SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page B1081-B1104, January 2021. The sensitivity equation method aims at estimating the derivative of the solution of partial differential equations with respect to a parameter of interest. The objective of this work is to investigate the ability of an isogeometric discontinuous Galerkin (DG) method to evaluate accurately sensitivities with respect

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3372-A3392, January 2021. We present an extension of the AAA (adaptive Antoulas--Anderson) algorithm for periodic functions, called “AAAtrig.” The algorithm uses the key steps of AAA approximation by (i) representing the approximant in (trigonometric) barycentric form and (ii) selecting the support points greedily. Accordingly, AAAtrig

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3357-A3371, January 2021. We construct integrators to be used in Hamiltonian (or Hybrid) Monte Carlo sampling. The new integrators are easily implementable and, for a given computational budget, may deliver five times as many accepted proposals as standard leapfrog/Verlet without impairing in any way the quality of the samples. They are

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3328-A3356, January 2021. We introduce a new theoretical framework for deriving lower bounds on data movement in bilinear algorithms. Bilinear algorithms are a general representation of fast algorithms for bilinear functions, which include computation of matrix multiplication, convolution, and symmetric tensor contractions. A bilinear

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page B1057-B1080, January 2021. It has recently been demonstrated that dynamical low-rank algorithms can provide robust and efficient approximations to a range of kinetic equations. This is true especially if the solution is close to some asymptotic limit where it is known that the solution is low-rank. A particularly interesting case is the

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3305-A3327, January 2021. In this paper, we propose and analyze a time stepping method for the time fractional Allen--Cahn equation. The key property of the proposed method is its unconditional stability for general meshes, including the graded mesh commonly used for this type of equation. The unconditional stability is proved through

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3269-A3304, January 2021. An $n$-cross field is a locally defined orthogonal coordinate system invariant with respect to the hyperoctahedral symmetry group (cubic for $n=3$). Cross fields are finding wide-spread use in mesh generation, computer graphics, and materials science among many applications. It was recently shown in [A. Chemin

SIAM Journal on Scientific Computing, Ahead of Print. PDE-constrained optimization problems are often treated using the reduced formulation where the PDE constraints are eliminated. This approach is known to be more computationally feasible than other alternatives at large scales. However, the elimination of the constraints forces the optimization process to fulfill the constraints at all times. In

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3244-A3268, January 2021. When using Laguerre and Hermite spectral methods to numerically solve PDEs in unbounded domains, the number of collocation points assigned inside the region of interest is often insufficient, particularly when the region is expanded or translated in order to safely capture the unknown solution. Simply increasing

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3212-A3243, January 2021. Well known to the machine learning community, the random feature model is a parametric approximation to kernel interpolation or regression methods. It is typically used to approximate functions mapping a finite-dimensional input space to the real line. In this paper, we instead propose a methodology for use of

SIAM Journal on Scientific Computing, Ahead of Print. We consider a second-order elliptic PDE discretized by the hybrid high-order method, for which globally coupled unknowns are located at faces. To efficiently solve the resulting linear system, we propose a geometric multigrid algorithm that keeps the degrees of freedom on the faces at every grid level. The core of the algorithm lies in the design

SIAM Journal on Scientific Computing, Ahead of Print. The hybrid ML-FETI-DP algorithm combines the advantages of adaptive coarse spaces in domain decomposition methods and certain supervised machine learning techniques. Adaptive coarse spaces ensure robustness of highly scalable domain decomposition solvers, even for highly heterogeneous coefficient distributions with arbitrary coefficient jumps. However

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3184-A3211, January 2021. The so-called fast inertial relaxation engine is a first-order method for unconstrained smooth optimization problems. It updates the search direction by a linear combination of the past search direction, the current gradient, and the normalized gradient direction. We explore more general combination rules and

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3155-A3183, January 2021. We present a novel particle management method using the characteristic mapping framework. In the context of explicit evolution of parametrized curves and surfaces, the surface distribution of marker points created from sampling the parametric space is controlled by the area element of the parametrization function

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3135-A3154, January 2021. In this paper, we introduce a numerical method for nonlinear parabolic partial differential equations (PDEs) that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3114-A3134, January 2021. This paper focuses on vector random fields defined on $\mathbb{S}^d\times \mathbb{R}^k$, $d \geq 2$ and $k \geq 1$, with covariance functions that depend on the geodesic distance in $\mathbb{S}^d$ and on the separation vector in $\mathbb{R}^k$. First, we propose parametric families of nonseparable covariance functions

