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On a New Class of BDF and IMEX Schemes for Parabolic Type Equations SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-07-16 Fukeng Huang, Jie Shen
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1609-1637, August 2024. Abstract. When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit
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Localized Implicit Time Stepping for the Wave Equation SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-07-15 Dietmar Gallistl, Roland Maier
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1589-1608, August 2024. Abstract. This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and
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Full-Spectrum Dispersion Relation Preserving Summation-by-Parts Operators SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-07-11 Christopher Williams, Kenneth Duru
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1565-1588, August 2024. Abstract. The dispersion error is currently the dominant error for computed solutions of wave propagation problems with high-frequency components. In this paper, we define and give explicit examples of interior [math]-dispersion-relation-preserving schemes, of interior order of accuracy 4, 5, 6, and 7, with a complete
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Robust Finite Elements for Linearized Magnetohydrodynamics SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-07-09 L. Beirão da Veiga, F. Dassi, G. Vacca
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1539-1564, August 2024. Abstract. We introduce a pressure robust finite element method for the linearized magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a nonconforming BDM approach with suitable DG terms for
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Randomized Least-Squares with Minimal Oversampling and Interpolation in General Spaces SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-07-09 Matthieu Dolbeault, Moulay Abdellah Chkifa
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1515-1538, August 2024. Abstract. In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected [math] sense, as soon as the sample size [math]
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Discrete Weak Duality of Hybrid High-Order Methods for Convex Minimization Problems SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-07-04 Ngoc Tien Tran
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1492-1514, August 2024. Abstract. This paper derives a discrete dual problem for a prototypical hybrid high-order method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with convergence rates under additional smoothness assumptions. This duality holds
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Uniform Substructuring Preconditioners for High Order FEM on Triangles and the Influence of Nodal Basis Functions SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-07-01 Mark Ainsworth, Shuai Jiang
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1465-1491, August 2024. Abstract. A robust substructuring type preconditioner is developed for high order approximation of problem for which the element matrix takes the form [math] where [math] and [math] are the mass and stiffness matrices, respectively. A standard preconditioner for the pure stiffness matrix results in a condition number
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A Kernel Machine Learning for Inverse Source and Scattering Problems SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-06-19 Shixu Meng, Bo Zhang
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1443-1464, June 2024. Abstract. In this work we connect machine learning techniques, in particular kernel machine learning, to inverse source and scattering problems. We show the proposed kernel machine learning has demonstrated generalization capability and has a rigorous mathematical foundation. The proposed learning is based on the Mercer
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A Finite Element Method for Hyperbolic Metamaterials with Applications for Hyperlens SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-06-17 Fuhao Liu, Wei Yang, Jichun Li
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1420-1442, June 2024. Abstract. In this paper, we first derive a time-dependent Maxwell’s equation model for simulating wave propagation in anisotropic dispersive media and hyperbolic metamaterials. The modeling equations are obtained by using the Drude–Lorentz model to approximate both the permittivity and permeability. Then we develop a
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The ([math], [math])-HDG Method for the Helmholtz Equation with Large Wave Number SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-06-12 Bingxin Zhu, Haijun Wu
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1394-1419, June 2024. Abstract. In this paper, we analyze a hybridizable discontinuous Galerkin method for the Helmholtz equation with large wave number, which uses piecewise polynomials of degree of [math] to approximate the potential [math] and its traces and piecewise polynomials of degree of [math] for the flux [math]. It is proved that
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Inverse Wave-Number-Dependent Source Problems for the Helmholtz Equation SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-06-06 Hongxia Guo, Guanghui Hu
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1372-1393, June 2024. Abstract. This paper is concerned with the multi-frequency factorization method for imaging the support of a wave-number-dependent source function. It is supposed that the source function is given by the inverse Fourier transform of some time-dependent source with a priori given radiating period. Using the multi-frequency
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Bilinear Optimal Control for the Fractional Laplacian: Analysis and Discretization SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-06-04 Francisco Bersetche, Francisco Fuica, Enrique Otárola, Daniel Quero
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1344-1371, June 2024. Abstract. We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic PDE as the state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are
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Solving PDEs with Incomplete Information SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-30 Peter Binev, Andrea Bonito, Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1278-1312, June 2024. Abstract. We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when the boundary data is unknown and instead one observes finitely many
