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Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems 用于涡流问题等距分析的同调发生器的任意阶样条表示法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-08-19 DOI: 10.1007/s10444-024-10181-0
Bernard Kapidani, Melina Merkel, Sebastian Schöps, Rafael Vázquez
{"title":"Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems","authors":"Bernard Kapidani, Melina Merkel, Sebastian Schöps, Rafael Vázquez","doi":"10.1007/s10444-024-10181-0","DOIUrl":"https://doi.org/10.1007/s10444-024-10181-0","url":null,"abstract":"<p>Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142002702","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
Convergence analysis of the Dirichlet-Neumann Waveform Relaxation algorithm for time fractional sub-diffusion and diffusion-wave equations in heterogeneous media 异质介质中时间分数次扩散和扩散波方程的迪里夏-诺伊曼波形松弛算法收敛性分析
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-08-14 DOI: 10.1007/s10444-024-10185-w
Soura Sana, Bankim C Mandal
{"title":"Convergence analysis of the Dirichlet-Neumann Waveform Relaxation algorithm for time fractional sub-diffusion and diffusion-wave equations in heterogeneous media","authors":"Soura Sana, Bankim C Mandal","doi":"10.1007/s10444-024-10185-w","DOIUrl":"https://doi.org/10.1007/s10444-024-10185-w","url":null,"abstract":"<p>This article presents a comprehensive study on the convergence behavior of the Dirichlet-Neumann Waveform Relaxation algorithm applied to solve the time fractional sub-diffusion and diffusion-wave equations in multiple subdomains, considering the presence of some heterogeneous media. Our analysis focuses on estimating the convergence rate of the algorithm and investigates how this estimate varies with different fractional orders. Furthermore, we extend our analysis to encompass the 2D sub-diffusion case. To validate our findings, we conduct numerical experiments to verify the estimated convergence rate. The results confirm the theoretical estimates and provide empirical evidence for the algorithm’s efficiency and reliability. Moreover, we push the boundaries of the algorithm’s applicability by extending it to solve the time fractional Allen-Chan equation, a problem that exceeds our initial theoretical estimates. Remarkably, we observe that the algorithm performs exceptionally well in this extended scenario for both short and long-time windows.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141980904","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
Frame-normalizable sequences 帧正则序列
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-08-09 DOI: 10.1007/s10444-024-10182-z
Pu-Ting Yu
{"title":"Frame-normalizable sequences","authors":"Pu-Ting Yu","doi":"10.1007/s10444-024-10182-z","DOIUrl":"https://doi.org/10.1007/s10444-024-10182-z","url":null,"abstract":"<p>Let <i>H</i> be a separable Hilbert space and let <span>({x_{n}})</span> be a sequence in <i>H</i> that does not contain any zero elements. We say that <span>({x_{n}})</span> is a <i>Bessel-normalizable</i> or <i>frame-normalizable</i> sequence if the normalized sequence <span>({bigl {frac{x_n}{Vert x_nVert }bigr }})</span> is a Bessel sequence or a frame for <i>H</i>, respectively. In this paper, several necessary and sufficient conditions for sequences to be frame-normalizable and not frame-normalizable are proved. Perturbation theorems for frame-normalizable sequences are also proved. As applications, we show that the Balazs–Stoeva conjecture holds for Bessel-normalizable sequences. Finally, we apply our results to partially answer the open question raised by Aldroubi et al. as to whether the iterative system <span>(bigl {frac{A^{n} x}{Vert A^{n}xVert }bigr }_{nge 0,, xin S})</span> associated with a normal operator <span>(A:Hrightarrow H)</span> and a countable subset <i>S</i> of <i>H</i>, is a frame for <i>H</i>. In particular, if <i>S</i> is finite, then we are able to show that <span>(bigl {frac{A^{n} x}{Vert A^{n}xVert }bigr }_{nge 0,, xin S})</span> is not a frame for <i>H</i> whenever <span>({A^{n}x}_{nge 0,,xin S})</span> is a frame for <i>H</i>.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141908893","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
SlabLU: a two-level sparse direct solver for elliptic PDEs SlabLU：椭圆 PDE 的两级稀疏直接求解器
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-08-09 DOI: 10.1007/s10444-024-10176-x
Anna Yesypenko, Per-Gunnar Martinsson
