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Sobolev regularity of bivariate isogeometric finite element spaces in case of a geometry map with degenerate corner 双变量等几何有限元空间在有退化角几何图形情况下的索波列夫正则性
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-10-24 DOI: 10.1007/s10444-024-10203-x
Ulrich Reif
{"title":"Sobolev regularity of bivariate isogeometric finite element spaces in case of a geometry map with degenerate corner","authors":"Ulrich Reif","doi":"10.1007/s10444-024-10203-x","DOIUrl":"10.1007/s10444-024-10203-x","url":null,"abstract":"<div><p>We investigate Sobolev regularity of bivariate functions obtained in Isogeometric Analysis when using geometry maps that are degenerate in the sense that the first partial derivatives vanish at isolated points. In particular, we show how the known <span>(C^1)</span>-conditions for D-patches have to be tightened to guarantee square integrability of second partial derivatives, as required when computing finite element approximations of elliptic fourth order PDEs like the biharmonic equation.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 6","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10203-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142488419","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 ansatz space for moving least squares approximation on spheres 球面移动最小二乘法近似的最佳解析空间
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-10-22 DOI: 10.1007/s10444-024-10201-z
Ralf Hielscher, Tim Pöschl
{"title":"An optimal ansatz space for moving least squares approximation on spheres","authors":"Ralf Hielscher,&nbsp;Tim Pöschl","doi":"10.1007/s10444-024-10201-z","DOIUrl":"10.1007/s10444-024-10201-z","url":null,"abstract":"<div><p>We revisit the moving least squares (MLS) approximation scheme on the sphere <span>(mathbb S^{d-1} subset {mathbb R}^d)</span>, where <span>(d&gt;1)</span>. It is well known that using the spherical harmonics up to degree <span>(L in {mathbb N})</span> as ansatz space yields for functions in <span>(mathcal {C}^{L+1}(mathbb S^{d-1}))</span> the approximation order <span>(mathcal {O}left( h^{L+1} right) )</span>, where <i>h</i> denotes the fill distance of the sampling nodes. In this paper, we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degrees up to <i>L</i>, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as <span>(h rightarrow 0)</span>. Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space of the sphere as ansatz space.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 6","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10201-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142453126","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 unified local projection-based stabilized virtual element method for the coupled Stokes-Darcy problem 斯托克斯-达西耦合问题的基于局部投影的统一稳定虚拟元素法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-10-21 DOI: 10.1007/s10444-024-10199-4
Sudheer Mishra, E. Natarajan
{"title":"A unified local projection-based stabilized virtual element method for the coupled Stokes-Darcy problem","authors":"Sudheer Mishra,&nbsp;E. Natarajan","doi":"10.1007/s10444-024-10199-4","DOIUrl":"10.1007/s10444-024-10199-4","url":null,"abstract":"<div><p>In this work, we propose and analyze a new stabilized virtual element method for the coupled Stokes-Darcy problem with Beavers-Joseph-Saffman interface condition on polygonal meshes. We derive two variants of local projection stabilization methods for the coupled Stokes-Darcy problem. The significance of local projection-based stabilization terms is that they provide reasonable control of the pressure component of the Stokes flow without involving higher-order derivative terms. The discrete inf-sup condition of the coupled Stokes-Darcy problem is established for the equal-order virtual element triplets involving velocity, hydraulic head, and pressure. The optimal error estimates are derived using the equal-order virtual elements in the energy and <span>(L^2)</span> norms. The proposed methods have several advantages: mass conservative, avoiding the coupling of the solution components, more accessible to implement, and performing efficiently on hybrid polygonal elements. Numerical experiments are conducted to depict the flexibility of the proposed methods, validating the theoretical results.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 6","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142451931","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 pressure-residual augmented GLS stabilized method for a type of Stokes equations with nonstandard boundary conditions 具有非标准边界条件的斯托克斯方程类型的压力-滞后增强 GLS 稳定方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-10-14 DOI: 10.1007/s10444-024-10204-w
Huoyuan Duan, Roger C. E. Tan, Duowei Zhu
