SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page B132-B166, January 2021. Mass-conservative reaction-diffusion systems have recently been proposed as a general framework to describe intracellular pattern formation. These systems have been used to model the conformational switching of proteins as they cycle from an inactive state in the cell cytoplasm, to an active state at the cell membrane

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A258-A277, January 2021. What is the fastest way to solve a linear system $Ax= b$ in arithmetic of a given precision when $A$ is symmetric positive definite and otherwise unstructured? The usual answer is by Cholesky factorization, assuming that $A$ can be factorized. We develop an algorithm that can be faster, given an arithmetic of precision

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page C31-C60, January 2021. A Fourier-based library of unbounded Poisson solvers (FLUPS) for 2D and 3D homogeneous distributed grids is presented. It is designed to handle every possible combination of periodic, symmetric, semi-unbounded, and fully unbounded boundary conditions for the Poisson equation on rectangular domains with uniform resolution

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A242-A257, January 2021. Mesh-free solvers for partial differential equations perform best on scattered quasi-uniform nodes. Computational efficiency can be improved by using nodes with greater spacing in regions of less activity. However, there is no ideal way to generate nodes for these solvers. We present an advancing front type method

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page C1-C30, January 2021. We present a study of the effectiveness of asynchronous incomplete LU factorization preconditioners for the time-implicit solution of compressible flow problems while exploiting thread-parallelism within a compute node. A block variant of the asynchronous fine-grained parallel preconditioner adapted to a finite volume

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page B108-B131, January 2021. The tensor train (TT) approximation of electronic wave functions lies at the core of the quantum chemistry density matrix renormalization group (QC-DMRG) method, a recent state-of-the-art method for numerically solving the $N$-electron Schrödinger equation. It is well known that the accuracy of TT approximations

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A221-A241, January 2021. This work develops novel time integration methods for the compressible Euler equations in the Lagrangian frame that are of arbitrary high order and exactly preserve the mass, momentum, and total energy of the system. The equations are considered in nonconservative form, that is, common for staggered grid hydrodynamics

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A193-A220, January 2021. We propose a fast algorithm for the calculation of the Wasserstein-1 distance, which is a particular type of optimal transport distance with transport cost homogeneous of degree one. Our algorithm is built on multilevel primal-dual algorithms. Several numerical examples and a complexity analysis are provided to

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page B82-B107, January 2021. In this work, we develop novel structure-preserving numerical schemes for a class of nonlinear Fokker--Planck equations with nonlocal interactions. Such equations can cover many cases of importance, such as porous medium equations with external potentials, optimal transport problems, and aggregation-diffusion models

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page B55-B81, January 2021. In this paper, we study nonnegative tensor data and propose an orthogonal nonnegative Tucker decomposition (ONTD). We discuss some properties of ONTD and develop a convex relaxation algorithm of the augmented Lagrangian function to solve the optimization problem. The convergence of the algorithm is given. We employ

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A164-A192, January 2021. Tensor network operators, such as the matrix product operator (MPO) and the projected entangled-pair operator (PEPO), can provide efficient representation of certain linear operators in high-dimensional spaces. This paper focuses on the efficient representation of tensor network operators with long-range pairwise

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A139-A163, January 2021. In this work a numerical method is proposed to compress a tensor by constructing a piecewise tensor approximation. This is constructed by partitioning a tensor into subtensors and by constructing a low-rank tensor approximation (in a given format) in each subtensor. Neither the partition nor the ranks are fixed

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page B30-B54, January 2021. In this work, we mainly develop a new numerical methodology to solve a PDE model recently proposed in the literature for pricing interest rate derivatives. More precisely, we use high-order-in-time AMFR-W-methods, which belong to a class of W-methods based on approximate matrix factorization (AMF) and are especially

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A109-A138, January 2021. Local Fourier analysis is a useful tool for predicting and analyzing the performance of many efficient algorithms for the solution of discretized PDEs, such as multigrid and domain decomposition methods. The crucial aspect of local Fourier analysis is that it can be used to minimize an estimate of the spectral radius

