SIAM Journal on Numerical Analysis最新文献

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Multiscale Hybrid-Mixed Methods for the Stokes and Brinkman Equations—A Priori Analysis Stokes和Brinkman方程的多尺度混合-混合方法-先验分析

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis

Pub Date : 2025-03-12 DOI: 10.1137/24m1649368

Rodolfo Araya, Christopher Harder, Abner H. Poza, Frédéric Valentin

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 588-618, April 2025.
Abstract. The multiscale hybrid-mixed (MHM) method for the Stokes operator was formally introduced in [R. Araya et al., Comput. Methods Appl. Mech. Engrg., 324, pp. 29–53, 2017] and numerically validated. The method has face degrees of freedom associated with multiscale basis functions computed from local Neumann problems driven by discontinuous polynomial spaces on skeletal meshes. The two-level MHM version approximates the multiscale basis using a stabilized finite element method. This work proposes the first numerical analysis for the one- and two-level MHM method applied to the Stokes/Brinkman equations within a new abstract framework relating MHM methods to discrete primal hybrid formulations. As a result, we generalize the two-level MHM method to include general second-level solvers and continuous polynomial interpolation on faces and establish abstract conditions to have those methods well-posed and optimally convergent on natural norms. We apply the abstract setting to analyze the MHM methods using stabilized and stable finite element methods as second-level solvers with (dis)continuous interpolation on faces. Also, we find that the discrete velocity and pressure variables preserve the balance of forces and conservation of mass at the element level. Numerical benchmarks assess theoretical results.

SIAM数值分析杂志，第63卷，第2期，第588-618页，2025年4月。摘要。本文正式引入Stokes算子的多尺度混合-混合（MHM）方法。Araya等人，计算机。方法:。动力机械。Engrg。科学进展，324,pp 29-53, 2017]并进行了数值验证。该方法具有与骨架网格上由不连续多项式空间驱动的局部诺伊曼问题计算的多尺度基函数相关联的面自由度。两级MHM版本使用稳定有限元法近似多尺度基。这项工作提出了应用于Stokes/Brinkman方程的一级和二级MHM方法的第一个数值分析，在一个新的抽象框架内将MHM方法与离散原始混合公式联系起来。因此，我们将两级MHM方法推广到包括一般的二级解和面上的连续多项式插值，并建立了这些方法在自然范数上的适定和最优收敛的抽象条件。本文应用抽象设置，分析了用稳定有限元法和稳定有限元法作为二级求解器，在面上（非）连续插值的MHM方法。此外，我们发现离散的速度和压力变量在单元水平上保持了力的平衡和质量的守恒。数值基准评估理论结果。

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引用次数: 0

Irrational-Window-Filter Projection Method and Application to Quasiperiodic Schrödinger Eigenproblems 无理性窗滤波投影法及其在拟周期Schrödinger特征问题中的应用

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis

Pub Date : 2025-03-11 DOI: 10.1137/24m1666197

Kai Jiang, Xueyang Li, Yao Ma, Juan Zhang, Pingwen Zhang, Qi Zhou

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 564-587, April 2025.
Abstract. In this paper, we propose a new algorithm, the irrational-window-filter projection method (IWFPM), for quasiperiodic systems with concentrated spectral point distribution. Based on the projection method (PM), IWFPM filters out dominant spectral points by defining an irrational window and uses a corresponding index-shift transform to make the FFT available. The error analysis on the function approximation level is also given. We apply IWFPM to one-dimensional, two-dimensional (2D), and three-dimensional (3D) quasiperiodic Schrödinger eigenproblems (QSEs) to demonstrate its accuracy and efficiency. IWFPM exhibits a significant computational advantage over PM for both extended and localized quantum states. More importantly, by using IWFPM, the existence of Anderson localization in 2D and 3D QSEs is numerically verified.

SIAM数值分析杂志，第63卷，第2期，第564-587页，2025年4月。摘要。本文针对具有集中谱点分布的准周期系统，提出了一种新的算法——无理性窗滤波投影法。基于投影法（PM）， IWFPM通过定义一个非理性窗口来滤除优势谱点，并使用相应的指数移位变换使FFT可用。给出了函数逼近级的误差分析。我们将IWFPM应用于一维，二维（2D）和三维（3D）准周期Schrödinger特征问题（qse），以证明其准确性和效率。IWFPM在扩展和局域量子态方面都比PM具有显著的计算优势。更重要的是，通过IWFPM，数值验证了二维和三维qse中Anderson定位的存在性。