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Error Bounds for Discrete Minimizers of the Ginzburg–Landau Energy in the High-[math] Regime SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-30 Benjamin Dörich, Patrick Henning
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1313-1343, June 2024. Abstract. In this work, we study discrete minimizers of the Ginzburg–Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg–Landau parameter [math]. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to
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On Bernoulli’s Method SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-24 Tamás Dózsa, Ferenc Schipp, Alexandros Soumelidis
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1259-1277, June 2024. Abstract. We generalize Bernoulli’s classical method for finding poles of rational functions using the rational orthogonal Malmquist–Takenaka system. We show that our approach overcomes the limitations of previous methods, especially their dependence on the existence of a so-called dominant pole, while significantly
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Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-22 Benedict J. Leimkuhler, Daniel Paulin, Peter A. Whalley
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1226-1258, June 2024. Abstract. We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for [math]-[math]Lipschitz, [math]-convex potentials. Our approach gives convergence rates of [math], with explicit step size restrictions, which are of the same order as the stability threshold for Gaussian targets and
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Pointwise Gradient Estimate of the Ritz Projection SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-21 Lars Diening, Julian Rolfes, Abner J. Salgado
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1212-1225, June 2024. Abstract. Let [math] be a convex polytope ([math]). The Ritz projection is the best approximation, in the [math]-norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely
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Mean Dimension of Radial Basis Functions SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-21 Christopher Hoyt, Art B. Owen
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1191-1211, June 2024. Abstract. We show that generalized multiquadric radial basis functions (RBFs) on [math] have a mean dimension that is [math] as [math] with an explicit bound for the implied constant, under moment conditions on their inputs. Under weaker moment conditions the mean dimension still approaches 1. As a consequence, these
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Total Variation Error Bounds for the Accelerated Exponential Euler Scheme Approximation of Parabolic Semilinear SPDEs SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-15 Charles-Edouard Bréhier
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1171-1190, June 2024. Abstract. We prove a new numerical approximation result for the solutions of semilinear parabolic stochastic partial differential equations, driven by additive space-time white noise in dimension 1. The temporal discretization is performed using an accelerated exponential Euler scheme, and we show that, under appropriate
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Implicit and Fully Discrete Approximation of the Supercooled Stefan Problem in the Presence of Blow-Ups SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-09 Christa Cuchiero, Christoph Reisinger, Stefan Rigger
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1145-1170, June 2024. Abstract.We consider two approximation schemes of the one-dimensional supercooled Stefan problem and prove their convergence, even in the presence of finite time blow-ups. All proofs are based on a probabilistic reformulation recently considered in the literature. The first scheme is a version of the time-stepping scheme
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Asymptotic Compatibility of a Class of Numerical Schemes for a Nonlocal Traffic Flow Model SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-07 Kuang Huang, Qiang Du
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1119-1144, June 2024. Abstract. This paper considers numerical discretization of a nonlocal conservation law modeling vehicular traffic flows involving nonlocal intervehicle interactions. The nonlocal model involves an integral over the range measured by a horizon parameter and it recovers the local Lighthill–Richards–Whitham model as the
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Kernel Interpolation of High Dimensional Scattered Data SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-06 Shao-Bo Lin, Xiangyu Chang, Xingping Sun
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1098-1118, June 2024. Abstract. Data sites selected from modeling high-dimensional problems often appear scattered in nonpaternalistic ways. Except for sporadic-clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These features defy any theoretical treatment that requires local or global
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An Asymptotic Preserving Discontinuous Galerkin Method for a Linear Boltzmann Semiconductor Model SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-06 Victor P. DeCaria, Cory D. Hauck, Stefan R. Schnake
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1067-1097, June 2024. Abstract. A key property of the linear Boltzmann semiconductor model is that as the collision frequency tends to infinity, the phase space density [math] converges to an isotropic function [math], called the drift-diffusion limit, where [math] is a Maxwellian and the physical density [math] satisfies a second-order parabolic
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A Novel Mixed Spectral Method and Error Estimates for Maxwell Transmission Eigenvalue Problems SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-03 Jing An, Waixiang Cao, Zhimin Zhang
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1039-1066, June 2024. Abstract. In this paper, a novel mixed spectral-Galerkin method is proposed and studied for a Maxwell transmission eigenvalue problem in a spherical domain. The method utilizes vector spherical harmonics to achieve dimension reduction. By introducing an auxiliary vector function, the original problem is rewritten as
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Gain Coefficients for Scrambled Halton Points SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-05-02 Art B. Owen, Zexin Pan
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1021-1038, June 2024. Abstract. Randomized quasi-Monte Carlo, via certain scramblings of digital nets, produces unbiased estimates of [math] with a variance that is [math] for any [math]. It also satisfies some nonasymptotic bounds where the variance is no larger than some [math] times the ordinary Monte Carlo variance. For scrambled Sobol’
