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Convergence and superconvergence of a fractional collocation method for weakly singular Volterra integro-differential equations 弱奇异 Volterra 积分微分方程的分数配位法的收敛性和超收敛性
IF 1.5 3区 数学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-02-12 DOI: 10.1007/s10543-024-01011-2

Abstract

A collocation method for the numerical solution of Volterra integro-differential equations with weakly singular kernels, based on piecewise polynomials of fractional order, is constructed and analysed. Typical exact solutions of this class of problems have a weak singularity at the initial time (t=0) . A rigorous error analysis of our method shows that, with an appropriate choice of the fractional-order polynomials and a suitably graded mesh, one can attain optimal orders of convergence to the exact solution and its derivative, and certain superconvergence results are also derived. In particular, our analysis shows that on a uniform mesh our method attains a higher order of convergence than standard piecewise polynomial collocation. Numerical examples are presented to demonstrate the sharpness of our theoretical results.

摘要 构建并分析了一种基于分数阶片化多项式的、用于数值求解具有弱奇异内核的 Volterra 积分微分方程的配位法。这类问题的典型精确解在初始时间 (t=0) 具有弱奇异性。对我们的方法进行的严格误差分析表明,只要选择适当的分数阶多项式和适当的分级网格,就能获得对精确解及其导数的最佳收敛阶数,同时还得出了某些超收敛结果。特别是,我们的分析表明,在均匀网格上,我们的方法比标准的分次多项式拼合方法获得更高的收敛阶数。为了证明我们理论结果的精确性,我们给出了一些数值示例。
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引用次数: 0
A convolution quadrature using derivatives and its application 使用导数的卷积正交及其应用
IF 1.5 3区 数学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-02-09 DOI: 10.1007/s10543-024-01009-w
Hao Ren, Junjie Ma, Huilan Liu

This paper is devoted to explore the convolution quadrature based on a class of two-point Hermite collocation methods. Incorporating derivatives into the numerical scheme enhances the accuracy while preserving stability, which is confirmed by the convergence analysis for the discretization of the initial value problem. Moreover, we employ the resulting quadrature to evaluate a class of highly oscillatory integrals. The frequency-explicit convergence analysis demonstrates that the proposed convolution quadrature surpasses existing convolution quadratures, achieving the highest convergence rate with respect to the oscillation among them. Numerical experiments involving convolution integrals with smooth, weakly singular, and highly oscillatory Bessel kernels illustrate the reliability and efficiency of the proposed convolution quadrature.

本文致力于探讨基于两点赫米特配位法的卷积正交。在数值方案中加入导数可以在保持稳定性的同时提高精度,这一点在初值问题离散化的收敛分析中得到了证实。此外,我们还利用由此产生的正交来评估一类高度振荡的积分。频率显式收敛分析表明,所提出的卷积正交超越了现有的卷积正交,在振荡方面达到了最高的收敛率。涉及平滑、弱奇异和高振荡贝塞尔核的卷积积分的数值实验说明了所提出的卷积正交的可靠性和效率。
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引用次数: 0
A posteriori error estimates for a dual finite element method for singularly perturbed reaction–diffusion problems 奇异扰动反应扩散问题二元有限元法的后验误差估计
IF 1.5 3区 数学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-02-05 DOI: 10.1007/s10543-024-01008-x
JaEun Ku, Martin Stynes

A posteriori error estimates are established for a two-step dual finite element method for singularly perturbed reaction–diffusion problems. The method can be considered as a modified least-squares finite element method. The least-squares functional is the basis for our residual-type a posteriori error estimators, which are shown to be reliable and efficient with respect to the error in an energy-type norm. Moreover, guaranteed upper bounds for the errors in the computed primary and dual variables are derived; these bounds are then used to drive an adaptive algorithm for our finite element method, yielding any desired accuracy. Our theory does not require the meshes generated to be shape-regular. Numerical experiments show the effectiveness of our a posteriori estimators.

