichlet and mixed boundary conditions 带 Dirichlet 和混合边界条件的线性弹性的 $$varvec{P}_1oplus varvec{RT}_0$ 有限元方法分析
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-02-05 DOI: 10.1007/s10444-024-10107-w
Hongpeng Li, Xu Li, Hongxing Rui
{"title":"Analysis of a $$varvec{P}_1oplus varvec{RT}_0$$ finite element method for linear elasticity with Dirichlet and mixed boundary conditions","authors":"Hongpeng Li, Xu Li, Hongxing Rui","doi":"10.1007/s10444-024-10107-w","DOIUrl":"https://doi.org/10.1007/s10444-024-10107-w","url":null,"abstract":"<p>In this paper, we investigate a low-order robust numerical method for the linear elasticity problem. The method is based on a Bernardi–Raugel-like <span>(varvec{H}(textrm{div}))</span>-conforming method proposed first for the Stokes flows in [Li and Rui, IMA J. Numer. Anal. <b>42</b> (2022) 3711–3734]. Therein, the lowest-order <span>(varvec{H}(textrm{div}))</span>-conforming Raviart–Thomas space (<span>(varvec{RT}_0)</span>) was added to the classical conforming <span>(varvec{P}_1times P_0)</span> pair to meet the inf-sup condition, while preserving the divergence constraint and some important features of conforming methods. Due to the inf-sup stability of the <span>(varvec{P}_1oplus varvec{RT}_0times P_0)</span> pair, a locking-free elasticity discretization with respect to the Lamé constant <span>(lambda )</span> can be naturally obtained. Moreover, our scheme is gradient-robust for the pure and homogeneous displacement boundary problem, that is, the discrete <span>(varvec{H}^1)</span>-norm of the displacement is <span>(mathcal {O}(lambda ^{-1}))</span> when the external body force is a gradient field. We also consider the mixed displacement and stress boundary problem, whose <span>(varvec{P}_1oplus varvec{RT}_0)</span> discretization should be carefully designed due to a consistency error arising from the <span>(varvec{RT}_0)</span> part. We propose both symmetric and nonsymmetric schemes to approximate the mixed boundary case. The optimal error estimates are derived for the energy norm and/or <span>(varvec{L}^2)</span>-norm. Numerical experiments demonstrate the accuracy and robustness of our schemes.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695830","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
Semi-Lagrangian finite element exterior calculus for incompressible flows 不可压缩流动的半拉格朗日有限元外部微积分
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-02-05 DOI: 10.1007/s10444-023-10092-6
Wouter Tonnon, Ralf Hiptmair
{"title":"Semi-Lagrangian finite element exterior calculus for incompressible flows","authors":"Wouter Tonnon, Ralf Hiptmair","doi":"10.1007/s10444-023-10092-6","DOIUrl":"https://doi.org/10.1007/s10444-023-10092-6","url":null,"abstract":"<p>We develop a semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions on arbitrary simplicial meshes. We recast the equations as a nonlinear transport problem for a momentum 1-form and discretize in space using methods from finite element exterior calculus. Numerical experiments show that the linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing viscosity limit and is applicable to inviscid incompressible Euler flows. We obtain second-order convergence and conservation of energy is achieved through a Lagrange multiplier.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695841","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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