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A84-A108, January 2021. In this work, we investigate a model-order reduction scheme for polynomial systems. We begin with defining the generalized multivariate transfer functions for the system. Based on this, we aim at constructing a reduced-order system, interpolating the defined generalized transfer functions at a given set of interpolation

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page B1-B29, January 2021. We consider integration with respect to a $d$-dimensional spherical Gaussian measure arising from computational finance. Importance sampling (IS) is one of the most important variance reduction techniques in Monte Carlo (MC) methods. In this paper, two kinds of IS are studied in randomized quasi-MC (RQMC) setting,

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A54-A83, January 2021. In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method benefits from a direct discretization approach and is called the “direct RBF partition of unity (D-RBF-PU)” method. Thanks to avoiding

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A26-A53, January 2021. As an important Markov chain Monte Carlo (MCMC) method, the stochastic gradient Langevin dynamics (SGLD) algorithm has achieved great success in Bayesian learning and posterior sampling. However, SGLD typically suffers from a slow convergence rate due to its large variance caused by the stochastic gradient. In order

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A1-A25, January 2021. Polynomial preconditioning can improve the convergence of the Arnoldi method for computing eigenvalues. Such preconditioning significantly reduces the cost of orthogonalization; for difficult problems, it can also reduce the number of matrix-vector products. Parallel computations can particularly benefit from the reduction

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page C436-C468, January 2020. In this work we formally derive and prove the correctness of the algorithms and data structures in a parallel, distributed-memory, generic finite element framework that supports $h$-adaptivity on computational domains represented as forest-of-trees. The framework is grounded on a rich representation of the adaptive

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1570-B1598, January 2020. The study of collision-induced nonlinear breakage phenomenon is mostly unexplored but is important in the area of particulate processes. In this work, the volume and time dependent collisional breakage kernel function is modeled based on the population balance modeling approach. To solve the nonlinear breakage

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A4063-A4083, January 2020. The standard iterative refinement procedure for improving an approximate solution to the least squares problem $\min_x\|b - Ax\|_2$, where $A\in\mathbb{R}^{m\times n}$ with $m \ge n$ has full rank, is based on solving the $(m+n)\times (m+n)$ augmented system with the aid of a QR factorization. In order to exploit

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A4046-A4062, January 2020. A rule for constructing interpolation nodes for (n)th degree polynomials on the simplex is presented. These nodes are simple to define recursively from families of 1D node sets, such as the Lobatto--Gauss--Legendre (LGL) nodes. The resulting nodes have attractive properties: they are fully symmetric, they match

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A4017-A4045, January 2020. In many applications throughout science and engineering, model reduction plays an important role replacing expensive large-scale linear dynamical systems by inexpensive reduced order models that capture key features of the original, full order model. One approach to model reduction finds reduced order models that

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A3979-A4016, January 2020. In this paper we present a convergence rate analysis of inexact variants of several randomized iterative methods for solving three closely related problems: a convex stochastic quadratic optimization problem, a best approximation problem, and its dual, a concave quadratic maximization problem. Among the methods

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page C410-C435, January 2020. This paper proposes a parallel domain decomposition Rayleigh--Ritz projection scheme to compute a selected number of eigenvalues (and, optionally, associated eigenvectors) of large and sparse symmetric pencils. The projection subspace associated with interface variables is built by computing a few of the eigenvectors

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page C384-C409, January 2020. Parallel implementations of linear iterative solvers generally alternate between phases of data exchange and phases of local computation. Increasingly large problem sizes and more heterogeneous compute architectures make load balancing and the design of low latency network interconnects that are able to satisfy

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1541-B1569, January 2020. In this paper, we propose a variational Lagrangian scheme for a modified phase-field model, which can compute the equilibrium states of the original Allen--Cahn type model. Our discretization is based on a prescribed energy-dissipation law in terms of the flow map. By employing a discrete energetic variational

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1517-B1540, January 2020. Recently the general synthetic iteration scheme (GSIS) was proposed for the Boltzmann equation [W. Su et al., J. Comput. Phys., 407 (2020), 109245], where various numerical simulations have shown that (i) the steady-state solution can be found within dozens of iterations at any Knudsen number $K$, and (ii) the

