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Generalized Dimension Truncation Error Analysis for High-Dimensional Numerical Integration: Lognormal Setting and Beyond SIAM J. Numer. Anal. (IF 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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 2.9) 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
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Asymptotic-Preserving and Energy Stable Dynamical Low-Rank Approximation SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2024-01-10 Lukas Einkemmer, Jingwei Hu, Jonas Kusch
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 73-92, February 2024. Abstract. Radiation transport problems are posed in a high-dimensional phase space, limiting the use of finely resolved numerical simulations. An emerging tool to efficiently reduce computational costs and memory footprint in such settings is dynamical low-rank approximation (DLRA). Despite its efficiency, numerical
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Algebraic Structure of the Weak Stage Order Conditions for Runge–Kutta Methods SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2024-01-04 Abhijit Biswas, David Ketcheson, Benjamin Seibold, David Shirokoff
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 48-72, February 2024. Abstract. Runge–Kutta (RK) methods may exhibit order reduction when applied to stiff problems. For linear problems with time-independent operators, order reduction can be avoided if the method satisfies certain weak stage order (WSO) conditions, which are less restrictive than traditional stage order conditions. This
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Lattice Green’s Functions for High-Order Finite Difference Stencils SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2024-01-04 James Gabbard, Wim M. van Rees
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 25-47, February 2024. Abstract. Lattice Green’s functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of existing numerical solvers that make use of LGFs rely on a second-order
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Learning High-Dimensional McKean–Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2024-01-04 Jiequn Han, Ruimeng Hu, Jihao Long
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 1-24, February 2024. Abstract. One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments
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Geometric Ergodicity for Hamiltonian Monte Carlo on Compact Manifolds SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-12-07 Kota Takeda, Takashi Sakajo
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2994-3013, December 2023. Abstract. We consider a Markov chain Monte Carlo method, known as Hamiltonian Monte Carlo (HMC), on compact manifolds in Euclidean space. It utilizes Hamiltonian dynamics to generate samples approximating a target distribution in high dimensions efficiently. The efficiency of HMC is characterized by its convergence
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Analysis of a sinc-Galerkin Method for the Fractional Laplacian SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-12-06 Harbir Antil, Patrick W. Dondl, Ludwig Striet
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2967-2993, December 2023. Abstract. We provide the convergence analysis for a [math]-Galerkin method to solve the fractional Dirichlet problem. This can be understood as a follow-up of [H. Antil, P. Dondl, and L. Striet, SIAM J. Sci. Comput., 43 (2021), pp. A2897–A2922], where the authors presented a [math]-function based method to solve
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The Discontinuous Galerkin Approximation of the Grad-Div and Curl-Curl Operators in First-Order Form Is Involution-Preserving and Spectrally Correct SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-12-06 Alexandre Ern, Jean-Luc Guermond
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2940-2966, December 2023. Abstract. The discontinuous Galerkin approximation of the grad-div and curl-curl problems formulated in conservative first-order form is investigated. It is shown that the approximation is spectrally correct, thereby confirming numerical observations made by various authors in the literature. This result hinges on
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Higher-Order Convergence of Perfectly Matched Layers in Three-Dimensional Biperiodic Surface Scattering Problems SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-12-05 Ruming Zhang
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2917-2939, December 2023. Abstract. The perfectly matched layer (PML) is a very popular tool in the truncation of wave scattering in unbounded domains. In [S. N. Chandler-Wilde and P. Monk, Appl. Numer. Math., 59 (2009), pp. 2131–2154], the author proposed a conjecture that for scattering problems with rough surfaces, the PML converges exponentially
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Convergent FEM for a Membrane Model of Liquid Crystal Polymer Networks SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-28 Lucas Bouck, Ricardo H. Nochetto, Shuo Yang
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2887-2916, December 2023. Abstract. We design a finite element method for a membrane model of liquid crystal polymer networks. This model consists of a minimization problem of a nonconvex stretching energy. We discuss properties of this energy functional such as lack of weak lower semicontinuity. We devise a discretization with regularization
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Implicit-Explicit Time Discretization for Oseen’s Equation at High Reynolds Number with Application to Fractional Step Methods SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-28 Erik Burman, Deepika Garg, Johnny Guzman
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2859-2886, December 2023. Abstract. In this paper we consider the application of implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations. The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly. Both the second-order backward differentiation