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A Two-Level Block Preconditioned Jacobi–Davidson Method for Multiple and Clustered Eigenvalues of Elliptic Operators SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-04-22 Qigang Liang, Wei Wang, Xuejun Xu
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 998-1019, April 2024. Abstract. In this paper, we propose a two-level block preconditioned Jacobi–Davidson (BPJD) method for efficiently solving discrete eigenvalue problems resulting from finite element approximations of [math]th ([math]) order symmetric elliptic eigenvalue problems. Our method works effectively to compute the first several
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Singularity Swapping Method for Nearly Singular Integrals Based on Trapezoidal Rule SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-04-08 Gang Bao, Wenmao Hua, Jun Lai, Jinrui Zhang
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 974-997, April 2024. Abstract. Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer potentials for arbitrarily close targets. Our method is based on the global
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Sequential Discretization Schemes for a Class of Stochastic Differential Equations and their Application to Bayesian Filtering SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-04-08 Ö. Deniz Akyildiz, Dan Crisan, Joaquin Miguez
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 946-973, April 2024. Abstract. We introduce a predictor-corrector discretization scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of
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A Posteriori Error Control for Fourth-Order Semilinear Problems with Quadratic Nonlinearity SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-04-03 Carsten Carstensen, Benedikt Gräßle, Neela Nataraj
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 919-945, April 2024. Abstract. A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semilinear problems with trilinear nonlinearity and a general source. A quasi-optimal smoother extends the source term to the discrete trial space and, more important, modifies the trilinear term in
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Cut Finite Element Method for Divergence-Free Approximation of Incompressible Flow: A Lagrange Multiplier Approach SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-04-01 Erik Burman, Peter Hansbo, Mats Larson
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 893-918, April 2024. Abstract. In this note, we design a cut finite element method for a low order divergence-free element applied to a boundary value problem subject to Stokes’ equations. For the imposition of Dirichlet boundary conditions, we consider either Nitsche’s method or a stabilized Lagrange multiplier method. In both cases, the
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Generalized Dimension Truncation Error Analysis for High-Dimensional Numerical Integration: Lognormal Setting and Beyond SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-28 Philipp A. Guth, Vesa Kaarnioja
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 872-892, April 2024. Abstract. Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional
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On the Approximability and Curse of Dimensionality of Certain Classes of High-Dimensional Functions SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-22 Christian Rieger, Holger Wendland
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 842-871, April 2024. Abstract. In this paper, we study the approximability of high-dimensional functions that appear, for example, in the context of many body expansions and high-dimensional model representation. Such functions, though high-dimensional, can be represented as finite sums of lower-dimensional functions. We will derive sampling
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wPINNs: Weak Physics Informed Neural Networks for Approximating Entropy Solutions of Hyperbolic Conservation Laws SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-14 Tim De Ryck, Siddhartha Mishra, Roberto Molinaro
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 811-841, April 2024. Abstract. Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed
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On Optimal Cell Average Decomposition for High-Order Bound-Preserving Schemes of Hyperbolic Conservation Laws SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-11 Shumo Cui, Shengrong Ding, Kailiang Wu
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 775-810, April 2024. Abstract. Cell average decomposition (CAD) plays a critical role in constructing bound-preserving (BP) high-order discontinuous Galerkin and finite volume methods for hyperbolic conservation laws. Seeking optimal CAD (OCAD) that attains the mildest BP Courant–Friedrichs–Lewy (CFL) condition is a fundamentally important
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On the Convergence of Continuous and Discrete Unbalanced Optimal Transport Models for 1-Wasserstein Distance SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-05 Zhe Xiong, Lei Li, Ya-Nan Zhu, Xiaoqun Zhang
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 749-774, April 2024. Abstract. We consider a Beckmann formulation of an unbalanced optimal transport (UOT) problem. The [math]-convergence of this formulation of UOT to the corresponding optimal transport (OT) problem is established as the balancing parameter [math] goes to infinity. The discretization of the problem is further shown to be
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Robust DPG Test Spaces and Fortin Operators—The [math] and [math] Cases SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-05 Thomas Führer, Norbert Heuer
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 718-748, April 2024. Abstract. At the fully discrete setting, stability of the discontinuous Petrov–Galerkin (DPG) method with optimal test functions requires local test spaces that ensure the existence of Fortin operators. We construct such operators for [math] and [math] on simplices in any space dimension and arbitrary polynomial degree
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Stable Lifting of Polynomial Traces on Triangles SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-04 Charles Parker, Endre Süli
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 692-717, April 2024. Abstract. We construct a right inverse of the trace operator [math] on the reference triangle [math] that maps suitable piecewise polynomial data on [math] into polynomials of the same degree and is bounded in all [math] norms with [math] and [math]. The analysis relies on new stability estimates for three classes of