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1490-B1516, January 2020. A system of Poisson--Nernst--Planck equations (PNP) is an important dielectric continuum model for simulating ion transport across biological membrane. In this paper, a PNP ion channel model with periodic boundary value conditions, denoted by PNPic, is presented and solved numerically with an effective finite

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A3957-A3978, January 2020. A new class of high-order maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The proposed method consists of a $k$th-order multistep exponential integrator in time and a lumped mass finite element method in space

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A3932-A3956, January 2020. A new adaptive diffusion central numerical flux within the framework of fifth-order characteristicwise alternative WENO-Z finite-difference schemes (A-WENO) with a modified local Lax--Friedrichs (LLF) flux for the Euler equations of gas dynamics is introduced. The new numerical flux adaptively adjusts the numerical

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A3907-A3931, January 2020. When solving linear systems arising from PDE discretizations, iterative methods (such as conjugate gradient (CG), GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be coupled with an efficient preconditioner. One approach to preconditioning

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A3878-A3906, January 2020. The problem of estimating numerically a distributed parameter from indirect measurements arises in many applications, and in that context the choice of the discretization plays an important role. In fact, guaranteeing a certain level of accuracy of the forward model that maps the unknown to the observations may

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A3859-A3877, January 2020. We construct smooth finite element de Rham complexes in two space dimensions. This leads to three families of curl-curl-conforming finite elements, two of which contain two existing families. The simplest triangular and rectangular finite elements have only six and eight degrees of freedom, respectively. Numerical

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A3825-A3858, January 2020. Fourier analysis has been shown to provide valuable insight into the dispersion and dissipation characteristics of numerical schemes for PDEs. Applying Fourier analysis to discontinuous Galerkin (DG) methods results in an eigenvalue problem with multiple eigenmodes. It was often relied on one of these modes, the

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A3812-A3824, January 2020. In this paper, we consider the tensor completion problem representing the solution in the tensor train (TT) format. It is assumed that the tensor is of high order, and tensor values are generated by an unknown smooth function. The assumption allows us to develop an efficient initialization scheme based on Gaussian

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page B1302-B1327, January 2020. Atmospheric weather and climate models must perform simulations very quickly to be useful. Therefore, modelers have traditionally focused on reducing computations as much as possible. However, in our new era of increasingly compute-capable hardware, data movement is now the prohibiting expense. This study examines

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page B1271-B1301, January 2020. This paper derives and analyzes new diffusion synthetic acceleration (DSA) preconditioners for the $S_N$ transport equation when discretized with a high-order (HO) discontinuous Galerkin (DG) discretization. DSA preconditioners address the need to accelerate the $S_N$ transport equation when the mean free path

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3516-A3539, January 2020. We present a method for converting tensors into the tensor train format based on actions of the tensor as a vector-valued multilinear function. Existing methods for constructing tensor trains require access to “array entries” of the tensor and are therefore inefficient or computationally prohibitive if the tensor

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3489-A3515, January 2020. This work introduces a method for learning low-dimensional models from data of high-dimensional black-box dynamical systems. The novelty is that the learned models are exactly the reduced models that are traditionally constructed with classical projection-based model reduction techniques. Thus, the proposed approach

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3462-A3488, January 2020. We propose a boundary method for the numerical simulation of time-dependent waves in three-dimensional (3D) exterior regions. The order of accuracy can be either second or fourth in both space and time. The method reduces a given initial boundary value problem for the wave equation to a set of operator equations

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3447-A3461, January 2020. In the context of linear time-invariant systems, the McMillan degree prescribes the smallest possible dimension of a system that reproduces the observed dynamics. When these observations take the form of impulse response measurements where the system evolves without input from an unknown initial condition, a result

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3427-A3446, January 2020. Standard backward error analyses for numerical linear algebra algorithms provide worst-case bounds that can significantly overestimate the backward error. Our recent probabilistic error analysis, which assumes rounding errors to be independent random variables [SIAM J. Sci. Comput., 41 (2019), pp. A2815--A2835]

