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Numerical Approximation of Gaussian Random Fields on Closed Surfaces 封闭曲面上高斯随机场的数值逼近
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2024-01-04 DOI: 10.1515/cmam-2022-0237
Andrea Bonito, Diane Guignard, Wenyu Lei
We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances in several numerical experiments.
我们考虑了封闭表面上高斯随机场的数值近似,其定义为带有加性白噪声的分数随机偏微分方程(SPDE)的解。SPDE 涉及两个参数,分别控制高斯随机场的平滑度和相关长度。所提出的数值方法依赖于解的 Balakrishnan 积分表示法,不需要对特征对进行近似。相反,它由 sinc 正交法和标准曲面有限元法组成。我们对该方法进行了完整的误差分析,并在几个数值实验中对其性能进行了说明。
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引用次数: 0
An Optimal Method for High-Order Mixed Derivatives of Bivariate Functions 双变量函数高阶混合导数的最优方法
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2024-01-01 DOI: 10.1515/cmam-2023-0137
Evgeniya V. Semenova, Sergiy G. Solodky
The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. Based on the truncation method, an algorithm for numerical differentiation is constructed, which is order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. Numerical examples are provided to illustrate the fact that our approach can be implemented successfully.
研究了有限平稳性双变量函数高阶混合导数的优化恢复问题。基于截断法,构建了一种数值微分算法,该算法在精度和所涉及的 Galerkin 信息量方面都是阶次最优的。我们提供了数值示例来说明我们的方法可以成功实施。
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引用次数: 0
Wave Propagation in High-Contrast Media: Periodic and Beyond 高对比度介质中的波传播:周期及其他
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2024-01-01 DOI: 10.1515/cmam-2023-0066
Élise Fressart, Barbara Verfürth
This work is concerned with the classical wave equation with a high-contrast coefficient in the spatial derivative operator. We first treat the periodic case, where we derive a new limit in the one-dimensional case. The behavior is illustrated numerically and contrasted to the higher-dimensional case. For general unstructured high-contrast coefficients, we present the Localized Orthogonal Decomposition and show a priori error estimates in suitably weighted norms. Numerical experiments illustrate the convergence rates in various settings.
这项研究涉及空间导数算子中具有高对比度系数的经典波方程。我们首先处理周期性情况,在一维情况下推导出一个新的极限。我们用数值说明了这一行为,并与高维情况进行了对比。对于一般的非结构化高对比度系数,我们提出了局部正交分解,并以适当的加权规范显示了先验误差估计。数值实验说明了各种情况下的收敛速度。
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引用次数: 0
Adaptive Image Compression via Optimal Mesh Refinement 通过优化网格细化实现自适应图像压缩
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2024-01-01 DOI: 10.1515/cmam-2023-0097
Michael Feischl, Hubert Hackl
The JPEG algorithm is a defacto standard for image compression. We investigate whether adaptive mesh refinement can be used to optimize the compression ratio and propose a new adaptive image compression algorithm. We prove that it produces a quasi-optimal subdivision grid for a given error norm with high probability. This subdivision can be stored with very little overhead and thus leads to an efficient compression algorithm. We demonstrate experimentally, that the new algorithm can achieve better compression ratios than standard JPEG compression with no visible loss of quality on many images. The mathematical core of this work shows that Binev’s optimal tree approximation algorithm is applicable to image compression with high probability, when we assume small additive Gaussian noise on the pixels of the image.
JPEG 算法是图像压缩的事实标准。我们研究了自适应网格细化是否可用于优化压缩比,并提出了一种新的自适应图像压缩算法。我们证明,对于给定的误差规范,它能高概率地生成准最优细分网格。这种细分网格的存储开销极小,因此是一种高效的压缩算法。我们通过实验证明,新算法可以实现比标准 JPEG 压缩更好的压缩率,而且在许多图像上没有明显的质量损失。这项工作的数学核心表明,当我们假定图像像素上的加性高斯噪声很小时,Binev 的最优树近似算法适用于高概率图像压缩。
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引用次数: 0
A Posteriori Error Estimation for the Optimal Control of Time-Periodic Eddy Current Problems 时间周期涡流问题最优控制的后验误差估计
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-11-27 DOI: 10.1515/cmam-2023-0119
Monika Wolfmayr
This work presents the multiharmonic analysis and derivation of functional type a posteriori estimates of a distributed eddy current optimal control problem and its state equation in a time-periodic setting. The existence and uniqueness of the solution of a weak space-time variational formulation for the optimality system and the forward problem are proved by deriving inf-sup and sup-sup conditions. Using the inf-sup and sup-sup conditions, we derive guaranteed, sharp and fully computable bounds of the approximation error for the optimal control problem and the forward problem in the functional type a posteriori estimation framework. We present here the first computational results on the derived estimates.
