SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1209-1231, June 2025. Abstract. Transparent (or exact or nonreflecting) boundary conditions are essential to truncate infinite computational domains. Since transparent boundary conditions are usually nonlocal and expensive, they must be approximated. In this paper, we study such an approximation for the Helmholtz equation on an infinite strip

SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1183-1208, June 2025. Abstract. We study a high order absorbing boundary condition, the so-called complete radiation boundary condition (CRBC), for a time-harmonic electromagnetic wave propagation problem in a waveguide in [math]. The CRBC has been designed for an absorbing boundary condition for simulating wave propagations governed by the

SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1160-1182, June 2025. Abstract. We construct a structure preserving nonconforming finite element approximation scheme for the biharmonic wave maps into spheres equations. It satisfies a discrete energy law and preserves the nonconvex sphere constraint of the continuous problem. The discrete sphere constraint is enforced at the mesh-points

SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1128-1159, June 2025. Abstract. A long-standing and formidable challenge faced by all conservative numerical schemes for relativistic magnetohydrodynamics (RMHD) equations is the recovery of primitive variables from conservative ones. This process involves solving highly nonlinear equations subject to physical constraints. An ideal solver

SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1105-1127, June 2025. Abstract. We propose a new stabilized finite element method for the classical Kolmogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and, crucially, is characterized by degenerate diffusion. The stabilization is constructed so that the resulting method admits a numerical

SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1078-1104, June 2025. Abstract. This paper addresses the challenge of constructing finite element [math] complexes in three dimensions. Tangential-normal continuity is introduced in order to develop distributional finite element [math] complexes. The spaces constructed are applied to discretize the quad curl problem, demonstrating optimal

SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1055-1077, June 2025. Abstract. We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous [math] coefficients. The methods are of Galerkin type and follow the Variational Multiscale and Localized Orthogonal Decomposition (LOD) approaches

SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1025-1054, June 2025. Abstract. In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncertainty quantification (UQ). The distribution and density functions correspond to a quantity of interest involving the solution to

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 1000-1024, April 2025. Abstract. In this paper, we propose a new class of splitting methods to solve the stochastic Langevin equation, which can simultaneously preserve the ergodicity and exponential integrability of the original equation. The central idea is to extract a stochastic subsystem that possesses the strict dissipation from the

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 976-999, April 2025. Abstract. This work is on a user-friendly reduced basis method for the solution of families of parametric partial differential equations by preconditioned Krylov subspace methods including the conjugate gradient method, generalized minimum residual method, and biconjugate gradient method. The proposed methods use a preconditioned

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 949-975, April 2025. Abstract. Interpolatory necessary optimality conditions for [math]-optimal reduced-order modeling of unstructured linear time-invariant (LTI) systems are well-known. Based on previous work on [math]-optimal reduced-order modeling of stationary parametric problems, in this paper, we develop and investigate optimality conditions

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 931-948, April 2025. Abstract. We present new convergence estimates of generalized empirical interpolation methods in terms of the entropy numbers of the parametrized function class. Our analysis is transparent and leads to sharper convergence rates than the classical analysis via the Kolmogorov [math]-width. In addition, we also derive novel

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 908-930, April 2025. Abstract. The structured [math]-stability radius is introduced as a quantity to assess the robustness of transient bounds of solutions to linear differential equations under structured perturbations of the matrix. This applies to general linear structures such as complex or real matrices with a given sparsity pattern

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 881-907, April 2025. Abstract. A simple iterative approach for solving a set of implicit kinetic moment equations is proposed. This implicit solve is a key component in the IMEX discretization of the multi-species Bhatnagar–Gross–Krook (M-BGK) model with nontrivial collision frequencies depending on individual species temperatures. We prove

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 854-880, April 2025. Abstract. In this paper, we propose two efficient fully discrete schemes for [math]-tensor flow of liquid crystals by using the first- and second-order stabilized exponential scalar auxiliary variable (sESAV) approach in time and the finite difference method for spatial discretization. The modified discrete energy dissipation

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 827-853, April 2025. Abstract. The hybrid high-order (HHO) scheme has many successful applications including linear elasticity as the first step towards computational solid mechanics. The striking advantage is the simplicity among other higher-order nonconforming schemes and its geometric flexibility as a polytopal method on the expanse of

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 800-826, April 2025. Abstract. This paper studies discretization of time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Most of the analysis in the literature has been performed on fully discrete methods using first order methods in time, typically the implicit Euler time

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 772-799, April 2025. Abstract. The unique solvability and error analysis of a scheme using the original Lagrange multiplier approach proposed in [Q. Cheng, C. Liu, and J. Shen, Comput. Methods Appl. Mech. Engrg., 367 (2020), 13070] for gradient flows is studied in this paper. We identify a necessary and sufficient condition that must be satisfied

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 744-771, April 2025. Abstract. This paper is concerned with the convergence theory of a perfectly matched layer (PML) method for wave scattering problems in a half plane bounded by a step-like surface. When a plane wave impinges upon the surface, the scattered waves are composed of an outgoing radiative field and two known parts. The first

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 716-743, April 2025. Abstract. In this paper we are concerned with a weighted additive Schwarz method with local impedance boundary conditions for a family of Helmholtz problems in two or three dimensions. These problems are discretized by the finite element method with conforming nodal finite elements. We design and analyze an adaptive coarse

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 685-715, April 2025. Abstract. Ensemble Kalman inversion (EKI) is an ensemble-based method to solve inverse problems. Its gradient-free formulation makes it an attractive tool for problems with involved formulation. However, EKI suffers from the “subspace property,” i.e., the EKI solutions are confined in the subspace spanned by the initial

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 661-684, April 2025. Abstract. The paper analyzes the discontinuous Galerkin approximation of Maxwell’s equations written in first-order form and with nonhomogeneous magnetic permeability and electric permittivity. Although the Sobolev smoothness index of the solution may be smaller than [math], it is shown that the approximation converges

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 641-660, April 2025. Abstract. This contribution presents an exponential stability criterion for linear systems with multiple pointwise and distributed delays. This result is obtained in the Lyapunov–Krasovskii framework via the approximations of the argument of the functional by projection on the first Legendre polynomials. The reduction

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 619-640, April 2025. Abstract. Mesh quality is crucial in the simulation of surface evolution equations using parametric finite element methods (FEMs). Energy-diminishing schemes may fail even when the surface remains smooth due to poor mesh distribution. In this paper, we aim to develop mesh-preserving and energy-stable parametric finite

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 588-618, April 2025. Abstract. The multiscale hybrid-mixed (MHM) method for the Stokes operator was formally introduced in [R. Araya et al., Comput. Methods Appl. Mech. Engrg., 324, pp. 29–53, 2017] and numerically validated. The method has face degrees of freedom associated with multiscale basis functions computed from local Neumann problems

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 564-587, April 2025. Abstract. In this paper, we propose a new algorithm, the irrational-window-filter projection method (IWFPM), for quasiperiodic systems with concentrated spectral point distribution. Based on the projection method (PM), IWFPM filters out dominant spectral points by defining an irrational window and uses a corresponding

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 542-563, April 2025. Abstract. Efficient simulation of stochastic partial differential equations (SPDEs) on general domains requires noise discretization. This paper employs piecewise linear interpolation of noise in a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 520-541, April 2025. Abstract. Crystalline materials exhibit long-range elastic fields due to the presence of defects, leading to significant domain size effects in atomistic simulations. A rigorous far-field expansion of these long-range fields identifies low-rank structure in the form of a sum of discrete multipole terms and continuum predictors

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