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

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.

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引用次数: 0

Irrational-Window-Filter Projection Method and Application to Quasiperiodic Schrödinger Eigenproblems

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.

引用次数: 0

Piecewise Linear Interpolation of Noise in Finite Element Approximations of Parabolic SPDEs

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.

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引用次数: 0

Higher-Order Far-Field Boundary Conditions for Crystalline Defects

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.

引用次数: 0

Gaussian Process Regression under Computational and Epistemic Misspecification

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.

引用次数: 0

On Polynomial Interpolation in the Monomial Basis

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

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.

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引用次数: 0

Corrigendum: Domain Decomposition Approaches for Mesh Generation via the Equidistribution Principle

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.

引用次数: 0

Rational Methods for Abstract, Linear, Nonhomogeneous Problems without Order Reduction

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.

引用次数: 0

Long Time Stability and Numerical Stability of Implicit Schemes for Stochastic Heat Equations

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.

引用次数: 0

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