SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page B894-B920, January 2020. In this paper we consider wave propagation problems in two-dimensional unbounded domains, including dissipative effects, reformulated in terms of space-time boundary integral equations. For their solution, we employ a convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the

SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A2037-A2061, January 2020. In the last few years, several methods have been developed to deal with jump singularities in parametric or stochastic hyperbolic PDEs. They typically use some alignment of the jump-sets in physical space before performing well-established reduced order modeling techniques such as reduced basis methods, proper

SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A2014-A2036, January 2020. In this paper we study elliptic partial differential equations with rapidly varying diffusion coefficient that can be represented as a perturbation of a reference coefficient. We develop a numerical method for efficiently solving multiple perturbed problems by reusing local computations performed with the reference

SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A1984-A2013, January 2020. A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization via the use of total variation. On the other hand, sparse or noisy data often demand a probabilistic approach to the reconstruction of images, to enable uncertainty quantification;

SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page B869-B893, January 2020. The discrete ordinates discontinuous Galerkin ($S_N$-DG) method is a well-established and practical approach for solving the radiative transport equation. In this paper, we study a low-memory variation of the upwind $S_N$-DG method. The proposed method uses a smaller finite element space that is constructed by coupling

SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page B847-B868, January 2020. The immersed boundary (IB) method is a numerical and mathematical formulation for solving fluid-structure interaction problems. It relies on solving fluid equations on an Eulerian fluid grid and interpolating the resulting velocity back onto immersed structures. To resolve slender fibers, the grid spacing must be

SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A1950-A1983, January 2020. A new iterative method, the WaveHoltz iteration, for solution of the Helmholtz equation is presented. WaveHoltz is a fixed point iteration that filters the solution to the solution of a wave equation with time periodic forcing and boundary data. The WaveHoltz iteration corresponds to a linear and coercive operator

SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A1935-A1949, January 2020. Nonlocal models provide exceptional simulation fidelity for a broad spectrum of scientific and engineering applications. However, wider deployment of nonlocal models is hindered by several modeling and numerical challenges. Among those, we focus on the nontrivial prescription of nonlocal boundary conditions, or

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page B520-B547, January 2020. In this paper, we consider the three dimensional Vlasov equation with an inhomogeneous, varying direction, strong magnetic field. Whenever the magnetic field has constant intensity, the oscillations generated by the stiff term are periodic. The homogenized model is then derived, and several state-of-the-art multiscale

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1428-A1457, January 2020. We propose a discontinuous Galerkin (DG) method to approximate the elliptic interface problem on unfitted mesh using a new approximation space. The approximation space is constructed by patch reconstruction with one degree of freedom per element. The optimal error estimates in both the $L^2$ norm and the DG energy

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1402-A1427, January 2020. Previous work on multilevel techniques for compression and reduction of scientific data is extended to the case of data given on unstructured meshes in two and three dimensions. The centerpiece of the work is a decomposition algorithm which is shown to be optimal, in terms of both storage and operational complexity

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1380-A1401, January 2020. In this work we present an algorithm to construct an infinitely differentiable smooth surface from an input consisting of a (rectilinear) triangulation of a surface of arbitrary shape. The original surface can have nontrivial genus and multiscale features, and our algorithm has computational complexity which is

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1348-A1379, January 2020. This paper proposes a frequency/time hybrid integral-equation method for the time-dependent wave equation in two- and three-dimensional spatial domains. Relying on Fourier transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate,

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1317-A1347, January 2020. Optimization-based samplers such as randomize-then-optimize (RTO) [J. M. Bardsley et al., SIAM J. Sci. Comput., 36 (2014), pp. A1895--A1910] provide an efficient and parallellizable approach to solving large-scale Bayesian inverse problems. These methods solve randomly perturbed optimization problems to draw samples

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1289-A1316, January 2020. We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in the formation of local stiffness matrices. To define our local finite

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1269-A1288, January 2020. In many inverse problems, a nonnegativity constraint is natural. Moreover, in some cases, we expect the vector of unknown parameters to have zero components. When a Bayesian approach is taken, this motivates a desire for prior probability density (and hence posterior probability density) functions that have positive

