SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 495-519, April 2025. Abstract. Gaussian process regression is a classical kernel method for function estimation and data interpolation. In large data applications, computational costs can be reduced using low-rank or sparse approximations of the kernel. This paper investigates the effect of such kernel approximations on the interpolation

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 469-494, April 2025. Abstract. In this paper, we show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. This leads to a practical algorithm for piecewise polynomial interpolation

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

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 437-460, February 2025. Abstract. We introduce discretizations of infinite-dimensional optimization problems with total variation regularization and integrality constraints on the optimization variables. We advance the discretization of the dual formulation of the total variation term with Raviart–Thomas functions, which is known from the

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

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 396-421, February 2025. Abstract. This paper studies the long time stability of both the solution of a stochastic heat equation on a bounded domain driven by a correlated noise and its approximations. It is popular for researchers to prove the intermittency of the solution, which means that the moments of solution to a stochastic heat equation

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 360-395, February 2025. Abstract. In this work, we propose a numerical method to compute the Wasserstein Hamiltonian flow (WHF), which is a Hamiltonian system on the probability density manifold. Many well-known PDE systems can be reformulated as WHFs. We use the parameterized function as a push-forward map to characterize the solution of

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 334-359, February 2025. Abstract. Hutchinson’s estimator is a randomized algorithm that computes an [math]-approximation to the trace of any positive semidefinite matrix using [math] matrix-vector products. An improvement of Hutchinson’s estimator, known as [math], only requires [math] matrix-vector products. In this paper, we propose a generalization

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 306-333, February 2025. Abstract. We consider the equilibrium equations for a linearized Cosserat material and provide two perspectives concerning well-posedness. First, the system can be viewed as the Hodge–Laplace problem on a differential complex. On the other hand, we show how the Cosserat materials can be analyzed by inheriting results

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 288-305, February 2025. Abstract. For linear differential equations of the form [math], [math], with a possibly unbounded operator [math], we construct and deduce error bounds for two families of second-order exponential splittings. The role of quadratures when integrating the twice-iterated Duhamel’s formula is reformulated: we show that

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 262-287, February 2025. Abstract. Stochastic PDEs of fluctuating hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a multilevel Monte Carlo (MLMC) scheme for the Dean–Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 239-261, February 2025. Abstract. A space–time–parameters structure of parametric parabolic PDEs motivates the application of tensor methods to define reduced order models (ROMs). Within a tensor-based ROM framework, the matrix SVD—a traditional dimension reduction technique—yields to a low-rank tensor decomposition (LRTD). Such tensor extension

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 214-238, February 2025. Abstract. A high-frequency recovered fully discrete low-regularity integrator is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method, with high-frequency recovery techniques, can capture the discontinuities of the solutions correctly without spurious

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 193-213, February 2025. Abstract. Motivated by optimization with differential equations, we consider optimization problems with Hilbert spaces as decision spaces. As a consequence of their infinite dimensionality, the numerical solution necessitates finite dimensional approximations and discretizations. We develop an approximation framework

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 149-192, February 2025. Abstract. We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish [math] stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The [math] estimate, crucial

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 122-148, February 2025. Abstract. Extended dynamic mode decomposition (EDMD) is a data-driven tool for forecasting and model reduction of dynamics, which has been extensively taken up in the physical sciences. While the method is conceptually simple, in deterministic chaos it is unclear what its properties are or even what it converges to

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 103-121, February 2025. Abstract. We propose an energy-stable parametric finite element method (PFEM) for the planar Willmore flow and establish its unconditional energy stability of the full discretization scheme. The key lies in the introduction of two novel geometric identities to describe the planar Willmore flow: the first involves the

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 81-102, February 2025. Abstract. This work targets the discretization of contact-mechanics accounting for small strains, linear elastic constitutive laws, and fractures or faults represented as a network of co-dimension one planar interfaces. This type of model coupled with Darcy flow plays an important role typically for the simulation of

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 54-80, February 2025. Abstract. This study addresses some previously unexplored issues concerning the stability and error bounds of the primal hybrid finite element method. This method relaxes the strong interelement continuity conditions on the unknown [math] of a boundary-value problem, set in terms of a second-order elliptic partial differential

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 23-53, February 2025. Abstract. In this paper, we design a simple and convergent [math] linear finite element method for the linear second-order elliptic equation in nondivergence form and extend it to the Hamilton–Jacobi–Bellman equation. Motivated by the Miranda–Talenti estimate, we establish a discrete analogue of the estimate for the

SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 1-22, February 2025. Abstract. In the analysis of the [math]-version of the finite-element method (FEM), with fixed polynomial degree [math], applied to the Helmholtz equation with wavenumber [math], the asymptotic regime is when [math] is sufficiently small and the sequence of Galerkin solutions are quasioptimal; here [math] is the [math]

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