为奇异扰动反应扩散问题的两步对偶有限元法建立了后验误差估计。该方法可视为修正的最小二乘有限元法。最小二乘函数是我们的残差型后验误差估计器的基础,它在能量型规范误差方面被证明是可靠和高效的。此外,我们还推导出了计算主变量和对偶变量误差的保证上限;这些上限可用于驱动有限元方法的自适应算法,从而获得任何所需的精度。我们的理论不要求生成的网格是形状规则的。数值实验证明了我们的后验估计器的有效性。
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引用次数: 0
Improved uniform error bounds on parareal exponential algorithm for highly oscillatory systems 高振荡系统准指数算法的改进均匀误差边界
IF 1.5 3区 数学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-01-31 DOI: 10.1007/s10543-023-01005-6
Bin Wang, Yaolin Jiang

For the well known parareal algorithm, we formulate and analyse a novel class of parareal exponential schemes with improved uniform accuracy for highly oscillatory system (ddot{q}+frac{1}{varepsilon ^2}M q =frac{1}{varepsilon ^{mu }}f(q)) with (mu =0) or 1. The solution of this considered system propagates waves with wavelength at (mathcal {O} (varepsilon )) in time and the value of (mu ) corresponds to the strength of nonlinearity. This brings significantly numerical burden in scientific computation for highly oscillatory systems with (0<varepsilon ll 1). The new proposed algorithm is formulated by using some reformulation approaches to the problem, Fourier pseudo-spectral methods, and parareal exponential integrators. The fast Fourier transform is incorporated in the implementation. We rigorously study the convergence, showing that for nonlinear systems, the algorithm has improved uniform accuracy (mathcal {O}big ( varepsilon ^{(2k+3)(1-mu )}Delta t^{2k+2}+varepsilon ^{5(1-mu )}delta t^4big )) in the position and (mathcal {O}big ( varepsilon ^{(2k+3)(1-mu )-1}Delta t^{2k+2}+varepsilon ^{4-5mu }delta t^4big )) in the momenta, where k is the number of parareal iterations, and (Delta t) and (delta t) are two time stepsizes used in the algorithm. The energy conservation is also discussed and the algorithm is shown to have an improved energy conservation. Numerical experiments are provided and the numerical results demonstrate the improved uniform accuracy and improved energy conservation of the obtained integrator through four Hamiltonian differential equations including nonlinear wave equations.

对于众所周知的抛物线算法,我们提出并分析了一类新的抛物线指数方案,该方案对于高度振荡系统 (ddot{q}+frac{1}{varepsilon ^2}M q =frac{1}{varepsilon ^{mu }}f(q)) with (mu =0) or 1 具有更高的均匀精度。这个系统的解会传播波长为 (mathcal {O} (varepsilon )) 的波,而 (mu ) 的值与非线性的强度相对应。这为具有 (0<varepsilon ll 1) 的高度振荡系统的科学计算带来了很大的数值负担。新提出的算法是通过对问题的一些重拟方法、傅立叶伪谱方法和准指数积分器来制定的。快速傅里叶变换被纳入了算法的实现过程。我们对收敛性进行了严格研究,结果表明,对于非线性系统、varepsilon ^{(2k+3)(1-mu )}Delta t^{2k+2}+varepsilon ^{5(1-.varepsilon ^{(2k+3)(1-mu )-1}Delta t^{2k+2}+varepsilon ^{4-5mu }delta t^^4big )) 中的位置和( ( (varepsilon ^{(2k+3)(1-mu )-1}Delta t^{2k+2}+varepsilon ^{4-5mu }delta t^^4big ) )中的矩、其中 k 是迭代次数,(delta t) 和 (delta t) 是算法中使用的两个时间步长。还讨论了能量守恒问题,并证明该算法具有更好的能量守恒。提供了数值实验,数值结果表明通过四个哈密顿微分方程(包括非线性波方程)得到的积分器具有更好的均匀精度和能量守恒。
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引用次数: 0
On the stability radius for linear time-delay systems 论线性时延系统的稳定半径
IF 1.5 3区 数学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-01-30 DOI: 10.1007/s10543-023-01006-5

Abstract

The exponential function that appears in the formula of the stability radius of linear time-delay differential systems is approximated by its Padé approximant. This reduces the computation of the level sets of singular values in the stability radius formula to the computation of imaginary eigenvalues of special matrix polynomials. Then a bisection method is used for computing lower and upper bounds on the stability radius. A rounding error analysis is presented. Several numerical examples are given to demonstrate the feasibility and efficiency of the bisection method.