SIAM Journal on Scientific Computing, Ahead of Print. This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues located inside a given disk, and the proposed algorithms approximate these eigenvalues and associated eigenvectors by harmonic Rayleigh--Ritz projections

SIAM Journal on Scientific Computing, Ahead of Print. We propose an unconditionally robust and highly effective preconditioner for general dense symmetric positive definite matrices based on structured incomplete factorization (SIF), called the enhanced SIF (eSIF) preconditioner. The original SIF strategy proposed recently derives a structured preconditioner by applying block diagonal preprocessing

SIAM Journal on Scientific Computing, Ahead of Print. Least squares form one of the most prominent classes of optimization problems with numerous applications in scientific computing and data fitting. When such formulations aim at modeling complex systems, the optimization process must account for nonlinear dynamics by incorporating constraints. In addition, these systems often incorporate a large

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3082-A3113, January 2021. This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3055-A3081, January 2021. The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constraints are often imposed as soft penalties during model training and effectively act as domain-specific regularizers

SIAM Journal on Scientific Computing, Volume 43, Issue 5, Page A3033-A3054, January 2021. A selection of algorithms for the rational approximation of matrix-valued functions are discussed, including variants of the interpolatory adaptive Antoulas--Anderson (AAA) method, the rational Krylov fitting (RKFIT) method based on approximate least squares fitting, vector fitting, and a method based on low-rank

SIAM Journal on Scientific Computing, Ahead of Print. In this paper, we study a posteriori error estimators which aid multilevel iterative solvers for linear systems of graph Laplacians. In earlier works such estimates were computed by solving a perturbed global optimization problem, which could be computationally expensive. We propose a novel strategy to compute these estimates by constructing a Helmholtz

SIAM Journal on Scientific Computing, Ahead of Print. We present PNKH-B, a projected Newton--Krylov method for iteratively solving large-scale optimization problems with bound constraints. PNKH-B is geared toward situations in which function and gradient evaluations are expensive, and the (approximate) Hessian is only available through matrix-vector products. This is commonly the case in large-scale

SIAM Journal on Scientific Computing, Ahead of Print. Linear solvers for large and sparse systems are a key element of scientific applications, and their efficient implementation is necessary to harness the computational power of current computers. Algebraic MultiGrid (AMG) preconditioners are a popular ingredient of such linear solvers; this is the motivation for the present work, where we examine

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A3009-A3031, January 2021. An accurate troubled-cell indicator (TCI) is always desired by the discontinuous Galerkin methods for solving nonlinear hyperbolic conservation laws. In this paper, we will design a new TCI using K-means clustering. The values of a troubled-cell indication variable are in general of different magnitudes between

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page B1029-B1055, January 2021. The positivity preservation or hyperbolicity preservation of numerical schemes is essential for robust computations of compressible two-fluid models. Due to the quasi-conservative (nonconservative) nature of the compressible two-fluid model and use of the stiffened-gas equation of state that allows liquid-gas

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2981-A3008, January 2021. We develop an adjoint approach for recovering the topographical function included in the source term of one-dimensional hyperbolic balance laws. We focus on a specific system, namely, the shallow water equations, in an effort to recover the riverbed topography. The novelty of this work is the ability to robustly

SIAM Journal on Scientific Computing, Ahead of Print. The paper considers supervised learning problems of labeled data with grouped input features. The groups are nonoverlapped such that the model coefficients corresponding to the input features form disjoint groups. The coefficients have group sparsity structure in the sense that coefficients corresponding to each group shall be simultaneously either

SIAM Journal on Scientific Computing, Ahead of Print. PDE-constrained optimization problems arise in a broad number of applications such as hyperthermia cancer treatment and blood flow simulation. Discretization of the optimization problem and using a Lagrangian approach result in a large-scale saddle-point system, which is challenging to solve, and acquiring a full space-time solution is often infeasible

SIAM Journal on Scientific Computing, Ahead of Print. In this paper we present a geometric multigrid method with Jacobi and Vanka relaxation for hybridized and embedded discontinuous Galerkin discretizations of the Laplacian. We present a local Fourier analysis (LFA) of the two-grid error-propagation operator and show that the multigrid method applied to an embedded discontinuous Galerkin (EDG) discretization