{"title":"A pressure-residual augmented GLS stabilized method for a type of Stokes equations with nonstandard boundary conditions","authors":"Huoyuan Duan,&nbsp;Roger C. E. Tan,&nbsp;Duowei Zhu","doi":"10.1007/s10444-024-10204-w","DOIUrl":"10.1007/s10444-024-10204-w","url":null,"abstract":"<div><p>With local pressure-residual stabilizations as an augmentation to the classical Galerkin/least-squares (GLS) stabilized method, a new locally evaluated residual-based stabilized finite element method is proposed for a type of Stokes equations from the incompressible flows. We focus on the study of a type of nonstandard boundary conditions involving the mixed tangential velocity and pressure Dirichlet boundary conditions. Unexpectedly, in sharp contrast to the standard no-slip velocity Dirichlet boundary condition, neither the discrete LBB inf-sup stable elements nor the stabilized methods such as the classical GLS method could certainly ensure a convergent finite element solution, because the velocity solution could be very weak with its gradient not being square integrable. The main purpose of this paper is to study the error estimates of the new stabilized method for approximating the very weak velocity solution; with the local pressure-residual stabilizations, we can manage to prove the error estimates with a reasonable convergence order. Numerical results are provided to illustrate the performance and the theoretical results of the proposed method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142431043","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 stochastic perturbation analysis of the QR decomposition and its applications QR 分解的随机扰动分析及其应用
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-10-02 DOI: 10.1007/s10444-024-10198-5
Tianru Wang, Yimin Wei
{"title":"A stochastic perturbation analysis of the QR decomposition and its applications","authors":"Tianru Wang,&nbsp;Yimin Wei","doi":"10.1007/s10444-024-10198-5","DOIUrl":"10.1007/s10444-024-10198-5","url":null,"abstract":"<div><p>The perturbation of the QR decompostion is analyzed from the probalistic point of view. The perturbation error is approximated by a first-order perturbation expansion with high probability where the perturbation is assumed to be random. Different from the previous normwise perturbation bounds using the Frobenius norm, our techniques are used to develop the spectral norm, as well as the entry-wise perturbation bounds for the stochastic perturbation of the QR decomposition. The statistics tends to be tighter (in the sense of the expectation) and more realistic than the classical worst-case perturbation bounds. The novel perturbation bounds are applicable to a wide range of problems in statistics and communications. In this paper, we consider the perturbation bound of the leverage scores under the Gaussian perturbation, the probability guarantees and the error bounds of the low rank matrix recovery, and the upper bound of the errors of the tensor CUR-type decomposition. We also apply our perturbation bounds to improve the robust design of the Tomlinson-Harashima precoding in the Multiple-Input Multiple-Output (MIMO) system.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142363094","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 electrical engineering perspective on naturality in computational physics 从电气工程角度看计算物理学的自然性
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-10-01 DOI: 10.1007/s10444-024-10197-6
P. Robert Kotiuga, Valtteri Lahtinen
{"title":"An electrical engineering perspective on naturality in computational physics","authors":"P. Robert Kotiuga,&nbsp;Valtteri Lahtinen","doi":"10.1007/s10444-024-10197-6","DOIUrl":"10.1007/s10444-024-10197-6","url":null,"abstract":"<div><p>We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We discuss elliptic complexes and highlight the category theoretical background and its role as a unifying language between algebraic topology, differential geometry, and modelling software design. In particular, the ubiquitous concept of naturality is central. Natural differential operators have functorial analogues on the cochains of triangulated manifolds. In order to establish this correspondence, we derive formulas involving simplices and barycentric coordinates, defining discrete vector fields and a discrete Lie derivative as a result of a discrete analogue of Cartan’s magic formula. This theorem is the main mathematical result of the paper.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142360098","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
Maximal volume matrix cross approximation for image compression and least squares solution 用于图像压缩和最小二乘法求解的最大体积矩阵交叉近似法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-09-16 DOI: 10.1007/s10444-024-10196-7
Kenneth Allen, Ming-Jun Lai, Zhaiming Shen
{"title":"Maximal volume matrix cross approximation for image compression and least squares solution","authors":"Kenneth Allen,&nbsp;Ming-Jun Lai,&nbsp;Zhaiming Shen","doi":"10.1007/s10444-024-10196-7","DOIUrl":"10.1007/s10444-024-10196-7","url":null,"abstract":"<div><p>We study the classic matrix cross approximation based on the maximal volume submatrices. Our main results consist of an improvement of the classic estimate for matrix cross approximation and a greedy approach for finding the maximal volume submatrices. More precisely, we present a new proof of the classic estimate of the inequality with an improved constant. Also, we present a family of greedy maximal volume algorithms to improve the computational efficiency of matrix cross approximation. The proposed algorithms are shown to have theoretical guarantees of convergence. Finally, we present two applications: image compression and the least squares approximation of continuous functions. Our numerical results at the end of the paper demonstrate the effective performance of our approach.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142234450","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
