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Dynamical low-rank approximation of the Vlasov–Poisson equation with piecewise linear spatial boundary 具有片状线性空间边界的弗拉索夫-泊松方程的动态低阶近似
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-04-07 DOI: 10.1007/s10543-024-01019-8
{"title":"Dynamical low-rank approximation of the Vlasov–Poisson equation with piecewise linear spatial boundary","authors":"","doi":"10.1007/s10543-024-01019-8","DOIUrl":"https://doi.org/10.1007/s10543-024-01019-8","url":null,"abstract":"<h3>Abstract</h3> <p>Dynamical low-rank approximation (DLRA) for the numerical simulation of Vlasov–Poisson equations is based on separation of space and velocity variables, as proposed in several recent works. The standard approach for the time integration in the DLRA model uses a splitting of the tangent space projector for the low-rank manifold according to the separated variables. It can also be modified to allow for rank-adaptivity. A less studied aspect is the incorporation of boundary conditions in the DLRA model. In this work, a variational formulation of the projector splitting is proposed which allows to handle inflow boundary conditions on spatial domains with piecewise linear boundary. Numerical experiments demonstrate the principle feasibility of this approach.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"194 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564916","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-explicit integration of second order for weakly coupled poroelasticity 弱耦合孔弹性的二阶半显式积分
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-04-07 DOI: 10.1007/s10543-024-01021-0
{"title":"Semi-explicit integration of second order for weakly coupled poroelasticity","authors":"","doi":"10.1007/s10543-024-01021-0","DOIUrl":"https://doi.org/10.1007/s10543-024-01021-0","url":null,"abstract":"<h3>Abstract</h3> <p>We introduce a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition. Here, semi-explicit means that the system, which needs to be solved in each step, decouples and hence improves the computational efficiency. The construction and the convergence proof are based on the connection to a differential equation with two time delays, namely one and two times the step size. Numerical experiments confirm the theoretical results and indicate the applicability to higher-order schemes.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"56 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564840","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
Accurate Horner methods in real and complex floating-point arithmetic 实数和复数浮点运算中的精确霍纳方法
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-03-27 DOI: 10.1007/s10543-024-01017-w
Thomas R. Cameron, Stef Graillat
{"title":"Accurate Horner methods in real and complex floating-point arithmetic","authors":"Thomas R. Cameron, Stef Graillat","doi":"10.1007/s10543-024-01017-w","DOIUrl":"https://doi.org/10.1007/s10543-024-01017-w","url":null,"abstract":"<p>In this article, we derive accurate Horner methods in real and complex floating-point arithmetic. In particular, we show that these methods are as accurate as if computed in <i>k</i>-fold precision and then rounded into the working precision. When <i>k</i> is two, our methods are comparable or faster than the existing compensated Horner routines. When compared to multi-precision software, such as MPFR and MPC, our methods are significantly faster, up to <i>k</i> equal to eight, that is, up to 489 bits in the significand.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"47 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140312952","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
Highly localized RBF Lagrange functions for finite difference methods on spheres 球面有限差分法的高度局部化 RBF 拉格朗日函数
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-03-15 DOI: 10.1007/s10543-024-01016-x
W. Erb, T. Hangelbroek, F. J. Narcowich, C. Rieger, J. D. Ward
{"title":"Highly localized RBF Lagrange functions for finite difference methods on spheres","authors":"W. Erb, T. Hangelbroek, F. J. Narcowich, C. Rieger, J. D. Ward","doi":"10.1007/s10543-024-01016-x","DOIUrl":"https://doi.org/10.1007/s10543-024-01016-x","url":null,"abstract":"<p>The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the sphere can be used to create a numerically feasible, stable finite difference method based on radial basis functions (an RBF-FD-like method). For certain classes of PDEs this approach leads to rigorous convergence estimates for stencils which grow moderately with increasing discretization fineness.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"62 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156271","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
Bounded perturbations resilient iterative methods for linear systems and least squares problems: operator-based approaches, analysis, and performance evaluation 线性系统和最小二乘问题的有界扰动弹性迭代法：基于算子的方法、分析和性能评估
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-03-05 DOI: 10.1007/s10543-024-01015-y
Mokhtar Abbasi, Touraj Nikazad
{"title":"Bounded perturbations resilient iterative methods for linear systems and least squares problems: operator-based approaches, analysis, and performance evaluation","authors":"Mokhtar Abbasi, Touraj Nikazad","doi":"10.1007/s10543-024-01015-y","DOIUrl":"https://doi.org/10.1007/s10543-024-01015-y","url":null,"abstract":"<p>We examine some bounded perturbations resilient iterative methods for addressing (constrained) consistent linear systems of equations and (constrained) least squares problems. We introduce multiple frameworks rooted in the operator of the Landweber iteration, adapting the operators to facilitate the minimization of absolute errors or residuals. We demonstrate that our operator-based methods exhibit comparable speed to powerful methods like CGLS, and we establish that the computational cost of our methods is nearly equal to that of CGLS. Furthermore, our methods possess the capability to handle constraints (e.g. non-negativity) and control the semi-convergence phenomenon. In addition, we provide convergence analysis of the methods when the current iterations are perturbed by summable vectors. This allows us to utilize these iterative methods for the superiorization methodology. We showcase their performance using examples drawn from tomographic imaging and compare them with CGLS, superiorized conjugate gradient (S-CG), and the non-negative flexible CGLS (NN-FCGLS) methods.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"14 2 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034982","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
The variable two-step BDF method for parabolic equations 抛物方程的可变两步 BDF 法
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-03-01 DOI: 10.1007/s10543-024-01007-y
Georgios Akrivis, Minghua Chen, Jianxing Han, Fan Yu, Zhimin Zhang
