SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2784-2787, December 2024. Abstract. This note is the correction of an error in the proof of Theorem 4.1 in [Q. Cheng and J. Shen, SIAM J. Numer. Anal., 60 (2022), pp. 970–998].

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2782-2783, December 2024. Abstract. We correct a mistake in the paper [P. Manns and C. Kirches, SIAM J. Numer. Anal., 58 (2020), pp. 3427–3447]. The grid refinement strategy in Definition 4.3 needs to ensure that the order of the (sets of) grid cells that are successively refined is preserved over all grid iterations. This was only partially

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2745-2781, December 2024. Abstract. We introduce a novel method, called swarm-based simulated annealing (SSA), for nonconvex optimization which is at the interface between the swarm-based gradient-descent (SBGD) [J. Lu et al., arXiv:2211.17157; E. Tadmor and A. Zenginoglu, Acta Appl. Math., 190 (2024)] and simulated annealing (SA) [V. Cerny

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2719-2744, December 2024. Abstract. A novel family of high-order structure-preserving methods is proposed for the nonlinear Schrödinger equation. The methods are developed by applying the multiple relaxation idea to the exponential Runge–Kutta methods. It is shown that the multiple relaxation exponential Runge–Kutta methods can achieve high-order

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2698-2718, December 2024. Abstract. The computation of global radial basis function (RBF) approximations requires the solution of a linear system which, depending on the choice of RBF parameters, may be ill-conditioned. We study the stability and accuracy of approximation methods using the Gaussian RBF in all scaling regimes of the associated

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2667-2697, December 2024. Abstract. We develop a two-stage, second-order, global-in-time energy stable implicit-explicit Runge–Kutta (IMEX RK(2, 2)) scheme for the phase field crystal equation with an [math] time step constraint, and without the global Lipschitz assumption. A linear stabilization term is introduced to the system with Fourier

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2640-2666, December 2024. Abstract. In this article, we show the existence of minimizers in the loss landscape for residual artificial neural networks (ANNs) with a multidimensional input layer and one hidden layer with ReLU activation. Our work contrasts with earlier results in [D. Gallon, A. Jentzen, and F. Lindner, preprint, arXiv:2211

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2611-2639, December 2024. Abstract. In recent years, stochastic partial differential equations (SPDEs) have become a well-studied field in mathematics. With their increase in popularity, it becomes important to efficiently approximate their solutions. Thus, our goal is a contribution towards the development of efficient and practical time-stepping

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2588-2610, December 2024. Abstract. We propose to use Neumann–Neumann algorithms for the time parallel solution of unconstrained linear parabolic optimal control problems. We study nine variants, analyze their convergence behavior, and determine the optimal relaxation parameter for each. Our findings indicate that while the most intuitive

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2549-2587, December 2024. Abstract. The ensemble Kalman filter is widely used in applications because, for high-dimensional filtering problems, it has a robustness that is not shared, for example, by the particle filter; in particular, it does not suffer from weight collapse. However, there is no theory which quantifies its accuracy as an

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2529-2548, December 2024. Abstract. We reformulate the Lanczos tau method for the discretization of time-delay systems in terms of a pencil of operators, allowing for new insights into this approach. As a first main result, we show that, for the choice of a shifted Legendre basis, this method is equivalent to Padé approximation in the frequency

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2506-2528, December 2024. Abstract. A spherical [math]-design is a set of points on the unit sphere, which provides an equal weight quadrature rule integrating exactly all spherical polynomials of degree at most [math] and has a sharp error bound for approximations on the sphere. This paper introduces a set of points called a spherical cap

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2484-2505, December 2024. Abstract. This paper aims to address two issues of integral equations for the scattering of time-harmonic electromagnetic waves by a perfect electric conductor with Lipschitz continuous boundary: ill-conditioned boundary element Galerkin discretization matrices on fine meshes and instability at spurious resonant

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2459-2483, December 2024. Abstract. This work develops an energy-based discontinuous Galerkin (EDG) method for the nonlinear Schrödinger equation with the wave operator. The focus of the study is on the energy-conserving or energy-dissipating behavior of the method with some simple mesh-independent numerical fluxes we designed. We establish

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2439-2458, December 2024. Abstract. An a posteriori error estimator based on an equilibrated flux reconstruction is proposed for defeaturing problems in the context of finite element discretizations. Defeaturing consists in the simplification of a geometry by removing features that are considered not relevant for the approximation of the

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2419-2438, December 2024. Abstract. In this paper, we study the a posteriori error estimates of the FEM for defective eigenvalues of non-self-adjoint eigenvalue problems. Using the spectral approximation theory, we establish the abstract a posteriori error formulas for the weighted average of approximate eigenvalues and approximate eigenspace

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2415-2417, October 2024. Abstract. We correct the proofs of Theorems 3.3 and 5.2 in [Y. A. P. Osborne and I. Smears, SIAM J. Numer. Anal., 62 (2024), pp. 138–166]. With the corrected proofs, Theorems 3.3 and 5.2 are shown to be valid without change to their hypotheses or conclusions.