摘要 线性时延微分系统稳定性半径公式中出现的指数函数用其帕代近似值来近似。这将稳定性半径公式中奇异值水平集的计算简化为特殊矩阵多项式虚特征值的计算。然后采用分段法计算稳定性半径的下限和上限。此外,还给出了四舍五入误差分析。给出了几个数值示例,以证明分段法的可行性和效率。
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引用次数: 0
Incremental algorithms for truncated higher-order singular value decompositions 截断高阶奇异值分解的增量算法
IF 1.5 3区 数学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-01-08 DOI: 10.1007/s10543-023-01004-7
Chao Zeng, Michael K. Ng, Tai-Xiang Jiang

We develop and study incremental algorithms for truncated higher-order singular value decompositions. By combining the SVD updating and different truncated higher-order singular value decompositions, two incremental algorithms are proposed. Not only the factor matrices but also the core tensor are updated in an incremental style. The costs of these algorithms are compared and the approximation errors are analyzed. Numerical results demonstrate that the proposed incremental algorithms have advantages in online computation.

我们开发并研究了截断高阶奇异值分解的增量算法。通过结合 SVD 更新和不同的截断高阶奇异值分解,我们提出了两种增量算法。不仅因子矩阵,核心张量也以增量方式更新。对这些算法的成本进行了比较,并分析了近似误差。数值结果表明,所提出的增量算法在在线计算中具有优势。
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引用次数: 0
A regularization–correction approach for adapting subdivision schemes to the presence of discontinuities 使细分方案适应不连续性的正则化修正方法
IF 1.5 3区 数学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-01-04 DOI: 10.1007/s10543-023-01003-8
Sergio Amat, David Levin, Juan Ruiz-Álvarez, Dionisio F. Yáñez

Linear approximation methods suffer from Gibbs oscillations when approximating functions with jumps. Essentially non oscillatory subcell-resolution (ENO-SR) is a local technique avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper jump or corner (jump in the first order derivative) discontinuity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. This new procedure can be applied as subdivision scheme to design curves and surfaces both in point-value and in cell-average contexts. Using the proposed algorithm, we are able to construct approximations with high precision, with high piecewise regularity, and without smearing nor oscillations in the presence of discontinuities. These are desired properties in real applications as computer aided design or car design, among others.

线性近似方法在近似有跳跃的函数时会受到吉布斯振荡的影响。本质上非振荡子单元求解(ENO-SR)是一种避免振荡的局部技术,具有全阶精度,但会损失近似值的正则性。本文的目标是引入一种同时具有全精度和规则性的新方法。为了获得这种方法,我们提出了一种三阶段算法:首先,通过减去适当的非光滑数据序列对数据进行平滑处理;然后,将选定的高阶线性近似算子应用于平滑数据;最后,通过用第一阶段使用的非光滑元素对光滑近似进行修正,恢复具有适当跳跃或拐角(一阶导数中的跳跃)不连续结构的近似。这一新程序可作为细分方案,用于设计点值和单元平均的曲线和曲面。利用所提出的算法,我们能够构建出高精度、高片状规则性的近似值,并且在存在不连续的情况下不会出现涂抹或振荡。这些都是计算机辅助设计或汽车设计等实际应用中所需的特性。
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引用次数: 0
Block diagonal Calderón preconditioning for scattering at multi-screens. 针对多屏幕散射的块对角卡尔德隆预处理。
IF 1.6 3区 数学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-01-01 Epub Date: 2024-09-03 DOI: 10.1007/s10543-024-01034-9
Kristof Cools, Carolina Urzúa-Torres

A preconditioner is proposed for Laplace exterior boundary value problems on multi-screens. To achieve this, the quotient-space boundary element method and operator preconditioning are combined. For a fairly general subclass of multi-screens, it is shown that this approach paves the way for block diagonal Calderón preconditioners which achieve a spectral condition number that grows only logarithmically with decreasing mesh size, just as in the case of simple screens. Since the resulting scheme contains many more degrees of freedom than strictly required, strategies are presented to remove almost all redundancy without significant loss of effectiveness of the preconditioner. The performance of this method is verified by providing representative numerical results. Further numerical experiments suggest that these results can be extended to a much wider class of multi-screens that cover essentially all geometries encountered in practice, leading to a significantly reduced simulation cost.