本文提出了分布式涡流最优控制问题的泛函型后验估计的多谐分析和推导及其在时间周期设置下的状态方程。通过推导出最优系统和正问题的一个弱空时变分公式解的存在唯一性,证明了其存在唯一性。利用if -sup和sup-sup条件,在函数型后验估计框架下,导出了最优控制问题和正演问题的近似误差的保证、尖锐和完全可计算的界。在这里,我们给出了推导估计的第一个计算结果。
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引用次数: 1
Discontinuous Galerkin Two-Grid Method for the Transient Navier–Stokes Equations 瞬态Navier-Stokes方程的不连续Galerkin双网格法
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-11-24 DOI: 10.1515/cmam-2023-0035
Kallol Ray, Deepjyoti Goswami, Saumya Bajpai
In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 𝐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce an approximate solution with desired accuracy. We establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and L 2 L^{2} -norms, respectively, for an appropriate choice of coarse and fine mesh parameters. We further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates. Finally, numerical results are presented to confirm the efficiency of the proposed scheme.
本文将两网格格式应用于二维瞬态Navier-Stokes模型的DG表达式。两网格算法包括以下步骤:步骤1涉及在网格尺寸为𝐻的粗网格上求解非线性系统,步骤2涉及通过在网格尺寸为的细网格上使用粗网格解对非线性系统进行线性化,并求解结果系统以产生所需精度的近似解。我们分别在能量范数和l2 L^{2}范数中建立了速度和压力的两网格DG近似的最佳误差估计,以适当选择粗网格和细网格参数。利用后向欧拉方法对两网格DG模型进行离散化,得到了完全离散的误差估计。最后给出了数值结果,验证了所提方案的有效性。
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引用次数: 0
A 𝐶1-𝑃7 Bell Finite Element on Triangle 一个𝐶1-𝑃7三角形上的贝尔有限元
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-11-22 DOI: 10.1515/cmam-2023-0068
Xuejun Xu, Shangyou Zhang
We construct a C 1 C^{1} - P 7 P_{7} Bell finite element by restricting its normal derivative from a P 6 P_{6} polynomial to a P 5 P_{5} polynomial, and its second normal derivative from a P 5 P_{5} polynomial to a P 4 P_{4} polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the P 6 P_{6} polynomial space. We show the method converges at order 7 in L 2
我们在每个三角形的三条边上,通过限制c1c ^{1} - p7p_ {7} Bell有限元的法向导数从p6p_{6}多项式到p5p_{5}多项式,以及它的二阶法向导数从p5p_{5}多项式到p4p_{4}多项式,构造了c1c ^{1} - p7p_ {7} Bell有限元。在一个三角形上,有限元空间包含p6p_{6}多项式空间。我们证明了该方法在l2 ^{2}范数下收敛于7阶。通过消除c1c ^{1} - p7p_ {7} Argyris有限元边缘上的所有自由度,新单元的整体自由度从27 ^ V 27V渐近地大幅降低到12 ^ V 12V,其中三角形网格中的顶点数。当c1 C^{1} - p6p_ {6} Argyris有限元的整体自由度为19 ^ V 19V时,新单元同样精确,但更经济。数值试验表明,新单元比现有单元精度更高,且具有更少的全局未知量。
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引用次数: 0
Optimal Pressure Recovery Using an Ultra-Weak Finite Element Method for the Pressure Poisson Equation and a Least-Squares Approach for the Gradient Equation 压力泊松方程的超弱有限元法和梯度方程的最小二乘法的最优压力恢复
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-11-20 DOI: 10.1515/cmam-2021-0242
Douglas R. Q. Pacheco, Olaf Steinbach
Reconstructing the pressure from given flow velocities is a task arising in various applications, and the standard approach uses the Navier–Stokes equations to derive a Poisson problem for the pressure p. That method, however, artificially increases the regularity requirements on both solution and data. In this context, we propose and analyze two alternative techniques to determine p L 2 ( Ω ) {pin L^{2}(Omega)} . The first is an ultra-weak variational formulation applying integration by parts to shift all derivatives to the test functions. We present conforming finite element discretizations and prove optimal convergence of the resulting Galerkin–Petrov method. The second approach is a least-squares method for the original gradient equation, reformulated and solved as an artificial Stokes system. To simplify the incorporation of the given velocity within the right-hand side, we assume in the derivations that the velocity field is solenoidal. Yet this assumption is not restrictive, as we can use non-divergence-free approximations and even compressible velocities. Numerical experiments confirm the optimal a priori error estimates for both methods considered.