引用次数: 0

Piecewise Linear Interpolation of Noise in Finite Element Approximations of Parabolic SPDEs 抛物型spde有限元逼近中噪声的分段线性插值

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis

Pub Date : 2025-03-10 DOI: 10.1137/23m1574117

Gabriel J. Lord, Andreas Petersson

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 542-563, April 2025.
Abstract. Efficient simulation of stochastic partial differential equations (SPDEs) on general domains requires noise discretization. This paper employs piecewise linear interpolation of noise in a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polyhedral domain. The Gaussian noise is white in time, spatially correlated, and modeled as a standard cylindrical Wiener process on a reproducing kernel Hilbert space. This paper provides the first rigorous analysis of the resulting noise discretization error for a general spatial covariance kernel. The kernel is assumed to be defined on a larger regular domain, allowing for sampling by the circulant embedding method. The error bound under mild kernel assumptions requires nontrivial techniques like Hilbert–Schmidt bounds on products of finite element interpolants, entropy numbers of fractional Sobolev space embeddings, and an error bound for interpolants in fractional Sobolev norms. Examples with kernels encountered in applications are illustrated in numerical simulations using the FEniCS finite element software. Key findings include the following: noise interpolation does not introduce additional errors for Matérn kernels in [math]; there exist kernels that yield dominant interpolation errors; and generating noise on a coarser mesh does not always compromise accuracy.

SIAM数值分析杂志，第63卷，第2期，第542-563页，2025年4月。摘要。一般域上的随机偏微分方程的有效模拟需要对噪声进行离散化。本文对凸多面体上半线性随机反应-平流-扩散方程的全离散有限元逼近，采用了噪声分段线性插值方法。高斯噪声在时间上是白色的，空间上是相关的，并在再现核希尔伯特空间上建模为标准的圆柱形维纳过程。本文首次对一般空间协方差核的噪声离散误差进行了严格的分析。假设核在更大的正则域上定义，允许循环嵌入方法进行采样。温和核假设下的误差界需要一些非平凡的技术，如有限元插值积的Hilbert-Schmidt界，分数Sobolev空间嵌入的熵数，分数Sobolev范数内插的误差界。利用FEniCS有限元软件对应用中遇到的核问题进行了数值模拟。主要发现包括：在[math]中，噪声插值不会给mat核引入额外的误差；存在产生显性插值误差的核；在粗糙的网格上产生噪声并不总是会影响精度。

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引用次数: 0

Higher-Order Far-Field Boundary Conditions for Crystalline Defects 晶体缺陷的高阶远场边界条件

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis

Pub Date : 2025-03-06 DOI: 10.1137/24m165836x

Julian Braun, Christoph Ortner, Yangshuai Wang, Lei Zhang

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 520-541, April 2025.
Abstract. Crystalline materials exhibit long-range elastic fields due to the presence of defects, leading to significant domain size effects in atomistic simulations. A rigorous far-field expansion of these long-range fields identifies low-rank structure in the form of a sum of discrete multipole terms and continuum predictors [J. Braun, T. Hudson, and C. Ortner, Arch. Ration. Mech. Anal., 245 (2022), pp. 1437–1490]. We propose a novel numerical scheme that exploits this low-rank structure to accelerate material defect simulations by minimizing the domain size effects. Our approach iteratively improves the boundary condition, systematically following the asymptotic expansion of the far field. We provide both rigorous error estimates for the method and a range of empirical numerical tests to assess its convergence and robustness.

SIAM数值分析杂志，第63卷，第2期，第520-541页，2025年4月。摘要。晶体材料由于缺陷的存在而表现出长程弹性场，导致原子模拟中显著的畴尺寸效应。对这些远场进行了严格的远场扩展，以离散多极项和连续预测因子的总和的形式识别低秩结构[J]。布劳恩，T.哈德森和C.奥特纳，Arch。配给。动力机械。分析的。中国科学，245 (2022),pp. 1437-1490]。我们提出了一种新的数值方案，利用这种低秩结构通过最小化畴尺寸效应来加速材料缺陷的模拟。我们的方法迭代地改进了边界条件，系统地遵循远场渐近展开。我们为该方法提供了严格的误差估计和一系列经验数值测试，以评估其收敛性和鲁棒性。

引用次数: 0

Gaussian Process Regression under Computational and Epistemic Misspecification 计算和认知错误规范下的高斯过程回归

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis

Pub Date : 2025-03-05 DOI: 10.1137/23m1624749

Daniel Sanz-Alonso, Ruiyi Yang

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 495-519, April 2025.
Abstract. Gaussian process regression is a classical kernel method for function estimation and data interpolation. In large data applications, computational costs can be reduced using low-rank or sparse approximations of the kernel. This paper investigates the effect of such kernel approximations on the interpolation error. We introduce a unified framework to analyze Gaussian process regression under important classes of computational misspecification: Karhunen–Loève expansions that result in low-rank kernel approximations, multiscale wavelet expansions that induce sparsity in the covariance matrix, and finite element representations that induce sparsity in the precision matrix. Our theory also accounts for epistemic misspecification in the choice of kernel parameters.