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page B1008-B1028, January 2021. The electro-quasistatic approximation of Maxwell equations is commonly used to model coupled resistive/capacitive phenomena at low frequencies. It neglects induction and becomes unstable in the stationary limit. We introduce a stabilization that prevents this low-frequency breakdown. It results in a system for

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2949-A2980, January 2021. It is well known that symplectic integrators lose their near energy preservation properties when variable time-steps are used. The most common approach to combining adaptive time-steps and symplectic integrators involves the Poincaré transformation of the original Hamiltonian. In this article, we provide a framework

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2923-A2948, January 2021. We consider a moving domain, fluid-porohyperelastic structure interaction problem in a dual-mixed formulation. The fluid is described using the Navier--Stokes equations, and the porohyperelastic structure is described using the Biot equations. To solve this problem numerically, we propose two novel, partitioned

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2897-A2922, January 2021. Fueled by many applications in random processes, imaging science, geophysics, etc., fractional Laplacians have recently received significant attention. The key driving force behind the success of this operator is its ability to capture nonlocal effects while enforcing less smoothness on functions. In this article

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2869-A2896, January 2021. Computing a flow system a number of times with different samples of flow parameters is a common practice in many uncertainty quantification applications, which can be prohibitively expensive for complex nonlinear flow problems. This report presents two second order, stabilized, scalar auxiliary variable (SAV)

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2840-A2868, January 2021. When solving partial differential equations (PDEs) with random fields as coefficients, the efficient sampling of random field realizations can be challenging. In this paper we focus on the fast sampling of Gaussian fields using quasi-random points in a finite element and multilevel quasi Monte Carlo (MLQMC) setting

SIAM Journal on Scientific Computing, Ahead of Print. Multiple network alignment is the problem of identifying similar and related regions in a given set of networks. While there are a large number of effective techniques for pairwise problems with two networks that scale in terms of edges, these cannot be readily extended to align multiple networks as the computational complexity will tend to grow

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2809-A2839, January 2021. Sparsity regularization has been widely adopted in many fields, such as signal and image processing and machine learning. In this paper, we mainly consider nonconvex minimization problems involving three terms, for applications such as sparse signal recovery and low rank matrix recovery. We employ a three-operator

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page C312-C334, January 2021. Discontinuous Galerkin (dG) methods on meshes consisting of polygonal/polyhedral (henceforth, collectively termed as polytopic) elements have received considerable attention in recent years. Due to the physical frame basis functions used typically and the quadrature challenges involved, the matrix-assembly step

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page C290-C311, January 2021. Alternating least squares is the most widely used algorithm for CANDECOMC/PARAFAC (CP) tensor decomposition. However, alternating least squares may exhibit slow or no convergence, especially when high accuracy is required. An alternative approach is to regard CP decomposition as a nonlinear least squares problem

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page B984-B1007, January 2021. The mechanical behavior of a poroelastic medium permeated by multiple interacting fluid networks can be described by a system of time-dependent partial differential equations known as the multiple-network poroelasticity (MPET) equations or multiporosity/multipermeability systems. These equations generalize Biot's

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page B961-B983, January 2021. We develop robust solvers for a class of perturbed saddle-point problems arising in the study of a second-order elliptic equation in mixed form (in terms of flux and potential), and of the four-field formulation of Biot's consolidation problem for linear poroelasticity (using displacement, filtration flux, total

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2785-A2808, January 2021. Hybrid quantum/molecular mechanics models (QM/MM methods) are widely used in material and molecular simulations when MM models do not provide sufficient accuracy but pure QM models are computationally prohibitive. Adaptive QM/MM coupling methods feature on-the-fly classification of atoms during the simulation

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2766-A2784, January 2021. In [McDonald, Pestana, and Wathen, SIAM J. Sci. Comput., 40 (2018), pp. A1012--A1033], a block circulant preconditioner is proposed for all-at-once linear systems arising from evolutionary partial differential equations, in which the preconditioned matrix is proven to be diagonalizable and to have identity-plus-low-rank

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2737-A2765, January 2021. We extend previously developed two-level coarsening procedures for graph Laplacian problems written in a mixed saddle point form to the fully recursive multilevel case. The resulting hierarchy of discretizations gives rise to a hierarchy of upscaled models, in the sense that they provide approximation in the natural