{"title":"SlabLU: a two-level sparse direct solver for elliptic PDEs","authors":"Anna Yesypenko, Per-Gunnar Martinsson","doi":"10.1007/s10444-024-10176-x","DOIUrl":"https://doi.org/10.1007/s10444-024-10176-x","url":null,"abstract":"<p>The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the domain into thin subdomains, or “slabs” and uses a two-level approach that is designed with parallelization in mind. The scheme takes advantage of <span>(varvec{mathcal {H}}^textbf{2})</span>-matrix structure emerging during factorization and utilizes randomized algorithms to efficiently recover this structure. As opposed to multi-level nested dissection schemes that incorporate the use of <span>(varvec{mathcal {H}})</span> or <span>(varvec{mathcal {H}}^textbf{2})</span> matrices for a hierarchy of front sizes, SlabLU is a two-level scheme which only uses <span>(varvec{mathcal {H}}^textbf{2})</span>-matrix algebra for fronts of roughly the same size. The simplicity allows the scheme to be easily tuned for performance on modern architectures and GPUs. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order accurate multidomain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size <span>(textbf{1000} varvec{lambda } times textbf{1000} varvec{lambda })</span> (for which <span>(varvec{N}~mathbf {=100} textbf{M})</span>) is solved in 15 min to 6 correct digits on a high-powered desktop with GPU acceleration.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141909211","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
Balanced truncation for quadratic-bilinear control systems 二次线性控制系统的平衡截断
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-08-08 DOI: 10.1007/s10444-024-10186-9
Peter Benner, Pawan Goyal
{"title":"Balanced truncation for quadratic-bilinear control systems","authors":"Peter Benner, Pawan Goyal","doi":"10.1007/s10444-024-10186-9","DOIUrl":"https://doi.org/10.1007/s10444-024-10186-9","url":null,"abstract":"<p>We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. <b>21</b>, 143-153 1993). These formulations of Gramians are not only challenging to compute for large-scale systems but hard to utilize also in the MOR framework. This work proposes algebraic Gramians for QB systems based on the underlying Volterra series representation of QB systems and their Hilbert adjoint systems. We then show their relation to a certain type of generalized quadratic Lyapunov equation. Furthermore, we quantify the reachability and observability subspaces based on the proposed Gramians. Consequently, we propose a balancing algorithm, allowing us to find those states that are simultaneously hard to reach and hard to observe. Truncating such states yields reduced-order systems. We also study sufficient conditions for the existence of Gramians, and a local stability of reduced-order models obtained using the proposed balanced truncation scheme. Finally, we demonstrate the proposed balancing-type MOR for QB systems using various numerical examples.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141904535","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 WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data 非光滑数据的后向分数费曼-卡克方程的 WSGD 方案分析
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-08-08 DOI: 10.1007/s10444-024-10188-7
Liyao Hao, Wenyi Tian
{"title":"Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data","authors":"Liyao Hao, Wenyi Tian","doi":"10.1007/s10444-024-10188-7","DOIUrl":"https://doi.org/10.1007/s10444-024-10188-7","url":null,"abstract":"<p>In this paper, we propose and analyze a second-order time-stepping numerical scheme for the inhomogeneous backward fractional Feynman-Kac equation with nonsmooth initial data. The complex parameters and time-space coupled Riemann-Liouville fractional substantial integral and derivative in the equation bring challenges on numerical analysis and computations. The nonlocal operators are approximated by using the weighted and shifted Grünwald difference (WSGD) formula. Then, a second-order WSGD scheme is obtained after making some initial corrections. Moreover, the error estimates of the proposed time-stepping scheme are rigorously established without the regularity requirement on the exact solution. Finally, some numerical experiments are performed to validate the efficiency and accuracy of the proposed numerical scheme.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141904533","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
Weights for moments’ geometrical localization: a canonical isomorphism 力矩几何定位的权重：典型同构
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-08-06 DOI: 10.1007/s10444-024-10183-y
Ana Alonso Rodríguez, Jessika Camaño, Eduardo De Los Santos, Francesca Rapetti
