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Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils. 奇异矩阵铅笔的一类随机数值方法的特征值条件数分析。
IF 1.6 3区 数学
BIT Numerical Mathematics Pub Date : 2024-01-01 Epub Date: 2024-07-15 DOI: 10.1007/s10543-024-01033-w
Daniel Kressner, Bor Plestenjak
{"title":"Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils.","authors":"Daniel Kressner, Bor Plestenjak","doi":"10.1007/s10543-024-01033-w","DOIUrl":"https://doi.org/10.1007/s10543-024-01033-w","url":null,"abstract":"<p><p>The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the <math><mi>δ</mi></math> -weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"64 3","pages":"32"},"PeriodicalIF":1.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11249782/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141635867","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
Well-posedness and finite element approximation of mixed dimensional partial differential equations 混合维偏微分方程的完备性和有限元逼近
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2023-12-29 DOI: 10.1007/s10543-023-01001-w
Fredrik Hellman, Axel Målqvist, Malin Mosquera
{"title":"Well-posedness and finite element approximation of mixed dimensional partial differential equations","authors":"Fredrik Hellman, Axel Målqvist, Malin Mosquera","doi":"10.1007/s10543-023-01001-w","DOIUrl":"https://doi.org/10.1007/s10543-023-01001-w","url":null,"abstract":"<p>In this article, a mixed dimensional elliptic partial differential equation is considered, posed in a bulk domain with a large number of embedded interfaces. In particular, well-posedness of the problem and regularity of the solution are studied. A fitted finite element approximation is also proposed and an a priori error bound is proved. For the solution of the arising linear system, an iterative method based on subspace decomposition is proposed and analyzed. Finally, numerical experiments are presented and rapid convergence using the proposed preconditioner is achieved, confirming the theoretical findings.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"178 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139070858","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
Randomized Kaczmarz algorithm with averaging and block projection 带有平均和块投影的随机卡兹马兹算法
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2023-12-12 DOI: 10.1007/s10543-023-01002-9
Zeyi Zhang, Dong Shen
{"title":"Randomized Kaczmarz algorithm with averaging and block projection","authors":"Zeyi Zhang, Dong Shen","doi":"10.1007/s10543-023-01002-9","DOIUrl":"https://doi.org/10.1007/s10543-023-01002-9","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"37 11","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138633210","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
Positivity-preserving truncated Euler–Maruyama method for generalised Ait-Sahalia-type interest model 广义ait - sahalia型利息模型的保正截断Euler-Maruyama方法
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2023-11-27 DOI: 10.1007/s10543-023-01000-x
Shounian Deng, Chen Fei, Weiyin Fei, Xuerong Mao
{"title":"Positivity-preserving truncated Euler–Maruyama method for generalised Ait-Sahalia-type interest model","authors":"Shounian Deng, Chen Fei, Weiyin Fei, Xuerong Mao","doi":"10.1007/s10543-023-01000-x","DOIUrl":"https://doi.org/10.1007/s10543-023-01000-x","url":null,"abstract":"<p>The well-known Ait-Sahalia-type interest model, arising in mathematical finance, has some typical features: polynomial drift that blows up at the origin, highly nonlinear diffusion, and positive solution. The known explicit numerical methods including truncated/tamed Euler–Maruyama (EM) applied to it do not preserve its positivity. The main interest of this work is to investigate the numerical conservation of positivity of the solution of generalised Ait-Sahalia-type model. By modifying the truncated EM method to generate positive sequences of numerical approximations, we obtain the rate of convergence of the numerical algorithm not only at time <i>T</i> but also over the time interval [0, <i>T</i>]. Numerical experiments confirm the theoretical results.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"16 3‐4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503713","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 flexible short recurrence Krylov subspace method for matrices arising in the time integration of port-Hamiltonian systems and ODEs/DAEs with a dissipative Hamiltonian 具有耗散哈密顿量的port- hamilton系统和ode /DAEs时间积分中矩阵的柔性短递归Krylov子空间方法
3区 数学
BIT Numerical Mathematics Pub Date : 2023-11-10 DOI: 10.1007/s10543-023-00999-3
Malak Diab, Andreas Frommer, Karsten Kahl