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2714-A2736, January 2021. We consider a space-time variational formulation of parabolic initial-boundary value problems in anisotropic Sobolev spaces in combination with a Hilbert-type transformation. This variational setting is the starting point for the space-time Galerkin finite element discretization that leads to a large global linear

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page B937-B960, January 2021. We develop a random batch Ewald (RBE) method for molecular dynamics simulations of particle systems with long-range Coulomb interactions, which achieves an $O(N)$ complexity in each step of simulating $N$-body systems. The RBE method is based on the Ewald splitting for the Coulomb kernel with a random “minibatch”

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2685-A2713, January 2021. Data points from many recent real applications usually have a multiview structure in the sense that they are drawn from a multivariate random variable $\mathtt{v}\in \mathbb{R}^n$ that can be partitioned into multiple, say, $m$, subvariables (i.e., multiview) $\mathtt{v}_i\in \mathbb{R}^{n_i}$ for $i=1,\ldots

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2660-A2684, January 2021. In recent years, contour-based eigensolvers have emerged as a standard approach for the solution of large and sparse eigenvalue problems. Building upon recent performance improvements through nonlinear least-squares optimization of so-called rational filters, we introduce a systematic method to design these filters

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2638-A2659, January 2021. Maxwell's equations are a system of partial differential equations that govern the laws of electromagnetic induction. We study a mimetic finite-difference (MFD) discretization of the equations which preserves important underlying physical properties. We show that, after mass-lumping and appropriate scaling, the

SIAM Journal on Scientific Computing, Ahead of Print. We present a numerical method for synchronous and concurrent solution of transient elastodynamics problem where the computational domain is divided into subdomains that may reside on separate computational platforms. This work employs the variational multiscale discontinuous Galerkin (VMDG) method to develop interdomain transmission conditions for

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2614-A2637, January 2021. In this paper, we propose a new stabilizer-free and pressure-robust weak Galerkin finite element method for the Stokes equations with superconvergence on polytopal mesh in the primary velocity-pressure formulation. Convergence rates with one order higher than the optimal order for velocity in both the energy norm

SIAM Journal on Scientific Computing, Ahead of Print. A new class of distance-two interpolation methods for algebraic multigrid (AMG) that can be formulated in terms of sparse matrix-matrix multiplications is presented and analyzed. Compared with similar distance-two prolongation operators [H. De Sterck et al., Numer. Linear Algebra Appl., 15 (2008), pp. 115--139], the proposed algorithms exhibit improved

SIAM Journal on Scientific Computing, Ahead of Print. For the solution of large sparse unsymmetric systems, Duff et al. [SIAM J. Sci. Comput., 37 (2015), pp. A1248--A1269] proposed an approach based on the block Cimmino iterations [Numer. Math., 35 (1980), pp. 1--12], in which the solution is computed in a single iteration, so we call it a pseudodirect solver. In this approach, matrices are augmented

SIAM Journal on Scientific Computing, Ahead of Print. We consider three mathematically equivalent variants of the conjugate gradient (CG) algorithm and how they perform in finite precision arithmetic. It was shown in [Greenbaum, Lin. Alg. Appl., 113 (1989), pp. 7--63] that under certain conditions involving local orthogonality and approximate satisfaction of a recurrence formula, that may be satisfied

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page B907-B936, January 2021. In this paper, we consider numerical simulations of the nonlocal optical response of metallic nanostructure arrays inside a dielectric host, which is of particular interest to the nanoplasmonics community due to many unusual properties and potential applications. Mathematically, it is described by Maxwell's equations

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2581-A2613, January 2021. We propose a nonlinear manifold learning technique based on deep convolutional autoencoders that is appropriate for model order reduction of physical systems in complex geometries. Convolutional neural networks have proven to be highly advantageous for compressing data arising from systems demonstrating a slow-decaying

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2555-A2580, January 2021. We consider the problem of parameter estimation for a class of continuous-time state space models (SSMs). In particular, we explore the case of a partially observed diffusion, with data also arriving according to a diffusion process. Based upon a standard identity of the score function, we consider two particle

SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2526-A2554, January 2021. In many multiphysics applications, an ultimate quantity of interest can be written as a linear functional of the solution to the discretized governing nonlinear partial differential equations and finding a sufficiently accurate pointwise solution may be regarded as a step toward that end. In this paper, we derive