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Pressure-robust approximation of the incompressible Navier–Stokes equations in a rotating frame of reference 旋转参照系中不可压缩纳维-斯托克斯方程的压力近似法
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-09-18 DOI: 10.1007/s10543-024-01037-6
Medine Demir, Volker John
{"title":"Pressure-robust approximation of the incompressible Navier–Stokes equations in a rotating frame of reference","authors":"Medine Demir, Volker John","doi":"10.1007/s10543-024-01037-6","DOIUrl":"https://doi.org/10.1007/s10543-024-01037-6","url":null,"abstract":"<p>A pressure-robust space discretization of the incompressible Navier–Stokes equations in a rotating frame of reference is considered. The discretization employs divergence-free, <span>(H^1)</span>-conforming mixed finite element methods like Scott–Vogelius pairs. An error estimate for the velocity is derived that tracks the dependency of the error bound on the coefficients of the problem, in particular on the angular velocity. Numerical examples support the theoretical results.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250073","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
Lower error bounds and optimality of approximation for jump-diffusion SDEs with discontinuous drift 具有不连续漂移的跳跃-扩散 SDE 的误差下限和近似最优性
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-09-09 DOI: 10.1007/s10543-024-01036-7
Paweł Przybyłowicz, Verena Schwarz, Michaela Szölgyenyi
{"title":"Lower error bounds and optimality of approximation for jump-diffusion SDEs with discontinuous drift","authors":"Paweł Przybyłowicz, Verena Schwarz, Michaela Szölgyenyi","doi":"10.1007/s10543-024-01036-7","DOIUrl":"https://doi.org/10.1007/s10543-024-01036-7","url":null,"abstract":"<p>In this paper sharp lower error bounds for numerical methods for jump-diffusion stochastic differential equations (SDEs) with discontinuous drift are proven. The approximation of jump-diffusion SDEs with non-adaptive as well as jump-adapted approximation schemes is studied and lower error bounds of order 3/4 for both classes of approximation schemes are provided. This yields optimality of the transformation-based jump-adapted quasi-Milstein scheme.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197729","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
Super-localized orthogonal decomposition for convection-dominated diffusion problems 对流主导扩散问题的超局部正交分解
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-08-05 DOI: 10.1007/s10543-024-01035-8
Francesca Bonizzoni, Philip Freese, Daniel Peterseim
{"title":"Super-localized orthogonal decomposition for convection-dominated diffusion problems","authors":"Francesca Bonizzoni, Philip Freese, Daniel Peterseim","doi":"10.1007/s10543-024-01035-8","DOIUrl":"https://doi.org/10.1007/s10543-024-01035-8","url":null,"abstract":"<p>This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the <span>(L^2)</span>-norm, the Galerkin projection onto this generalized finite element space even yields <span>(varepsilon )</span>-independent error bounds, <span>(varepsilon )</span> being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate <span>(varepsilon )</span>-robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969259","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
Substructuring the Hiptmair-Xu preconditioner for positive definite $$textbf{H}(varvec{curl},Omega )$$ problems 为正定 $$textbf{H}(varvec{curl},Omega )$$ 问题构建 Hiptmair-Xu 预处理子结构
IF 1.6 3区 数学
BIT Numerical Mathematics Pub Date : 2024-07-15 DOI: 10.1007/s10543-024-01031-y
R. Delville-Atchekzai, Xavier Claeys, M. Lecouvez
{"title":"Substructuring the Hiptmair-Xu preconditioner for positive definite $$textbf{H}(varvec{curl},Omega )$$ problems","authors":"R. Delville-Atchekzai, Xavier Claeys, M. Lecouvez","doi":"10.1007/s10543-024-01031-y","DOIUrl":"https://doi.org/10.1007/s10543-024-01031-y","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141647744","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 robust second-order low-rank BUG integrator based on the midpoint rule 基于中点规则的稳健二阶低阶 BUG 积分器
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-07-13 DOI: 10.1007/s10543-024-01032-x
Gianluca Ceruti, Lukas Einkemmer, Jonas Kusch, Christian Lubich
{"title":"A robust second-order low-rank BUG integrator based on the midpoint rule","authors":"Gianluca Ceruti, Lukas Einkemmer, Jonas Kusch, Christian Lubich","doi":"10.1007/s10543-024-01032-x","DOIUrl":"https://doi.org/10.1007/s10543-024-01032-x","url":null,"abstract":"<p>Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update &amp; Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint quadrature rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141610703","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
Weak convergence of tamed exponential integrators for stochastic differential equations 随机微分方程驯服指数积分器的弱收敛性
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-07-11 DOI: 10.1007/s10543-024-01029-6
Utku Erdoğan, Gabriel J. Lord
{"title":"Weak convergence of tamed exponential integrators for stochastic differential equations","authors":"Utku Erdoğan, Gabriel J. Lord","doi":"10.1007/s10543-024-01029-6","DOIUrl":"https://doi.org/10.1007/s10543-024-01029-6","url":null,"abstract":"<p>We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschitz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard exponential schemes. The numerical performance of both the GBM and exponential tamed methods through four different multi-level Monte Carlo techniques are compared. We observe that for linear noise the standard exponential tamed method requires severe restrictions on the step size unlike the GBM tamed method.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584730","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
