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Error bound analysis of the stochastic parareal algorithm 随机拟面算法的误差界分析
Pub Date : 2022-11-10 DOI: 10.48550/arXiv.2211.05496
K. Pentland, M. Tamborrino, Timothy John Sullivan
Stochastic parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as parareal. Similarly to parareal, it combines fine- and coarse-grained solutions to an ordinary differential equation (ODE) using a predictor-corrector (PC) scheme. The key difference is that carefully chosen random perturbations are added to the PC to try to accelerate the location of a stochastic solution to the ODE. In this paper, we derive superlinear and linear mean-square error bounds for SParareal applied to nonlinear systems of ODEs using different types of perturbations. We illustrate these bounds numerically on a linear system of ODEs and a scalar nonlinear ODE, showing a good match between theory and numerics.
随机并行并行算法(parareal)是流行的并行实时算法(parareal)的一种概率变体。与平行相似,它使用预测校正器(PC)方案组合了常微分方程(ODE)的细粒度和粗粒度解。关键的区别在于,仔细选择的随机扰动被添加到PC中,以试图加速ODE随机解的位置。在本文中,我们导出了应用于不同类型扰动的非线性微分方程系统的超线性和线性均方误差界。我们在线性ODE系统和标量非线性ODE系统上对这些边界进行了数值说明,证明了理论与数值之间的良好匹配。
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引用次数: 0
A Div FOSLS Method Suitable for Quadrilateral RT and Hexahedral RTNH(rmdiv)-elements 一种适用于四边形RTNH和六面体RTNH(rmdiv)元素的Div FOSLS方法
Pub Date : 2022-11-10 DOI: 10.1007/s10915-022-02043-y
Huoyuan Duan, Can Wang, Zhijie Du
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引用次数: 0
Construction and Evaluation of Pythagorean Hodograph Curves in Exponential-Polynomial Spaces 指数-多项式空间中毕达哥拉斯曲线的构造与评价
Pub Date : 2022-11-09 DOI: 10.1137/21m1455711
Lucia Romani, Alberto Viscardi
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引用次数: 2
Linearly Implicit Multistep Methods for Time Integration 时间积分的线性隐式多步方法
Pub Date : 2022-11-03 DOI: 10.1137/20m133748x
S. R. Glandon, M. Narayanamurthi, Adrian Sandu
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引用次数: 0
Diameter, Eccentricities and Distance Oracle Computations on H-Minor Free Graphs and Graphs of Bounded (Distance) Vapnik-Chervonenkis Dimension H-Minor自由图和有界(距离)Vapnik-Chervonenkis维图的直径、偏心率和距离Oracle计算
Pub Date : 2022-10-28 DOI: 10.1137/20m136551x
G. Ducoffe, M. Habib, L. Viennot
9 Under the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs 10 cannot be computed in truly subquadratic time (in the size n + m of the input), as shown 11 by Roditty and Williams. Nevertheless there are several graph classes for which this can be 12 done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We 13 propose to study unweighted graphs of constant distance VC-dimension as a broad generalization 14 of many such classes – where the distance VC-dimension of a graph G is defined as the VC-15 dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers 16 in G . In particular for any fixed H , the class of H -minor free graphs has distance VC-dimension 17 at most | V ( H ) | − 1. 18 • Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension 19 at most d , for any fixed k , either computes the diameter or concludes that it is larger than 20 k in time ˜ O ( k · mn 1 − ε d ), where ε d ∈ (0; 1) only depends on d 1 . We thus obtain a truly 21 subquadratic-time parameterized algorithm for computing the diameter on such graphs. 22 • Then as a byproduct of our approach, we get a truly subquadratic-time randomized algo-23 rithm for constant diameter computation on all the nowhere dense graph classes. The latter 24 classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of 25 bounded expansion. Before our work, the only known such algorithm was resulting from 26 an application of Courcelle’s theorem, see Grohe et al. [47]. 27
9在强指数时间假设下,一般未加权图10的直径不能在真正的次二次时间内计算(输入的大小为n + m),如Roditty和Williams所示11。然而,有几个图类可以这样做,如有界树宽图,区间图和平面图,仅举几例。我们13建议研究恒定距离vc维的无权图,作为许多此类图的广义推广14 -其中图G的距离vc维定义为其球超图的VC-15维:其超边是G中所有可能半径和中心16的球。特别是对于任意固定的H, H次自由图类的距离vc -维数最多为17 | V (H) |−1。18•我们的第一个主要结果是一个蒙特卡罗算法,该算法在距离vc维最多为19的图上,对于任何固定k,要么计算直径,要么得出它在时间上大于20k (k·mn 1 - ε d),其中ε d∈(0;1)只依赖于d1。因此,我们得到了一个真正的21次二次时间参数化算法来计算这种图上的直径。22•然后,作为我们方法的副产品,我们得到了一个真正的次二次时间随机算法-23算法,用于所有无处密集图类的常直径计算。后24类包括所有适当的小闭图类、有界度图和25有界展开图。在我们的工作之前,唯一已知的这种算法是由Courcelle定理的应用产生的,参见Grohe等人[47]。27
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引用次数: 1
On multiscale quasi-interpolation of scattered scalar- and manifold-valued functions 离散标量和流形函数的多尺度拟插值
Pub Date : 2022-10-25 DOI: 10.48550/arXiv.2210.14333
N. Sharon, Rafael Sherbu Cohen, H. Wendland
We address the problem of approximating an unknown function from its discrete samples given at arbitrarily scattered sites. This problem is essential in numerical sciences, where modern applications also highlight the need for a solution to the case of functions with manifold values. In this paper, we introduce and analyze a combination of kernel-based quasi-interpolation and multiscale approximations for both scalar- and manifold-valued functions. While quasi-interpolation provides a powerful tool for approximation problems if the data is defined on infinite grids, the situation is more complicated when it comes to scattered data. Here, higher-order quasi-interpolation schemes either require derivative information or become numerically unstable. Hence, this paper principally studies the improvement achieved by combining quasi-interpolation with a multiscale technique. The main contributions of this paper are as follows. First, we introduce the multiscale quasi-interpolation technique for scalar-valued functions. Second, we show how this technique can be carried over using moving least-squares operators to the manifold-valued setting. Third, we give a mathematical proof that converging quasi-interpolation will also lead to converging multiscale quasi-interpolation. Fourth, we provide ample numerical evidence that multiscale quasi-interpolation has superior convergence to quasi-interpolation. In addition, we will provide examples showing that the multiscale quasi-interpolation approach offers a powerful tool for many data analysis tasks, such as denoising and anomaly detection. It is especially attractive for cases of massive data points and high dimensionality.