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2393-2414, October 2024. Abstract. We consider the problem of estimating an expectation [math] by quasi-Monte Carlo (QMC) methods, where [math] is an unbounded smooth function and [math] is a standard normal random vector. While the classical Koksma–Hlawka inequality cannot be directly applied to unbounded functions, we establish a novel

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2370-2392, October 2024. Abstract. Numerical integration over the real line for analytic functions is studied. Our main focus is on the sharpness of the error bounds. We first derive two general lower estimates for the worst-case integration error, and then apply these to establish lower bounds for various quadrature rules. These bounds turn

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2349-2369, October 2024. Abstract. In this article, we impose fractal features onto classical multiquadric (MQ) functions. This generates a novel class of fractal functions, called fractal MQ functions, where the symmetry of the original MQ function with respect to the origin is maintained. This construction requires a suitable extension

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2331-2348, October 2024. Abstract. From the literature, it is known that the choice of basis functions in hp-FEM heavily influences the computational cost in order to obtain an approximate solution. Depending on the choice of the reference element, suitable tensor product like basis functions of Jacobi polynomials with different weights lead

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2308-2330, October 2024. Abstract. In this paper, we study the convergence properties of the parareal algorithm with uniform coarse time grid and arbitrarily distributed (nonuniform) fine time grid, which may be changed at each iteration. We employ the backward-Euler method as the coarse propagator and a general single-step time-integrator

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2276-2307, October 2024. Abstract. The focus of this paper is on the concurrent reconstruction of both the diffusion and potential coefficients present in an elliptic/parabolic equation, utilizing two internal measurements of the solutions. A decoupled algorithm is constructed to sequentially recover these two parameters. In the first step

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2249-2275, October 2024. Abstract. Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the selection of optimal function sampling locations is a central problem

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2222-2248, October 2024. Abstract. In this paper, we present the stability and error analysis of two fully discrete IMEX-LDG schemes, combining local discontinuous Galerkin spatial discretization with implicit-explicit Runge–Kutta temporal discretization, for the linearized one-dimensional KdV equations. The energy stability analysis begins

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2196-2221, October 2024. Abstract. We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2172-2195, October 2024. Abstract. A novel evolving surface finite element method, based on a novel equivalent formulation of the continuous problem, is proposed for computing the evolution of a closed hypersurface moving under a prescribed velocity field in two- and three-dimensional spaces. The method improves the mesh quality of the approximate

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2143-2171, October 2024. Abstract. The coupling of nonconservative hyperbolic systems at a static interface has been a delicate issue as common approaches rely on the Lax-curves of the systems, which are not always available. To address this a new linear relaxation system is introduced, in which a nonlocal source term accounts for the nonconservative

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2121-2142, October 2024. Abstract. This paper studies the discretization of a homogenization and dimension reduction model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Velčić [Calc. Var. Partial Differential Equations, 51 (2014), pp. 677–699]. Thereby, a nonlinear bending energy is based on

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2087-2120, October 2024. Abstract. Along with the practical success of the discovery of dynamics using deep learning, the theoretical analysis of this approach has attracted increasing attention. Prior works have established the grid error estimation with auxiliary conditions for the discovery of dynamics using linear multistep methods and

SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2071-2086, October 2024. Abstract. For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2048-2070, August 2024. Abstract. We present new Dirichlet–Neumann and Neumann–Dirichlet algorithms with a time domain decomposition applied to unconstrained parabolic optimal control problems. After a spatial semidiscretization, we use the Lagrange multiplier approach to derive a coupled forward-backward optimality system, which can then

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2025-2047, August 2024. Abstract. Statistical inverse learning aims at recovering an unknown function [math] from randomly scattered and possibly noisy point evaluations of another function [math], connected to [math] via an ill-posed mathematical model. In this paper we blend statistical inverse learning theory with the classical regularization

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 2004-2024, August 2024. Abstract. We propose and analyze a novel approach to construct structure preserving approximations for the Poisson–Nernst–Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1979-2003, August 2024. Abstract. We introduce a new overlapping domain decomposition method (DDM) to solve fully nonlinear elliptic partial differential equations (PDEs) approximated with monotone schemes. While DDMs have been extensively studied for linear problems, their application to fully nonlinear PDEs remains limited in the literature

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1956-1978, August 2024. Abstract. In this article, we employ the construction of the time-marching discontinuous Petrov–Galerkin (DPG) scheme we developed for linear problems to derive high-order multistage DPG methods for nonlinear systems of ordinary differential equations. The methodology extends to abstract evolution equations in Banach

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1929-1955, August 2024. Abstract. In this work is considered an elliptic problem, referred to as the Ventcel problem, involving a second-order term on the domain boundary (the Laplace–Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a finite element discretization. The focus is on the construction

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1901-1928, August 2024. Abstract. We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schrödinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form [math] with [math] being the wave function and [math] being the exponent of the nonlinearity

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1874-1900, August 2024. Abstract. We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions, where the coefficients (and hence the solution) may depend on a parameter. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1844-1873, August 2024. Abstract. Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1814-1843, August 2024. Abstract. We propose and analyze discontinuous Galerkin (dG) approximations to 3D−1D coupled systems which model diffusion in a 3D domain containing a small inclusion reduced to its 1D centerline. Convergence to weak solutions of a steady state problem is established via deriving a posteriori error estimates and bounds

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1782-1813, August 2024. Abstract. Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluid-structure interactions (FSIs) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1759-1781, August 2024. Abstract. Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1736-1758, August 2024. Abstract. We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490–505] by introducing a unifying abstract framework, and we derive a finite-dimensional approximating system via pseudospectral discretization

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1713-1735, August 2024. Abstract. Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay or translation invariance. How to accurately recover these systems, especially for low-regularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1687-1712, August 2024. Abstract. In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in [math], for [math] in two spatial dimensions. This new analysis treats the positive and negative parts of

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1660-1686, August 2024. Abstract. We propose a finite element discretization for the steady, generalized Navier–Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1638-1659, August 2024. Abstract. Maximal regularity is a kind of a priori estimate for parabolic-type equations, and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping

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

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

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

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

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]

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

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

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

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

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

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

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