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1245-A1268, January 2020. 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. The proposed methods are based on a specific subset of explicit one-step exponential integrators. More precisely, starting from an explicit exponential

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1226-A1244, January 2020. This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem. Moreover, we prove the convergence of discretizations in two-dimensional situations

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page C124-C141, January 2020. Computing units that carry out a fused multiply-add (FMA) operation with matrix arguments, referred to as tensor units by some vendors, have great potential for use in scientific computing. However, these units are inherently mixed precision, and existing rounding error analyses do not support them. We consider

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page C97-C123, January 2020. With the gap between processor clock speeds and memory bandwidth speeds continuing to increase, the use of arithmetically intense schemes, such as high-order finite element methods, continues to be of considerable interest. In particular, the use of matrix-free formulations of finite element operators for tensor-product

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page C69-C96, January 2020. High-order discontinuous Galerkin (DG) methods promise to be an excellent discretization paradigm for hyperbolic differential equation solvers running on supercomputers, since they combine high arithmetic intensity with localized data access, since they straightforwardly translate into nonoverlapping domain decomposition

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B816-B845, January 2020. This work introduces an extension of the residual distribution (RD) framework to stiff relaxation problems. The RD is a class of schemes which is used to solve a hyperbolic system of partial differential equations. To our knowledge, it has been used only for systems with mild source terms, such as gravitation problems

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B792-B815, January 2020. We propose a Hermite--Galerkin spectral method to numerically solve the spatially homogeneous Fokker--Planck--Landau equation with a singular quadratic collision model. To overcome the difficulty of the computationally expensive collision model, we adopt a novel approximation formulated by a combination of a simple

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B761-B791, January 2020. In this paper, we present block preconditioners for a stabilized discretization of the poroelastic equations developed in [C. Rodrigo, X. Hu, P. Ohm, J. Adler, F. Gaspar, and L. Zikatanov, Comput. Methods Appl. Mech. Engrg., 341 (2018), pp. 467--484]. The discretization is proved to be well-posed with respect to

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B738-B760, January 2020. Adaptive second-order Crank--Nicolson time-stepping methods using the recent scalar auxiliary variable (SAV) approach are developed for the time-fractional molecular beam epitaxial models with Caputo's fractional derivative. Based on the piecewise linear interpolation, the Caputo's derivative is approximated by

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B722-B737, January 2020. This paper is concerned with the electromagnetic inverse scattering problem that aims to determine the location and shape of anisotropic scatterers from far field data (at a fixed frequency). We study the orthogonality sampling method, which is a simple, fast, and robust imaging method for solving the inverse scattering

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B696-B721, January 2020. Geophysical electromagnetic (EM) methods are an important technique for investigating the subsurface of the Earth, particularly when exploring for metallic ore deposits but also when delineating hydrocarbon reserves and in hydrological and geotechnical applications. Geophysical EM methods provide information on

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B675-B695, January 2020. We present a $d$-dimensional exponential analysis algorithm that offers a range of advantages compared to other methods. The technique does not suffer the curse of dimensionality and only needs $O((d+1)n)$ samples for the analysis of an $n$-sparse expression. It does not require a prior estimate of the sparsity

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B656-B674, January 2020. In this paper, we develop a novel spectral method for solving the time-dependent Kohn--Sham equations in the semiclassical regime. Our strategy is to use a single-step predictor-corrector algorithm, where the propagator is derived using a Fourier integral operator commonly known as the frozen Gaussian approximation

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B630-B655, January 2020. In this paper, we consider an exponential scalar auxiliary variable (E-SAV) approach to obtain energy stable schemes for a class of phase field models. This novel auxiliary variable method based on the exponential form of the nonlinear free energy potential is more effective and applicable than the traditional SAV

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B608-B629, January 2020. In this paper, we develop a pressure robust weak Galerkin finite element scheme for Stokes equations on polygonal mesh. The major idea for achieving a pressure-independent energy-error estimate is to use a divergence preserving velocity reconstruction operator in the discretization of the right-hand side body force