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On the Convergence of Sobolev Gradient Flow for the Gross–Pitaevskii Eigenvalue Problem SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-04 Ziang Chen, Jianfeng Lu, Yulong Lu, Xiangxiong Zhang
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 667-691, April 2024. Abstract. We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross–Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross–Pitaevskii energy functional with respect to the [math]-metric and two other equivalent metrics on [math], including the
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Homogenization of Nondivergence-Form Elliptic Equations with Discontinuous Coefficients and Finite Element Approximation of the Homogenized Problem SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-01 Timo Sprekeler
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 646-666, April 2024. Abstract. We study the homogenization of the equation [math] posed in a bounded convex domain [math] subject to a Dirichlet boundary condition and the numerical approximation of the corresponding homogenized problem, where the measurable, uniformly elliptic, periodic, and symmetric diffusion matrix [math] is merely assumed
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A Numerical Framework for Nonlinear Peridynamics on Two-Dimensional Manifolds Based on Implicit P-(EC)[math] Schemes SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-01 Alessandro Coclite, Giuseppe M. Coclite, Francesco Maddalena, Tiziano Politi
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 622-645, April 2024. Abstract. In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily shaped two-dimensional (2D) closed manifolds is proposed. When dealing with non-parameterized 2D manifolds at the discrete scale, the problem of computing geodesic distances between two non-adjacent points arise
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Numerical Analysis for Convergence of a Sample-Wise Backpropagation Method for Training Stochastic Neural Networks SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-03-01 Richard Archibald, Feng Bao, Yanzhao Cao, Hui Sun
SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 593-621, April 2024. Abstract. The aim of this paper is to carry out convergence analysis and algorithm implementation of a novel sample-wise backpropagation method for training a class of stochastic neural networks (SNNs). The preliminary discussion on such an SNN framework was first introduced in [Archibald et al., Discrete Contin. Dyn
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Virtual Element Methods Without Extrinsic Stabilization SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-02-20 Chunyu Chen, Xuehai Huang, Huayi Wei
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 567-591, February 2024. Abstract. Virtual element methods (VEMs) without extrinsic stabilization in an arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension. The key is to construct local [math]-conforming macro finite element spaces such
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A Universal Median Quasi-Monte Carlo Integration SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-02-16 Takashi Goda, Kosuke Suzuki, Makoto Matsumoto
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 533-566, February 2024. Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC
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High Order Splitting Methods for SDEs Satisfying a Commutativity Condition SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-02-15 James M. Foster, Gonçalo dos Reis, Calum Strange
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 500-532, February 2024. Abstract. In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal, and scalar noise types. The central idea is to view the splitting method as a replacement of the
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On Uncertainty Quantification of Eigenvalues and Eigenspaces with Higher Multiplicity SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-02-07 Jürgen Dölz, David Ebert
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 422-451, February 2024. Abstract. We consider generalized operator eigenvalue problems in variational form with random perturbations in the bilinear forms. This setting is motivated by variational forms of partial differential equations with random input data. The considered eigenpairs can be of higher but finite multiplicity. We investigate
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Convergence Analysis for Bregman Iterations in Minimizing a Class of Landau Free Energy Functionals SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-02-07 Chenglong Bao, Chang Chen, Kai Jiang, Lingyun Qiu
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 476-499, February 2024. Abstract. Finding stationary states of Landau free energy functionals has to solve a nonconvex infinite-dimensional optimization problem. In this paper, we develop a Bregman distance based optimization method for minimizing a class of Landau energy functionals and focus on its convergence analysis in the function space
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Space-Time Finite Element Methods for Distributed Optimal Control of the Wave Equation SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-02-07 Richard Löscher, Olaf Steinbach
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 452-475, February 2024. Abstract. We consider space-time tracking-type distributed optimal control problems for the wave equation in the space-time domain [math], where the control is assumed to be in the energy space [math], rather than in [math], which is more common. While the latter ensures a unique state in the Sobolev space [math],
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Frequency-Explicit A Posteriori Error Estimates for Discontinuous Galerkin Discretizations of Maxwell’s Equations SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-02-06 Théophile Chaumont-Frelet, Patrick Vega
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 400-421, February 2024. Abstract. We propose a new residual-based a posteriori error estimator for discontinuous Galerkin discretizations of time-harmonic Maxwell’s equations in first-order form. We establish that the estimator is reliable and efficient, and the dependency of the reliability and efficiency constants on the frequency is analyzed
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Structure Preserving Primal Dual Methods for Gradient Flows with Nonlinear Mobility Transport Distances SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-02-05 José A. Carrillo, Li Wang, Chaozhen Wei