{"title":"Weights for moments’ geometrical localization: a canonical isomorphism","authors":"Ana Alonso Rodríguez, Jessika Camaño, Eduardo De Los Santos, Francesca Rapetti","doi":"10.1007/s10444-024-10183-y","DOIUrl":"https://doi.org/10.1007/s10444-024-10183-y","url":null,"abstract":"<p>This paper deals with high order Whitney forms. We define a canonical isomorphism between two sets of degrees of freedom. This allows to geometrically localize the classical degrees of freedom, the moments, over the elements of a simplicial mesh. With such a localization, it is thus possible to associate, even with moments, a graph structure relating a field with its potential.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141895604","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
Online identification and control of PDEs via reinforcement learning methods 通过强化学习方法对 PDE 进行在线识别和控制
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-08-01 DOI: 10.1007/s10444-024-10167-y
Alessandro Alla, Agnese Pacifico, Michele Palladino, Andrea Pesare
{"title":"Online identification and control of PDEs via reinforcement learning methods","authors":"Alessandro Alla, Agnese Pacifico, Michele Palladino, Andrea Pesare","doi":"10.1007/s10444-024-10167-y","DOIUrl":"https://doi.org/10.1007/s10444-024-10167-y","url":null,"abstract":"<p>We focus on the control of unknown partial differential equations (PDEs). The system dynamics is unknown, but we assume we are able to observe its evolution for a given control input, as typical in a reinforcement learning framework. We propose an algorithm based on the idea to control and identify on the fly the unknown system configuration. In this work, the control is based on the state-dependent Riccati approach, whereas the identification of the model on Bayesian linear regression. At each iteration, based on the observed data, we obtain an estimate of the <i>a-priori</i> unknown parameter configuration of the PDE and then we compute the control of the correspondent model. We show by numerical evidence the convergence of the method for infinite horizon control problems.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862339","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
Averaging property of wedge product and naturality in discrete exterior calculus 离散外部微积分中的楔积平均特性和自然性
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-07-31 DOI: 10.1007/s10444-024-10179-8
Mark D. Schubel, Daniel Berwick-Evans, Anil N. Hirani
{"title":"Averaging property of wedge product and naturality in discrete exterior calculus","authors":"Mark D. Schubel, Daniel Berwick-Evans, Anil N. Hirani","doi":"10.1007/s10444-024-10179-8","DOIUrl":"https://doi.org/10.1007/s10444-024-10179-8","url":null,"abstract":"<p>In exterior calculus on smooth manifolds, the exterior derivative and wedge products are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and an antisymmetrized cup-like product provides a discrete wedge product. We show that these discrete operations in DEC are natural with respect to abstract simplicial maps. A second contribution is a new averaging interpretation of the discrete wedge product in DEC. We also show that this wedge product is the same as Wilson’s cochain product defined using Whitney and de Rham maps.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141857629","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
Two-grid stabilized finite element methods with backtracking for the stationary Navier-Stokes equations 静态纳维-斯托克斯方程的双网格稳定有限元法与回溯法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-07-30 DOI: 10.1007/s10444-024-10180-1
Jing Han, Guangzhi Du
{"title":"Two-grid stabilized finite element methods with backtracking for the stationary Navier-Stokes equations","authors":"Jing Han, Guangzhi Du","doi":"10.1007/s10444-024-10180-1","DOIUrl":"https://doi.org/10.1007/s10444-024-10180-1","url":null,"abstract":"<p>Based on local Gauss integral technique and backtracking technique, this paper presents and studies three kinds of two-grid stabilized finite element algorithms for the stationary Navier-Stokes equations. The proposed methods consist of deducing a coarse solution on the nonlinear system, updating the solution on a fine mesh via three different methods, and solving a linear correction problem on the coarse mesh to obtain the final solution. The error estimates are derived for the solution approximated by the proposed algorithms. A series of numerical experiments are illustrated to test the applicability and efficiency of our proposed methods, and support the theoretical analysis results.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141857628","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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