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B581-B607, January 2020. In this paper we consider a numerical homogenization technique for curl-curl problems that is based on the framework of the localized orthogonal decomposition and which was proposed in [D. Gallistl, P. Henning, and B. Verfürth, SIAM J. Numer. Anal., 56 (2018), pp. 1570--1596] for problems with essential boundary

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B549-B580, January 2020. We present a novel formulation for the calibration of a biophysical tumor growth model from a single-time snapshot, multiparametric magnetic resonance imaging (MRI) scan of a glioblastoma patient. Tumor growth models are typically nonlinear parabolic partial differential equations (PDEs). Thus, we have to generate

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1913-A1934, January 2020. The Strang splitting method, formally of order two, can suffer from order reduction when applied to semilinear parabolic problems with inhomogeneous boundary conditions. The recent work [L. Einkemmer and A. Ostermann, SIAM J. Sci. Comput., 37, 2015; SIAM J. Sci. Comput., 38, 2016] introduces a modification of

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1885-A1912, January 2020. This paper investigates gradient recovery schemes for data defined on discretized manifolds. The proposed method, parametric polynomial preserving recovery (PPPR), does not require the tangent spaces of the exact manifolds which have been assumed for some significant gradient recovery methods in the literature

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1860-A1884, January 2020. In two dimensions, we propose and analyze an a posteriori error estimator for finite element approximations of the stationary Navier--Stokes equations with singular sources on Lipschitz, but not necessarily convex, polygonal domains. Under a smallness assumption on the continuous and discrete solutions, we prove

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1836-A1859, January 2020. We build upon R. Estrin et al., [SIAM J. Sci. Comput., 42 (2020), pp. A1809--A1835] to develop a general constrained nonlinear optimization algorithm based on a smooth penalty function proposed by R. Fletcher [Integer and Nonlinear Programming, J. Abadie, ed., North-Holland, Amsterdam, (1970), pp. 157--175; Math

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1809-A1835, January 2020. We develop a general equality-constrained nonlinear optimization algorithm based on a smooth penalty function proposed by Fletcher in 1970. Although it was historically considered to be computationally prohibitive in practice, we demonstrate that the computational kernels required are no more expensive than other

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1789-A1808, January 2020. This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method reformulates the equation as a collection of second-kind integral equations

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1765-A1788, January 2020. Markov chain Monte Carlo (MCMC) samplers are numerical methods for drawing samples from a given target probability distribution. We discuss one particular MCMC sampler, the MALA-within-Gibbs sampler, from the theoretical and practical perspectives. We first show that the acceptance ratio and step size of this

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1741-A1764, January 2020. We modify the well-known interior penalty finite element discretization method so that it allows for element-by-element assembly. This is possible due to the introduction of additional unknowns associated with the interfaces between neighboring elements. The resulting bilinear form, and a Schur complement (reduced)

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1714-A1740, January 2020. We consider optimal design of PDE-based Bayesian linear inverse problems with infinite-dimensional parameters. We focus on the A-optimal design criterion, defined as the average posterior variance and quantified by the trace of the posterior covariance operator. We propose using structure exploiting randomized

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1686-A1713, January 2020. In this paper, we study fixed-point schemes with certain low-rank structures arising from matrix optimization problems. Traditional first-order methods depend on the eigenvalue decomposition at each iteration, which may take most of the computational time. In order to reduce the cost, we propose an inexact algorithmic

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1663-A1685, January 2020. We present a new numerical method for transporting arbitrary sets in a velocity field. The method computes a deformation mapping of the domain and advects particular sets by function composition with the map. This also allows for the transport of multiple sets at low computational cost. Our strategy is to separate

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1639-A1662, January 2020. Dynamic mode decomposition (DMD), which belongs to the family of singular-value decompositions (SVDs), is a popular tool of data-driven regression. While multiple numerical tests demonstrated the power and efficiency of DMD in representing data (i.e., in the interpolation mode), applications of DMD as a predictive

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1609-A1638, January 2020. Traditional time discretization methods use a single timestep for the entire system of interest and can perform poorly when the dynamics of the system exhibits a wide range of time scales. Multirate infinitesimal step (MIS) methods [O. Knoth and R. Wolke, Appl. Numer. Math., 28 (1998), pp. 327--341] offer an elegant