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3397-A3426, January 2020. Hessian operators arising in inverse problems governed by partial differential equations (PDEs) play a critical role in delivering efficient, dimension-independent convergence for Newton solution of deterministic inverse problems, as well as Markov chain Monte Carlo sampling of posteriors in the Bayesian setting

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3367-A3396, January 2020. This work introduces nodal auxiliary space preconditioners for discretizations of mixed-dimensional partial differential equations. We first consider the continuous setting and generalize the regular decomposition to this setting. With the use of conforming mixed finite element spaces, we then expand these results

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3340-A3366, January 2020. We define a distance metric between partitions of a graph using machinery from optimal transport. Our metric is built from a linear assignment problem that matches partition components, with assignment cost proportional to transport distance over graph edges. We show that our distance can be computed using a single

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3313-A3339, January 2020. We study for the first time Schwarz domain decomposition methods for the solution of the Navier equations modeling the propagation of elastic waves. These equations in the time-harmonic regime are difficult to solve by iterative methods, even more so than the Helmholtz equation. We first prove that the classical

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3285-A3312, January 2020. In this paper, we propose a phase shift deep neural network (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. The PhaseDNN makes use of the fact that common deep neural networks (DNNs) often achieve convergence in the low frequency

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3250-A3284, January 2020. This paper proposes a new computational method for solving the structured least squares problem that arises in the process of identification of coherent structures in dynamic processes, such as, e.g., fluid flows. It is deployed in combination with dynamic mode decomposition (DMD), which provides a nonorthogonal

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3233-A3249, January 2020. We explore the computational issues concerning a new algorithm for the classical least-squares approximation of $N$ samples by an algebraic polynomial of degree at most $n$ when the number $N$ of the samples is very large. The algorithm is based on a recent idea about accurate numerical approximations of sums

SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3210-A3232, January 2020. Recently, there has been interest in high precision approximations of the first eigenvalue of the Laplace--Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page C359-C383, January 2020. We present an approach to the numerical solution of steady Navier--Stokes equations. Approximation by the finite element method (FEM) leads to a nonlinear saddle-point system. The system is linearized by the Picard iteration, which leads to a sequence of linear saddle-point systems with nonsymmetric matrices. In

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page C335-C358, January 2020. The conjugate gradient (CG) method is the most widely used iterative scheme for the solution of large sparse systems of linear equations when the matrix is symmetric positive definite. Although more than 60 years old, it is still a serious candidate for extreme-scale computations on large computing platforms. On

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page C313-C334, January 2020. In this work, we deal with the $QR$ factorization of block-tridiagonal matrices, where the blocks are dense and rectangular. This work is motivated by a novel method for computing geodesics over Riemannian manifolds. If blocks are reduced sequentially along the diagonal, only limited parallelism is available. We

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1462-B1489, January 2020. We consider the strictly correlated electron (SCE) limit of the fermionic quantum many-body problem in the second-quantized formalism. This limit gives rise to a multimarginal optimal transport (MMOT) problem. Here the marginal state space for our MMOT problem is the binary set 0,1, and the number of marginals

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1429-B1461, January 2020. Matrix-free finite element implementations for large applications provide an attractive alternative to standard sparse matrix data formats due to the significantly reduced memory consumption. Here, we show that they are also competitive with respect to the run-time in the low-order case if combined with suitable

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1404-B1428, January 2020. We develop a multilayer nonlinear elimination preconditioned inexact Newton method for a nonlinear algebraic system of equations, and a target application is the three-dimensional steady-state incompressible Navier--Stokes equations at high Reynolds numbers. Nonlinear steady-state problems are often more difficult

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1378-B1403, January 2020. We implement and test kernel averaging nonuniform fast Fourier transform (NUFFT) methods to enhance the performance of correlation and covariance estimation on asynchronously sampled event data using the Malliavin--Mancino Fourier estimator. The methods are benchmarked for Dirichlet and Fejér Fourier basis kernels

SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page B1350-B1377, January 2020. Finding the stationary states of a free energy functional is an important problem in phase field crystal (PFC) models. Many efforts have been devoted to designing numerical schemes with energy dissipation and mass conservation properties. However, most existing approaches are time-consuming due to the requirement