{"title":"A flexible short recurrence Krylov subspace method for matrices arising in the time integration of port-Hamiltonian systems and ODEs/DAEs with a dissipative Hamiltonian","authors":"Malak Diab, Andreas Frommer, Karsten Kahl","doi":"10.1007/s10543-023-00999-3","DOIUrl":"https://doi.org/10.1007/s10543-023-00999-3","url":null,"abstract":"Abstract For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for discretizations of dissipative Hamiltonian ODEs, DAEs and port-Hamiltonian systems where, in addition, the Hermitian part is positive definite or semi-definite. It is then possible to develop short recurrence optimal Krylov subspace methods in which the Hermitian part is used as a preconditioner. In this paper, we develop new, right preconditioned variants of this approach which, as their crucial new feature, allow the systems with the Hermitian part to be solved only approximately in each iteration while keeping the short recurrences. This new class of methods is particularly efficient as it allows, for example, to use few steps of a multigrid solver or a (preconditioned) CG method for the Hermitian part in each iteration. We illustrate this with several numerical experiments for large scale systems.","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"47 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135092615","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: 1
Symmetric-conjugate splitting methods for linear unitary problems 线性酉问题的对称共轭分裂方法
3区 数学
BIT Numerical Mathematics Pub Date : 2023-11-10 DOI: 10.1007/s10543-023-00998-4
Joackim Bernier, S Blanes, Fernando Casas, A Escorihuela-Tomàs
{"title":"Symmetric-conjugate splitting methods for linear unitary problems","authors":"Joackim Bernier, S Blanes, Fernando Casas, A Escorihuela-Tomàs","doi":"10.1007/s10543-023-00998-4","DOIUrl":"https://doi.org/10.1007/s10543-023-00998-4","url":null,"abstract":"Abstract We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficients and are conjugated to unitary transformations for sufficiently small values of the time step-size. New and efficient methods up to order six are constructed and tested on the linear Schrödinger equation.","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"78 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087342","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: 1
An hp-version interior penalty discontinuous Galerkin method for the quad-curl eigenvalue problem 求解四旋度特征值问题的hp型内罚不连续Galerkin方法
3区 数学
BIT Numerical Mathematics Pub Date : 2023-11-10 DOI: 10.1007/s10543-023-00996-6
Jiayu Han, Zhimin Zhang
{"title":"An hp-version interior penalty discontinuous Galerkin method for the quad-curl eigenvalue problem","authors":"Jiayu Han, Zhimin Zhang","doi":"10.1007/s10543-023-00996-6","DOIUrl":"https://doi.org/10.1007/s10543-023-00996-6","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"37 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135093357","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 note on approximate Jacobians of implicit Runge–Kutta methods and convergence of modified Newton iterations 隐式龙格-库塔方法的近似雅可比矩阵及修正牛顿迭代的收敛性
3区 数学
BIT Numerical Mathematics Pub Date : 2023-11-01 DOI: 10.1007/s10543-023-00994-8
Laurent O. Jay, Olga Sokratova
{"title":"A note on approximate Jacobians of implicit Runge–Kutta methods and convergence of modified Newton iterations","authors":"Laurent O. Jay, Olga Sokratova","doi":"10.1007/s10543-023-00994-8","DOIUrl":"https://doi.org/10.1007/s10543-023-00994-8","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135325977","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 structure-preserving mixed finite element method for the stationary MHD equations with magnetic-current formulation 具有磁流公式的平稳MHD方程的一种新的保结构混合有限元方法
3区 数学
BIT Numerical Mathematics Pub Date : 2023-11-01 DOI: 10.1007/s10543-023-00995-7
Xiaodi Zhang, Shitian Dong
{"title":"New structure-preserving mixed finite element method for the stationary MHD equations with magnetic-current formulation","authors":"Xiaodi Zhang, Shitian Dong","doi":"10.1007/s10543-023-00995-7","DOIUrl":"https://doi.org/10.1007/s10543-023-00995-7","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"18 2-3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135270974","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
Gaussian rule for integrals involving Bessel functions 涉及贝塞尔函数积分的高斯规则
3区 数学
BIT Numerical Mathematics Pub Date : 2023-11-01 DOI: 10.1007/s10543-023-00997-5
Eleonora Denich, Paolo Novati
{"title":"Gaussian rule for integrals involving Bessel functions","authors":"Eleonora Denich, Paolo Novati","doi":"10.1007/s10543-023-00997-5","DOIUrl":"https://doi.org/10.1007/s10543-023-00997-5","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"31 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136103242","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: 1
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