针对多屏幕上的拉普拉斯外部边界值问题提出了一种预处理方法。为此,结合了商空间边界元方法和算子预处理方法。对于相当普遍的多屏幕子类,研究表明这种方法为块对角线卡尔德龙预处理铺平了道路,这种预处理可以实现谱条件数只随网格尺寸的减小而对数增长,就像在简单屏幕的情况下一样。由于由此产生的方案所包含的自由度比严格要求的要多得多,因此提出了一些策略来消除几乎所有的冗余,而不会明显降低预处理的有效性。通过提供具有代表性的数值结果,验证了该方法的性能。进一步的数值实验表明,这些结果可以扩展到更广泛的多屏幕类别,基本上涵盖了实践中遇到的所有几何形状,从而大大降低了模拟成本。
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引用次数: 0
From low-rank retractions to dynamical low-rank approximation and back. 从低阶回缩到动态低阶近似,再回到低阶近似。
IF 1.6 3区 数学 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-01-01 Epub Date: 2024-06-17 DOI: 10.1007/s10543-024-01028-7
Axel Séguin, Gianluca Ceruti, Daniel Kressner

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston-Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.

在求解受光滑流形约束的优化问题的算法中,回缩是一种行之有效的工具,可确保迭代保持在流形上。最近的研究表明,对于流形上的其他计算任务,包括插值任务,回撤也是一个有用的概念。在这项工作中,我们考虑将缩回应用于固定阶矩阵流形上微分方程的数值积分。这与动态低阶近似(DLRA)技术密切相关。事实上,任何回缩都会导致数值积分,反之亦然,某些 DLRA 技术与回缩有直接关系。作为后者的一个例子,我们介绍了一种新的回缩方法,称为 KLS 回缩,它是从所谓的 DLRA 非常规积分器中衍生出来的。我们还说明了如何利用回缩来恢复已知的 DLRA 技术和设计新技术。本研究特别介绍了两种适用于一般流形上微分方程的新型数值积分方案:加速前向欧拉(AFE)方法和投影拉尔斯顿-赫米特(PRH)方法。这两种方法都建立在缩回的基础上,将其作为逼近流形上曲线的工具。这两种方法被证明具有三阶的局部截断误差。经典 DLRA 例子的数值实验凸显了这些新方法的优势和不足。
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引用次数: 0
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 Q3 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-01-01 Epub Date: 2024-07-15 DOI: 10.1007/s10543-024-01033-w
Daniel Kressner, Bor Plestenjak

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 δ -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.

由于奇异矩阵铅笔的特征值不连续,因此其广义特征值问题的数值求解具有挑战性。通常,解决这类问题的方法是先通过阶梯形式提取正则部分,然后对正则部分应用标准求解器,如 QZ 算法。最近,人们提出了几种新方法,通过相对简单的随机修改将奇异铅笔转化为正则铅笔。在这项研究中,我们分析了 Hochstenbach、Mehl 和 Plestenjak 使用随机矩阵修改、投影或增强铅笔的三种方法。这三种方法都依赖于正常秩,不会改变原始铅笔的有限特征值。我们的研究表明,变换后的铅笔的特征值条件数不可能比 Lotz 和 Noferini 引入的原始铅笔的 δ 弱特征值条件数大很多。这不仅表明了良好的数值稳定性,而且再次证实了这些条件数是检测简单有限特征值的可靠标准。我们还提供证据表明,从数值稳定性的角度来看,即使对于实奇异矩阵铅笔和实特征值,使用复随机矩阵而非实随机矩阵也是可取的。作为一个附带结果,我们为两个独立随机变量的乘积提供了尖锐的左尾边界,这两个随机变量的分布是广义贝塔第一种分布或库马拉斯瓦米分布。
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引用次数: 0
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BIT Numerical Mathematics
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