SIAM数值分析杂志，第63卷，第2期，第495-519页，2025年4月。摘要。高斯过程回归是函数估计和数据插值的经典核方法。在大数据应用程序中，可以使用核的低秩或稀疏近似来减少计算成本。本文研究了这种核近似对插值误差的影响。我们引入了一个统一的框架来分析高斯过程回归中重要的计算错误类别：导致低秩核近似的karhunen - lo展开式，导致协方差矩阵稀疏的多尺度小波展开式，以及导致精度矩阵稀疏的有限元表示。我们的理论还解释了核参数选择中的认知错误。

引用次数: 0

On Polynomial Interpolation in the Monomial Basis 关于单项式基上的多项式插值

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis

Pub Date : 2025-03-05 DOI: 10.1137/23m1623215

Zewen Shen, Kirill Serkh

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 469-494, April 2025.
Abstract. In this paper, we show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. This leads to a practical algorithm for piecewise polynomial interpolation over general regions in the complex plane using the monomial basis. Our analysis also yields a new upper bound for the condition number of an arbitrary Vandermonde matrix, which generalizes several previous results.

SIAM 数值分析期刊》，第 63 卷，第 2 期，第 469-494 页，2025 年 4 月。摘要在本文中，我们证明了只要范德蒙德矩阵的条件数小于机器ε的倒数，单项式基在插值方面通常与条件良好的多项式基一样好。这就为使用单项式基础在复平面内的一般区域进行片断多项式插值提供了一种实用算法。我们的分析还得出了任意 Vandermonde 矩阵条件数的新上界，这概括了之前的几个结果。

引用次数: 0

Discretization of Total Variation in Optimization with Integrality Constraints 具有完整性约束的优化中总变分的离散化

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis

Pub Date : 2025-02-25 DOI: 10.1137/24m164608x

Annika Schiemann, Paul Manns

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 437-460, February 2025.
Abstract. We introduce discretizations of infinite-dimensional optimization problems with total variation regularization and integrality constraints on the optimization variables. We advance the discretization of the dual formulation of the total variation term with Raviart–Thomas functions, which is known from the literature for certain convex problems. Since we have an integrality constraint, the previous analysis from Caillaud and Chambolle [IMA J. Numer. Anal., 43 (2022), pp. 692–736] no longer holds. Even weaker [math]-convergence results no longer hold because the recovery sequences generally need to attain noninteger values to recover the total variation of the limit function. We solve this issue by introducing a discretization of the input functions on an embedded, finer mesh. A superlinear coupling of the mesh sizes implies an averaging on the coarser mesh of the Raviart–Thomas ansatz, which enables us to recover the total variation of integer-valued limit functions with integer-valued discretized input functions. Moreover, we are able to estimate the discretized total variation of the recovery sequence by the total variation of its limit and an error depending on the mesh size ratio. For the discretized optimization problems, we additionally add a constraint that vanishes in the limit and enforces compactness of the sequence of minimizers, which yields their convergence to a minimizer of the original problem. This constraint contains a degree of freedom whose admissible range is determined. Its choice may have a strong impact on the solutions in practice as we demonstrate with an example from imaging.

SIAM数值分析杂志，第63卷，第1期，第437-460页，2025年2月。摘要。引入了具有全变分正则化和优化变量完整性约束的无限维优化问题的离散化方法。利用文献中已知的关于某些凸问题的Raviart-Thomas函数，提出了全变分项的对偶形式的离散化。由于我们有一个完整性约束，以前的分析由Caillaud和Chambolle [IMA J. number]。分析的。， 43 (2022), pp. 692-736]不再成立。甚至较弱的[数学]收敛结果也不再成立，因为恢复序列通常需要获得非整数值才能恢复极限函数的总变化。我们通过在嵌入的更细的网格上引入输入函数的离散化来解决这个问题。网格尺寸的超线性耦合意味着对Raviart-Thomas ansatz的粗网格进行平均，这使我们能够恢复整值极限函数与整值离散输入函数的总变化。此外，我们还可以通过恢复序列的极限总变分和依赖于网格尺寸比的误差来估计恢复序列的离散化总变分。对于离散优化问题，我们额外增加了一个约束，该约束在极限中消失，并强制最小化序列的紧性，从而使它们收敛到原始问题的最小化。这个约束包含一个可接受范围已确定的自由度。它的选择可能对实践中的解决方案有很大的影响，正如我们用成像的例子所展示的那样。