Multilevel approximation of Gaussian random fields: Covariance compression, estimation, and spatial prediction 高斯随机场的多级近似：协方差压缩、估计和空间预测
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-09-14 DOI: 10.1007/s10444-024-10187-8
Helmut Harbrecht, Lukas Herrmann, Kristin Kirchner, Christoph Schwab
{"title":"Multilevel approximation of Gaussian random fields: Covariance compression, estimation, and spatial prediction","authors":"Helmut Harbrecht,&nbsp;Lukas Herrmann,&nbsp;Kristin Kirchner,&nbsp;Christoph Schwab","doi":"10.1007/s10444-024-10187-8","DOIUrl":"10.1007/s10444-024-10187-8","url":null,"abstract":"<div><p>The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension <i>p</i> of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number <i>p</i> of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters <i>p</i> of the sample-wise approximation of the GRF in Sobolev scales.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10187-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142231551","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
Improved a posteriori error bounds for reduced port-Hamiltonian systems 改进的还原端口-哈密尔顿系统后验误差边界
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-09-11 DOI: 10.1007/s10444-024-10195-8
Johannes Rettberg, Dominik Wittwar, Patrick Buchfink, Robin Herkert, Jörg Fehr, Bernard Haasdonk
{"title":"Improved a posteriori error bounds for reduced port-Hamiltonian systems","authors":"Johannes Rettberg,&nbsp;Dominik Wittwar,&nbsp;Patrick Buchfink,&nbsp;Robin Herkert,&nbsp;Jörg Fehr,&nbsp;Bernard Haasdonk","doi":"10.1007/s10444-024-10195-8","DOIUrl":"10.1007/s10444-024-10195-8","url":null,"abstract":"<div><p>Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) <i>a hierarchical error bound</i> and (b) <i>an error bound based on an auxiliary linear problem</i>, to the case of port-Hamiltonian systems. The approaches rely on a secondary approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are adapted to port-Hamiltonian systems. The mathematical relationship between the two methods is discussed both theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid–structure interaction.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10195-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142166256","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
Interpolating refinable functions and (n_s)-step interpolatory subdivision schemes 可细化函数内插和 $$n_s$$ 步内插细分方案
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-09-05 DOI: 10.1007/s10444-024-10192-x
Bin Han
{"title":"Interpolating refinable functions and (n_s)-step interpolatory subdivision schemes","authors":"Bin Han","doi":"10.1007/s10444-024-10192-x","DOIUrl":"10.1007/s10444-024-10192-x","url":null,"abstract":"<div><p>Standard interpolatory subdivision schemes and their underlying interpolating refinable functions are of interest in CAGD, numerical PDEs, and approximation theory. Generalizing these notions, we introduce and study <span>(n_s)</span>-step interpolatory <span>(textsf{M})</span>-subdivision schemes and their interpolating <span>(textsf{M})</span>-refinable functions with <span>(n_sin mathbb {N}cup {infty })</span> and a dilation factor <span>(textsf{M}in mathbb {N}backslash {1})</span>. We completely characterize <span>(mathscr {C}^m)</span>-convergence and smoothness of <span>(n_s)</span>-step interpolatory subdivision schemes and their interpolating <span>(textsf{M})</span>-refinable functions in terms of their masks. Inspired by <span>(n_s)</span>-step interpolatory stationary subdivision schemes, we further introduce the notion of <i>r</i>-mask quasi-stationary subdivision schemes, and then we characterize their <span>(mathscr {C}^m)</span>-convergence and smoothness properties using only their masks. Moreover, combining <span>(n_s)</span>-step interpolatory subdivision schemes with <i>r</i>-mask quasi-stationary subdivision schemes, we can obtain <span>(r n_s)</span>-step interpolatory subdivision schemes. Examples and construction procedures of convergent <span>(n_s)</span>-step interpolatory <span>(textsf{M})</span>-subdivision schemes are provided to illustrate our results with dilation factors <span>(textsf{M}=2,3,4)</span>. In addition, for the dyadic dilation <span>(textsf{M}=2)</span> and <span>(r=2,3)</span>, using <i>r</i> masks with only two-ring stencils, we provide examples of <span>(mathscr {C}^r)</span>-convergent <i>r</i>-step interpolatory <i>r</i>-mask quasi-stationary dyadic subdivision schemes.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142138151","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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