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Leapfrog Methods for Relativistic Charged-Particle Dynamics SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-27 Ernst Hairer, Christian Lubich, Yanyan Shi
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2844-2858, December 2023. Abstract. A basic leapfrog integrator and its energy-preserving and variational/symplectic variants are proposed and studied for the numerical integration of the equations of motion of relativistic charged particles in an electromagnetic field. The methods are based on a four-dimensional formulation of the equations
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Fast Krasnosel’skiĭ–Mann Algorithm with a Convergence Rate of the Fixed Point Iteration of [math] SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-20 Radu Ioan Boţ, Dang-Khoa Nguyen
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2813-2843, December 2023. Abstract. The Krasnosel’skiĭ–Mann (KM) algorithm is the most fundamental iterative scheme designed to find a fixed point of an averaged operator in the framework of a real Hilbert space, since it lies at the heart of various numerical algorithms for solving monotone inclusions and convex optimization problems. We
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Harmonic Functions on Finitely Connected Tori SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-20 Chiu-Yen Kao, Braxton Osting, Édouard Oudet
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2795-2812, December 2023. Abstract. In this paper, we prove a logarithmic conjugation theorem on finitely connected tori. The theorem states that a harmonic function can be written as the real part of a function whose derivative is analytic and a finite sum of terms involving the logarithm of the modulus of a modified Weierstrass sigma function
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Relaxed Kačanov Scheme for the [math]-Laplacian with Large Exponent SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-17 Anna Kh. Balci, Lars Diening, Johannes Storn
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2775-2794, December 2023. Abstract. We introduce a novel relaxed Kačanov scheme for the computation of the discrete minimizer to the [math]-Laplace problem with [math]. The iterative scheme is easy to implement since each iterate results only from the solve of a weighted, linear Poisson problem. It neither requires an additional line search
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Monte Carlo Gradient in Optimization Constrained by Radiative Transport Equation SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-17 Qin Li, Li Wang, Yunan Yang
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2744-2774, December 2023. Abstract. Can Monte Carlo (MC) solvers be directly used in gradient-based methods for PDE-constrained optimization problems? In these problems, a gradient of the loss function is typically presented as a product of two PDE solutions, one for the forward equation and the other for the adjoint. When MC solvers are
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An Energy Stable and Maximum Bound Principle Preserving Scheme for the Dynamic Ginzburg–Landau Equations under the Temporal Gauge SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-16 Limin Ma, Zhonghua Qiao
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2695-2717, December 2023. Abstract. This paper proposes a decoupled numerical scheme of the time-dependent Ginzburg–Landau equations under the temporal gauge. For the magnetic potential and the order parameter, the discrete scheme adopts the second type Nedélec element and the linear element for spatial discretization, respectively; and a
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A Probabilistic Scheme for Semilinear Nonlocal Diffusion Equations with Volume Constraints SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-15 Minglei Yang, Guannan Zhang, Diego Del-Castillo-Negrete, Yanzhao Cao
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2718-2743, December 2023. Abstract. This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integro-differential operator
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Convergence of a Decoupled Splitting Scheme for the Cahn–Hilliard–Navier–Stokes System SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-15 Chen Liu, Rami Masri, Beatrice Riviere
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2651-2694, December 2023. Abstract. This paper is devoted to the analysis of an energy-stable discontinuous Galerkin algorithm for solving the Cahn–Hilliard–Navier–Stokes equations within a decoupled splitting framework. We show that the proposed scheme is uniquely solvable and mass conservative. The energy dissipation and the [math] stability
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Numerical Solution of Free Stochastic Differential Equations SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-15 Georg Schlüchtermann, Michael Wibmer
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2623-2650, December 2023. Abstract. This paper derives a free analogue of the Euler–Maruyama method (fEMM) to numerically approximate solutions of free stochastic differential equations (fSDEs). Simply speaking, fSDEs are SDEs in the context of noncommutative random variables (e.g., large random matrices). By applying the theory of multiple
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Exponential Convergence of [math]-FEM for the Integral Fractional Laplacian in Polygons SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-15 Markus Faustmann, Carlo Marcati, Jens Markus Melenk, Christoph Schwab
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2601-2622, December 2023. Abstract. We prove exponential convergence in the energy norm of [math]-finite element discretizations for the integral fractional Laplacian of order [math] subject to homogeneous Dirichlet boundary conditions in bounded polygonal domains [math]. Key ingredients in the analysis are the weighted analytic regularity