{"title":"The variable two-step BDF method for parabolic equations","authors":"Georgios Akrivis, Minghua Chen, Jianxing Han, Fan Yu, Zhimin Zhang","doi":"10.1007/s10543-024-01007-y","DOIUrl":"https://doi.org/10.1007/s10543-024-01007-y","url":null,"abstract":"<p>The two-step backward difference formula (BDF) method on variable grids for parabolic equations with self-adjoint elliptic part is considered. Standard stability estimates for adjacent time-step ratios <span>(r_j:=k_j/k_{j-1}leqslant 1.8685)</span> and 1.9104, respectively, have been proved by Becker (BIT 38:644–662, 1998) and Emmrich (J Appl Math Comput 19:33–55, 2005) by the energy technique with a single multiplier. Even slightly improving the ratio is cumbersome. In this paper, we present a novel technique to examine the positive definiteness of banded matrices that are neither Toeplitz nor weakly diagonally dominant; this result can be viewed as a variant of the Grenander–Szegő theorem. Then, utilizing the energy technique with two multipliers, we establish stability for adjacent time-step ratios up to 1.9398.\u0000</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"12 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140017990","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 unified immersed finite element error analysis for one-dimensional interface problems 一维界面问题的统一沉浸式有限元误差分析
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-29 DOI: 10.1007/s10543-024-01014-z
Slimane Adjerid, Tao Lin, Haroun Meghaichi
{"title":"A unified immersed finite element error analysis for one-dimensional interface problems","authors":"Slimane Adjerid, Tao Lin, Haroun Meghaichi","doi":"10.1007/s10543-024-01014-z","DOIUrl":"https://doi.org/10.1007/s10543-024-01014-z","url":null,"abstract":"<p>It has been known that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"28 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001645","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
Admissible subspaces and the subspace iteration method 可容许子空间和子空间迭代法
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-28 DOI: 10.1007/s10543-024-01012-1
{"title":"Admissible subspaces and the subspace iteration method","authors":"","doi":"10.1007/s10543-024-01012-1","DOIUrl":"https://doi.org/10.1007/s10543-024-01012-1","url":null,"abstract":"<h3>Abstract</h3> <p>In this work we revisit the convergence analysis of the Subspace Iteration Method (SIM) for the computation of approximations of a matrix <em>A</em> by matrices of rank <em>h</em>. Typically, the analysis of convergence of these low-rank approximations has been obtained by first estimating the (angular) distance between the subspaces produced by the SIM and the dominant subspaces of <em>A</em>. It has been noticed that this approach leads to upper bounds that overestimate the approximation error in case the <em>h</em>th singular value of <em>A</em> lies in a cluster of singular values. To overcome this difficulty we introduce a substitute for dominant subspaces, which we call admissible subspaces. We develop a proximity analysis of subspaces produced by the SIM to admissible subspaces; in turn, this analysis allows us to obtain novel estimates for the approximation error by low-rank matrices obtained by the implementation of the deterministic SIM. Our results apply in the case when the <em>h</em>th singular value of <em>A</em> belongs to a cluster of singular values. Indeed, our approach allows us to consider the case when the <em>h</em>th and the <span> <span>((h+1))</span> </span>st singular values of <em>A</em> coincide, which does not seem to be covered by previous works in the deterministic setting.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"44 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001840","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
Singularity swap quadrature for nearly singular line integrals on closed curves in two dimensions 二维封闭曲线上近奇异线积分的奇异交换正交
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-27 DOI: 10.1007/s10543-024-01013-0
Ludvig af Klinteberg
{"title":"Singularity swap quadrature for nearly singular line integrals on closed curves in two dimensions","authors":"Ludvig af Klinteberg","doi":"10.1007/s10543-024-01013-0","DOIUrl":"https://doi.org/10.1007/s10543-024-01013-0","url":null,"abstract":"<p>This paper presents a quadrature method for evaluating layer potentials in two dimensions close to periodic boundaries, discretized using the trapezoidal rule. It is an extension of the method of singularity swap quadrature, which recently was introduced for boundaries discretized using composite Gauss–Legendre quadrature. The original method builds on swapping the target singularity for its preimage in the complexified space of the curve parametrization, where the source panel is flat. This allows the integral to be efficiently evaluated using an interpolatory quadrature with a monomial basis. In this extension, we use the target preimage to swap the singularity to a point close to the unit circle. This allows us to evaluate the integral using an interpolatory quadrature with complex exponential basis functions. This is well-conditioned, and can be efficiently evaluated using the fast Fourier transform. The resulting method has exponential convergence, and can be used to accurately evaluate layer potentials close to the source geometry. We report experimental results on a simple test geometry, and provide a baseline Julia implementation that can be used for further experimentation.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"17 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979315","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 eigenvalue backward errors for rational matrix functions with symmetry structures 具有对称结构的有理矩阵函数的结构化特征值后向误差
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-17 DOI: 10.1007/s10543-024-01010-3
Anshul Prajapati, Punit Sharma
{"title":"Structured eigenvalue backward errors for rational matrix functions with symmetry structures","authors":"Anshul Prajapati, Punit Sharma","doi":"10.1007/s10543-024-01010-3","DOIUrl":"https://doi.org/10.1007/s10543-024-01010-3","url":null,"abstract":"<p>We derive computable formulas for the structured backward errors of a complex number <span>(lambda )</span> when considered as an approximate eigenvalue of rational matrix functions that carry a symmetry structure. We consider symmetric, skew-symmetric, Hermitian, skew-Hermitian, <span>(*)</span>-palindromic, T-even, T-odd, <span>(*)</span>-even, and <span>(*)</span>-odd structures. Numerical experiments show that the backward errors with respect to structure-preserving and arbitrary perturbations are significantly different.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"200 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139764879","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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