Optimal convergence analysis of the virtual element methods for viscoelastic wave equations with variable coefficients on polygonal meshes 多边形网格上可变系数粘弹性波方程虚拟元素方法的最佳收敛分析
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-07-05 DOI: 10.1007/s10543-024-01030-z
Gouranga Pradhan, Bhupen Deka
{"title":"Optimal convergence analysis of the virtual element methods for viscoelastic wave equations with variable coefficients on polygonal meshes","authors":"Gouranga Pradhan, Bhupen Deka","doi":"10.1007/s10543-024-01030-z","DOIUrl":"https://doi.org/10.1007/s10543-024-01030-z","url":null,"abstract":"<p>The objective of this work is to develop a conforming virtual element method for viscoelastic wave equations with variable coefficients on polygonal meshes. For problems where the coefficients are variable, the standard virtual element discrete forms do not work efficiently and require modification. For the optimal convergence estimate of the semi-discrete approximation in the <span>(L^{2})</span> norm, a special projection operator is used. In the fully discrete scheme, the implicit second-order Newmark method is employed to approximate the temporal derivatives. Numerical experiments are presented to support the theoretical results. The proposed numerical algorithm can be applied to various problems arising in the engineering and medical fields.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141575926","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 hp-version of the discontinuous Galerkin method for fractional integro-differential equations with weakly singular kernels 弱奇异内核分式积分微分方程的非连续伽勒金方法 hp 版本
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-07-04 DOI: 10.1007/s10543-024-01026-9
Yanping Chen, Zhenrong Chen, Yunqing Huang
{"title":"An hp-version of the discontinuous Galerkin method for fractional integro-differential equations with weakly singular kernels","authors":"Yanping Chen, Zhenrong Chen, Yunqing Huang","doi":"10.1007/s10543-024-01026-9","DOIUrl":"https://doi.org/10.1007/s10543-024-01026-9","url":null,"abstract":"<p>This paper suggests an <i>hp</i>-discontinuous Galerkin approach for the fractional integro-differential equations with weakly singular kernels. The key idea behind our method is to first convert the fractional integro-differential equations into the second kind of Volterra integral equations, and then solve the equivalent integral equations using the <i>hp</i>-discontinuous Galerkin method. We establish prior error bounds in the <span>(L^{2})</span>-norm that is entirely explicit about the local mesh sizes, local polynomial degrees, and local regularities of the exact solutions. The use of geometrically refined meshes and linearly increasing approximation orders demonstrates, in particular, that exponential convergence is achievable for solutions with endpoint singularities. Numerical results indicate the usefulness of the proposed method.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141546974","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
Stochastic iterative methods for online rank aggregation from pairwise comparisons 通过成对比较进行在线排名汇总的随机迭代法
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-06-21 DOI: 10.1007/s10543-024-01024-x
Benjamin Jarman, Lara Kassab, Deanna Needell, Alexander Sietsema
{"title":"Stochastic iterative methods for online rank aggregation from pairwise comparisons","authors":"Benjamin Jarman, Lara Kassab, Deanna Needell, Alexander Sietsema","doi":"10.1007/s10543-024-01024-x","DOIUrl":"https://doi.org/10.1007/s10543-024-01024-x","url":null,"abstract":"<p>In this paper, we consider large-scale ranking problems where one is given a set of (possibly non-redundant) pairwise comparisons and the underlying ranking explained by those comparisons is desired. We show that stochastic gradient descent approaches can be leveraged to offer convergence to a solution that reveals the underlying ranking while requiring low-memory operations. We introduce several variations of this approach that offer a tradeoff in speed and convergence when the pairwise comparisons are noisy (i.e., some comparisons do not respect the underlying ranking). We prove theoretical results for convergence almost surely and study several regimes including those with full observations, partial observations, and noisy observations. Our empirical results give insights into the number of observations required as well as how much noise in those measurements can be tolerated.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141546975","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 class of monotonicity-preserving variable-step discretizations for Volterra integral equations Volterra 积分方程的一类单调性保留变步离散法
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-06-14 DOI: 10.1007/s10543-024-01027-8
Yuanyuan Feng, Lei Li
{"title":"A class of monotonicity-preserving variable-step discretizations for Volterra integral equations","authors":"Yuanyuan Feng, Lei Li","doi":"10.1007/s10543-024-01027-8","DOIUrl":"https://doi.org/10.1007/s10543-024-01027-8","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141341995","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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