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引用次数: 0

Corrigendum: Domain Decomposition Approaches for Mesh Generation via the Equidistribution Principle 勘误：通过均分原理生成网格的域分解方法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis

Pub Date : 2025-02-25 DOI: 10.1137/24m1693453

Martin J. Gander, Ronald D. Haynes, Felix Kwok

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 461-467, February 2025.
Abstract. Various nonlinear Schwarz domain decomposition methods were proposed to solve the one-dimensional equidistribution principle in [M. J. Gander and R. D. Haynes, SIAM J. Numer. Anal., 50 (2012), pp. 2111-2135]. A corrected proof of convergence for the linearized Schwarz algorithm presented in section 3.2, under additional hypotheses, is presented here. An alternative linearized Schwarz algorithm for equidistributed grid generation is also provided.

SIAM数值分析杂志，第63卷，第1期，第461-467页，2025年2月。摘要。提出了多种非线性Schwarz域分解方法来求解[M]中的一维均匀分布原理。J.甘德和R. D.海恩斯，SIAM J. number。分析的。， 50 (2012), pp. 2111-2135]。第3.2节中给出的线性化Schwarz算法在附加假设下的收敛性的修正证明，在这里给出。本文还提供了一种可选的用于等分布网格生成的线性化Schwarz算法。

引用次数: 0

Rational Methods for Abstract, Linear, Nonhomogeneous Problems without Order Reduction 无阶约抽象、线性、非齐次问题的有理方法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis

Pub Date : 2025-02-24 DOI: 10.1137/24m165942x

Carlos Arranz-Simón, César Palencia

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 422-436, February 2025.
Abstract. Starting from an A-stable rational approximation to [math] of order [math], [math], families of stable methods are proposed to time discretize abstract IVPs of the type [math]. These numerical procedures turn out to be of order [math], thus overcoming the order reduction phenomenon, and only one evaluation of [math] per step is required.

SIAM数值分析杂志，第63卷，第1期，第422-436页，2025年2月。摘要。从阶[math]， [math]的[math]的a稳定有理逼近出发，提出了对类型[math]的抽象ivp进行时间离散的稳定方法族。这些数值过程被证明是有序的（数学），从而克服了降阶现象，并且每一步只需要对[数学]进行一次评估。

引用次数: 0

Long Time Stability and Numerical Stability of Implicit Schemes for Stochastic Heat Equations 随机热方程隐式格式的长时间稳定性和数值稳定性

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis

Pub Date : 2025-02-18 DOI: 10.1137/24m1636691

Xiaochen Yang, Yaozhong Hu

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 396-421, February 2025.
Abstract. This paper studies the long time stability of both the solution of a stochastic heat equation on a bounded domain driven by a correlated noise and its approximations. It is popular for researchers to prove the intermittency of the solution, which means that the moments of solution to a stochastic heat equation usually grow to infinity exponentially fast and this hints that the solution to stochastic heat equation is generally not stable in long time. However, quite surprisingly in this paper we show that when the domain is bounded and when the noise is not singular in spatial variables, the system can be long time stable and we also prove that we can approximate the solution by its finite dimensional spectral approximation, which is also long time stable. The idea is to use eigenfunction expansion of the Laplacian on a bounded domain to write a stochastic heat equation as a system of infinite many stochastic differential equations. We also present numerical experiments which are consistent with our theoretical results.

SIAM数值分析杂志，第63卷，第1期，第396-421页，2025年2月。摘要。本文研究了由相关噪声驱动的有界区域上随机热方程解及其近似解的长时间稳定性。研究人员普遍认为，解的间歇性是一个普遍的问题，这意味着随机热方程的解的矩通常以指数速度增长到无穷大，这暗示了随机热方程的解在长时间内通常是不稳定的。然而，令人惊讶的是，在本文中，我们证明了当域是有界的，当噪声在空间变量中不是奇异时，系统可以长时间稳定，并且我们还证明了我们可以用它的有限维谱近似来近似解，这也是长时间稳定的。其思想是利用拉普拉斯函数在有界域上的特征函数展开，将一个随机热方程写成一个由无穷多个随机微分方程组成的系统。我们还提出了与理论结果一致的数值实验。

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