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Resolution of Singularities by Rational Functions SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-15 Astrid Herremans, Daan Huybrechs, Lloyd N. Trefethen
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2580-2600, December 2023. Abstract. Results on the rational approximation of functions containing singularities are presented. We build further on the “lightning method,” recently proposed by Trefethen and Gopal [SIAM J. Numer. Anal., 57 (2019), pp. 2074–2094], based on exponentially clustering poles close to the singularities. Our results
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High-Order BDF Convolution Quadrature for Subdiffusion Models with a Singular Source Term SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-11-03 Jiankang Shi, Minghua Chen
SIAM Journal on Numerical Analysis, Volume 61, Issue 6, Page 2559-2579, December 2023. Abstract. Anomalous diffusion is often modelled in terms of the subdiffusion equation, which can involve a weakly singular source term. For this case, many predominant time-stepping methods, including the correction of high-order backward differentiation formula (BDF) schemes [B. Jin, B. Y. Li, and Z. Zhou, SIAM
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The Unique Continuation Problem for the Heat Equation Discretized with a High-Order Space-Time Nonconforming Method SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-10-25 Erik Burman, Guillaume Delay, Alexandre Ern
SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2534-2557, October 2023. Abstract. We are interested in solving the unique continuation problem for the heat equation, i.e., we want to reconstruct the solution of the heat equation in a target space-time subdomain given its (noised) value in a subset of the computational domain. Both initial and boundary data can be unknown. We discretize
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Convergence of a Time Discrete Scheme for a Chemotaxis-Consumption Model SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-10-24 Francisco Guillén-González, André Luiz Corrêa Vianna Filho
SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2509-2533, October 2023. Abstract. In the present work we propose and study a time discrete scheme for the following chemotaxis-consumption model (for any [math]): [math] endowed with isolated boundary conditions and initial conditions, where [math] model cell density and chemical signal concentration. The proposed scheme is defined via a
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Mixed and Multipoint Finite Element Methods for Rotation-Based Poroelasticity SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-10-20 Wietse M. Boon, Alessio Fumagalli, Anna Scotti
SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2485-2508, October 2023. Abstract. This work proposes a mixed finite element method for the Biot poroelasticity equations that employs the lowest-order Raviart–Thomas finite element space for the solid displacement and piecewise constants for the fluid pressure. The method is based on the formulation of linearized elasticity as a weighted
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Nonconforming Virtual Elements for the Biharmonic Equation with Morley Degrees of Freedom on Polygonal Meshes SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-10-19 Carsten Carstensen, Rekha Khot, Amiya K. Pani
SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2460-2484, October 2023. Abstract. The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution [math] to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces [math] and a smoother allows rough source
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Convergence of the Hiptmair–Xu Preconditioner for [math]-Elliptic Problems with Jump Coefficients (ii): Main Results SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-10-19 Qiya Hu
SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2434-2459, October 2023. Abstract. This paper is the second of two articles, in which we aim to prove the convergence of the Hiptmair–Xu (HX) preconditioner (originally proposed by [R. Hiptmair and J. Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483–2509]) for [math]-elliptic boundary value problems with jump coefficients. In this paper, based
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Stability and Error Analysis of a Second-Order Consistent Splitting Scheme for the Navier–Stokes Equations SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-10-18 Fukeng Huang, Jie Shen
SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2408-2433, October 2023. Abstract. We present in this paper a new second-order consistent splitting scheme for the Navier–Stokes equations with no-slip boundary conditions based on (i) the Taylor expansions at time [math] which offer better stability than the usual expansion at time [math], and (ii) the generalized scalar auxiliary variable
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Numerical Attractors for Rough Differential Equations SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-10-18 Luu Hoang Duc, Peter Kloeden
SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2381-2407, October 2023. Abstract. We study the explicit Euler scheme to approximate the solutions of rough differential equations under a bounded or linear diffusion term, where the drift term satisfies a local Lipschitz continuity and a one-sided linear growth condition. The Euler scheme is then proved to converge for a given solution,
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A Posteriori Error Estimates for Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes SIAM J. Numer. Anal. (IF 2.9) Pub Date : 2023-10-17 Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis
SIAM Journal on Numerical Analysis, Volume 61, Issue 5, Page 2352-2380, October 2023. Abstract. We present a new residual-type energy-norm a posteriori error analysis for interior penalty discontinuous Galerkin (dG) methods for linear elliptic problems. The new error bounds are also applicable to dG methods on meshes consisting of elements with very general polygonal/polyhedral shapes. The case of