从给定的流速重建压力是一项在各种应用中出现的任务,标准方法使用Navier-Stokes方程来推导压力p的泊松问题。然而,这种方法人为地增加了对解和数据的规则性要求。在这种情况下,我们提出并分析了两种替代技术来确定p∈l2¹(Ω) {p In L^{2}(Omega)}。第一个是一个超弱变分公式,应用分部积分法将所有导数平移到测试函数。给出了合适的有限元离散化方法,并证明了Galerkin-Petrov方法的最佳收敛性。第二种方法是对原始梯度方程采用最小二乘法,将其重新表述并求解为人工Stokes系统。为了简化将给定的速度并入右边,我们在推导中假定速度场是螺线形的。然而,这个假设不是限制性的,因为我们可以使用非散度近似,甚至是可压缩速度。数值实验证实了两种方法的最优先验误差估计。
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引用次数: 0
Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation 非线性Schrödinger方程的自适应吸收边界层
4区 数学 Q2 Mathematics Pub Date : 2023-11-01 DOI: 10.1515/cmam-2023-0096
Hans Peter Stimming, Xin Wen, Norbert J. Mauser
Abstract We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential function is employed for adaptive parameter selection. This formula is obtained by an extension of the analysis in [R. Kosloff and D. Kosloff, Absorbing boundaries for wave propagation problems, J. Comput. Phys. 63 1986, 2, 363–376]. We also show that our imaginary potential function is more efficient than what is used in the literature. Numerical examples show that our ansatz is significantly better than existing approaches. We show that our approach can very accurately compute the solutions of the NLS equations in one dimension, including in the case of multi-dominant wave number solutions.
摘要提出了一种非线性Schrödinger方程的自适应吸收边界层技术,该技术与分时傅立叶谱方法(TSSP)相结合,用于NLS方程的离散化。本文提出了一种新的基于高阶多项式的复吸收势函数,主要改进是采用显式公式对势函数中的系数进行自适应参数选择。该公式由[R]中的分析推广而得。波传播问题的吸收边界,J.计算机学报。物理学报,2002,23(2):363-376。我们还证明了我们的虚势函数比文献中使用的函数更有效。数值算例表明,我们的方法明显优于现有的方法。我们表明,我们的方法可以非常准确地计算一维NLS方程的解,包括在多主导波数解的情况下。
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引用次数: 0
A Conforming Virtual Element Method for Parabolic Integro-Differential Equations 抛物型积分-微分方程的一致虚元法
4区 数学 Q2 Mathematics Pub Date : 2023-10-11 DOI: 10.1515/cmam-2023-0061
Sangita Yadav, Meghana Suthar, Sarvesh Kumar
Abstract This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and L 2 L^{2} projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings.
摘要结合时间离散的后向欧拉格式,提出并分析了抛物型积分微分方程空间离散化的符合虚元格式。利用Ritz-Voltera算子和l2l ^{2}投影算子,建立了最优先验误差估计。最后,通过数值实验验证了该方法的计算效率和理论结果。
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引用次数: 0
期刊
Computational Methods in Applied Mathematics
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