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 376-399, February 2024. Abstract. We develop structure preserving schemes for a class of nonlinear mobility continuity equation. When the mobility is a concave function, this equation admits a form of gradient flow with respect to a Wasserstein-like transport metric. Our numerical schemes build upon such formulation and utilize modern large-scale
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Numerical Methods and Analysis of Computing Quasiperiodic Systems SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-02-01 Kai Jiang, Shifeng Li, Pingwen Zhang
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 353-375, February 2024. Abstract. Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is a great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256 (2014), pp. 428–440], has been proposed
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Numerical Integration of Schrödinger Maps via the Hasimoto Transform SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-01-31 Valeria Banica, Georg Maierhofer, Katharina Schratz
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 322-352, February 2024. Abstract. We introduce a numerical approach to computing the Schrödinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schrödinger (NLS) equation. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators
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An Adaptive Spectral Method for Oscillatory Second-Order Linear ODEs with Frequency-Independent Cost SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-01-29 Fruzsina J. Agocs, Alex H. Barnett
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 295-321, February 2024. Abstract. We introduce an efficient numerical method for second-order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose
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A Positive and Moment-Preserving Fourier Spectral Method SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-01-25 Zhenning Cai, Bo Lin, Meixia Lin
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 273-294, February 2024. Abstract. This paper presents a novel Fourier spectral method that utilizes optimization techniques to ensure the positivity and conservation of moments in the space of trigonometric polynomials. We rigorously analyze the accuracy of the new method and prove that it maintains spectral accuracy. To solve the optimization
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A Tangential and Penalty-Free Finite Element Method for the Surface Stokes Problem SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-01-25 Alan Demlow, Michael Neilan
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 248-272, February 2024. Abstract. Surface Stokes and Navier–Stokes equations are used to model fluid flow on surfaces. They have attracted significant recent attention in the numerical analysis literature because approximation of their solutions poses significant challenges not encountered in the Euclidean context. One challenge comes from
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Higher-Order Monte Carlo through Cubic Stratification SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-01-24 Nicolas Chopin, Mathieu Gerber
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 229-247, February 2024. Abstract. We propose two novel unbiased estimators of the integral [math] for a function [math], which depend on a smoothness parameter [math]. The first estimator integrates exactly the polynomials of degrees [math] and achieves the optimal error [math] (where [math] is the number of evaluations of [math]) when [math]
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Space-Time Virtual Elements for the Heat Equation SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-01-18 Sergio Gomez, Lorenzo Mascotto, Andrea Moiola, Ilaria Perugia
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 199-228, February 2024. Abstract. We propose and analyze a space-time virtual element method for the discretization of the heat equation in a space-time cylinder, based on a standard Petrov–Galerkin formulation. Local discrete functions are solutions to a heat equation problem with polynomial data. Global virtual element spaces are nonconforming
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A Lagrange–Galerkin Scheme for First Order Mean Field Game Systems SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-01-16 Elisabetta Carlini, Francisco J. Silva, Ahmad Zorkot
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 167-198, February 2024. Abstract. In this work, we consider a first order mean field game system with nonlocal couplings. A Lagrange–Galerkin scheme for the continuity equation, coupled with a semi-Lagrangian scheme for the Hamilton–Jacobi–Bellman equation, is proposed to discretize the mean field games system. The convergence of solutions
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Analysis and Numerical Approximation of Stationary Second-Order Mean Field Game Partial Differential Inclusions SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-01-12 Yohance A. P. Osborne, Iain Smears
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 138-166, February 2024. Abstract. The formulation of mean field games (MFG) typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov–Fokker–Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may
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Error Analysis of a First-Order IMEX Scheme for the Logarithmic Schrödinger Equation SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-01-11 Li-Lian Wang, Jingye Yan, Xiaolong Zhang
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 119-137, February 2024. Abstract. The logarithmic Schrödinger equation (LogSE) has a logarithmic nonlinearity [math] that is not differentiable at [math]. Compared with its counterpart with a regular nonlinear term, it possesses richer and unusual dynamics, though the low regularity of the nonlinearity brings about significant challenges
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Optimal Error Bounds on the Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity SIAM J. Numer. Anal. (IF 2.8) Pub Date : 2024-01-11 Weizhu Bao, Chushan Wang
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 93-118, February 2024. Abstract. We establish optimal error bounds for the exponential wave integrator (EWI) applied to the nonlinear Schrödinger equation (NLSE) with [math]-potential and/or locally Lipschitz nonlinearity under the assumption of [math]-solution of the NLSE. For the semidiscretization in time by the first-order Gautschi-type