我们解决了从任意分散位置给出的离散样本近似未知函数的问题。这个问题在数值科学中是必不可少的,在现代应用中也强调需要解决具有流形值的函数的情况。本文介绍并分析了标量函数和流形函数的基于核的拟插值和多尺度逼近的组合。当数据定义在无限网格上时,准插值为逼近问题提供了一个强大的工具,但当涉及到分散数据时,情况就更加复杂了。在这里,高阶准插值方案要么需要导数信息,要么在数值上变得不稳定。因此,本文主要研究拟插值与多尺度技术相结合的改进方法。本文的主要贡献如下:首先,我们介绍了标量函数的多尺度拟插值技术。其次,我们将展示如何使用将最小二乘算子移动到流形值设置来延续此技术。第三,给出了收敛拟插值也会导致收敛多尺度拟插值的数学证明。第四,我们提供了大量的数值证据,证明了多尺度拟插值比拟插值具有更好的收敛性。此外,我们还将提供一些例子,表明多尺度准插值方法为许多数据分析任务提供了强大的工具,例如去噪和异常检测。它对大量数据点和高维的情况特别有吸引力。
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引用次数: 0
Hamiltonian-Preserving Discontinuous Galerkin Methods for the Liouville Equation With Discontinuous Potential 具有不连续势的Liouville方程的保持哈密顿间断Galerkin方法
Pub Date : 2022-10-01 DOI: 10.1137/22m147952x
Boyang Ye, Shi Jin, Y. Xing, Xinghui Zhong
. Numerically solving the Liouville equation in classical mechanics with a discontinuous potential often leads to the 4 challenges of how to preserve the Hamiltonian across the potential barrier and a severe time step constraint according to the CFL 5 condition. Motivated by the Hamiltonian-preserving finite volume schemes by Jin and Wen [19], we introduce a Hamiltonian- 6 preserving discontinuous Galerkin (DG) scheme for the Liouville equation with discontinuous potential in this paper. The DG 7 method can be designed with arbitrary order of accuracy, and offers many advantages including easy adaptivity, compact stencils 8 and the ability of handling complicated boundary condition and interfaces. We propose to carefully design the numerical fluxes 9 of the DG methods to build the behavior of a classical particle at the potential barrier into the numerical scheme, which ensures 10 the continuity of the Hamiltonian across the potential barrier and the correct transmission and reflection condition. Our scheme 11 is proved to be positive and stable in L 1 norm if the positivity-preserving limiter is applied. Numerical examples are provided to 12 illustrate the accuracy and effectiveness of the proposed numerical scheme. 13 results show 2D2V test discontinuity of HPDG
. 对经典力学中具有不连续势的Liouville方程进行数值求解,往往会遇到如何保持势垒上的哈密顿量和cfl5条件下严格的时间步长约束的挑战。在Jin和Wen[19]的保持哈密顿有限体积格式的启发下,本文引入了具有不连续势的Liouville方程的保持哈密顿- 6间断Galerkin (DG)格式。DG - 7方法可以设计成任意精度级,具有易于自适应、模板紧凑和处理复杂边界条件和界面的能力等优点。我们建议仔细设计DG方法的数值通量,以将经典粒子在势垒处的行为建立到数值格式中,从而确保哈密顿量在势垒处的连续性和正确的透射和反射条件。在使用保正极限器的情况下,证明了方案11在L 1范数上是正稳定的。数值算例说明了所提数值格式的准确性和有效性。13 .结果显示HPDG的2D2V试验不连续
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引用次数: 0
Invariant-Domain-Preserving High-Order Time Stepping: I. Explicit Runge-Kutta Schemes 保持不变域的高阶时间步进:1 .显式Runge-Kutta格式
Pub Date : 2022-10-01 DOI: 10.1137/21m145793x
A. Ern, J. Guermond
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引用次数: 6
Frequency Extraction for BEM Matrices Arising From the 3D Scalar Helmholtz Equation 基于三维标量亥姆霍兹方程的边界元矩阵频率提取
Pub Date : 2022-10-01 DOI: 10.1137/20m1382957
Simon Dirckx, D. Huybrechs, K. Meerbergen
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引用次数: 0
Space-Split Algorithm for Sensitivity Analysis of Discrete Chaotic Systems With Multidimensional Unstable Manifolds 具有多维不稳定流形的离散混沌系统灵敏度分析的空间分割算法
Pub Date : 2022-10-01 DOI: 10.1137/21m1452135
Adam A. Śliwiak, Qiqi Wang
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引用次数: 1
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SIAM J. Sci. Comput.
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