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1582-A1608, January 2020. The discrete empirical interpolation method (DEIM) is a popular technique for nonlinear model reduction, and it has two main ingredients: an interpolating basis that is computed from a collection of snapshots of the solution, and a set of indices which determine the nonlinear components to be simulated. The computation

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1558-A1581, January 2020. Reliable tracking of moving boundaries is important for the simulation of compressible fluid flows, and there are a lot of contributions in literature. We recognize from the classical piston problem, a typical moving boundary problem in gas dynamics, that acceleration is a key element in the description of the

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1541-A1557, January 2020. The need to sum floating-point numbers is ubiquitous in scientific computing. Standard recursive summation of $n$ summands, often implemented in a blocked form, has a backward error bound proportional to $nu$, where $u$ is the unit roundoff. With the growing interest in low precision floating-point arithmetic

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1510-A1540, January 2020. In this paper, we propose a new efficient preconditioner for iteratively solving the large-scale indefinite saddle-point sparse linear system, which arises from discretizing the optimality system in optimal control problems of wave equations with a one-shot second-order finite difference scheme in both space and

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1486-A1509, January 2020. We propose a fast potential splitting Markov chain Monte Carlo method which costs $O(1)$ time each step for sampling from equilibrium distributions (Gibbs measures) corresponding to particle systems with singular interacting kernels. We decompose the interacting potential into two parts; one is of long range but

SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1459-A1485, January 2020. We present extensions to hierarchical probing, a method developed in [A. Stathopoulos, J. Laeuchli, and K. Orginos, SIAM J. Sci. Comput., 35 (2013), pp. S299--S322] to reduce the variance of the Monte Carlo estimation of the trace or the diagonal of the inverse of a large, sparse matrix. In that context, probing

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1200-A1225, January 2020. We consider shape optimization problems subject to elliptic partial differential equations. In the context of the finite element method, the geometry to be optimized is represented by the computational mesh, and the optimization proceeds by repeatedly updating the mesh node positions. It is well known that such

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1174-A1199, January 2020. For discretized elliptic equations, we develop a new factorization update algorithm that is suitable for incorporating coefficient updates with large support and large magnitude in subdomains. When a large number of local updates are involved, in addition to the standard factors in various (interior) subdomains

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1147-A1173, January 2020. We present algebraic multigrid (AMG) methods for the efficient solution of the linear system of equations stemming from high-order discontinuous Galerkin (DG) discretizations of second-order elliptic problems. For DG methods, standard multigrid approaches cannot be employed because of redundancy of the degrees

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1116-A1146, January 2020. Several reduced order models have been successfully developed for nonlinear dynamical systems. To achieve a considerable speed-up, a hyper-reduction step is needed to reduce the computational complexity due to nonlinear terms. Many hyper-reduction techniques require the construction of nonlinear term basis, which

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1097-A1115, January 2020. This paper introduces a “kernel-independent” interpolative decomposition butterfly factorization (IDBF) as a data-sparse approximation for matrices that satisfy a complementary low-rank property. The IDBF can be constructed in $O(N\log N)$ operations for an $N\times N$ matrix via hierarchical interpolative decompositions

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1071-A1096, January 2020. In this paper, we develop the ball-approximated characteristics (B-char) method, which is an algorithm for efficiently implementing characteristic-based schemes in two and three dimensions. Core to the implementation of numerical schemes is the evaluation of integrals, which in the context of characteristic-based

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page C43-C68, January 2020. Matrices with hierarchical low-rank structure, including HODLR and HSS matrices, constitute a versatile tool to develop fast algorithms for addressing large-scale problems. While existing software packages for such matrices often focus on linear systems, their scope of applications is in fact much wider and includes

SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1052-A1070, January 2020. We present a general purpose method for solving partial differential equations on a closed surface, based on a technique for discretizing the surface introduced by Wenjun Ying and Wei-Cheng Wang [J. Comput. Phys., 252 (2013), pp. 606--624] which uses projections on coordinate planes. Assuming it is given as a