SIAM Journal on Scientific Computing, Volume 43, Issue 3, Page B570-B597, January 2021. In ptychography experiments, redundant scanning is usually required to guarantee the stable recovery, such that a huge number of frames are generated, and thus it poses a great demand on parallel computing to solve this large-scale inverse problem. In this paper, we propose the overlapping domain decomposition methods

SIAM Journal on Scientific Computing, Volume 43, Issue 3, Page A1671-A1691, January 2021. Numerical resolution of exterior Helmholtz problems requires some approach to domain truncation. As an alternative to approximate nonreflecting boundary conditions and invocation of the Dirichlet-to-Neumann map, we introduce a new, nonlocal boundary condition. This condition is exact and requires the evaluation

SIAM Journal on Scientific Computing, Volume 43, Issue 3, Page A1651-A1670, January 2021. We introduce Nitsche's method for the numerical approximation of the Kirchhoff--Love plate equation under general Robin-type boundary conditions. We analyze the method by presenting a priori and a posteriori error estimates in mesh-dependent norms. Several numerical examples are given to validate the approach

SIAM Journal on Scientific Computing, Volume 43, Issue 3, Page A1625-A1650, January 2021. A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton--Jacobi--Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The method combines a tensor train approximation for the value function together with a Newton-like iterative method

SIAM Journal on Scientific Computing, Volume 43, Issue 3, Page B539-B569, January 2021. In this paper, we consider a nonlinear and nonlocal parabolic model for multispecies ionic fluids and introduce a semi-implicit finite volume scheme, which is second-order accurate in space, is first-order in time, and satisfies the positivity preserving and mass conservation properties. The scheme also satisfies

SIAM Journal on Scientific Computing, Volume 43, Issue 3, Page A1607-A1624, January 2021. We present a numerical framework for recovering unknown nonautonomous dynamical systems with time-dependent inputs. To circumvent the difficulty presented by the nonautonomous nature of the system, our method transforms the solution state into piecewise integration of the system over a discrete set of time instances

SIAM Journal on Scientific Computing, Volume 43, Issue 3, Page A1583-A1606, January 2021. The aim of this paper is to study the numerical approximation of the displacement formulation of the acoustic eigenvalue problem in the axisymmetric case. We show that spurious eigenvalues appear when lowest order triangular Raviart--Thomas elements are used to discretize the problem. We propose an alternative

SIAM Journal on Scientific Computing, Volume 43, Issue 3, Page A1555-A1582, January 2021. Topology optimization problems often support multiple local minima due to a lack of convexity. Typically, gradient-based techniques combined with continuation in model parameters are used to promote convergence to more optimal solutions; however, these methods can fail even in the simplest cases. In this paper

SIAM Journal on Scientific Computing, Ahead of Print. We developed a hybrid restarted Lanczos method that combines thick-restarting with Ritz vectors with iteratively refined Ritz vectors to compute a few of the extreme eigenvalues and associated eigenvectors of a large sparse symmetric matrix $A$. The iterative refined Ritz vectors use a scheme, where we replace the approximate eigenvalue in the original

SIAM Journal on Scientific Computing, Ahead of Print. We propose two new algebraic reconstruction techniques based on Kaczmarz's method that produce a regularized solution to noisy tomography problems. Tomography problems exhibit semiconvergence when iterative methods are employed, and the aim is therefore to stop near the semiconvergence point. Our approach is based on an error gauge that is constructed

SIAM Journal on Scientific Computing, Ahead of Print. Iterative hybrid projection methods have proven to be very effective for solving large linear inverse problems due to their inherent regularizing properties as well as the added flexibility to select regularization parameters adaptively. In this work, we develop Golub--Kahan-based hybrid projection methods that can exploit compression and recycling

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page C177-C202, January 2021. Based on the parallel cyclic reduction technique, a promising new parallel algorithm is designed for pentadiagonal systems. Subject to fulfilling stability conditions, this highly parallelizable algorithm works very well for systems of any size. The solver is implemented on a graphics processing unit using the CUDA

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1527-A1554, January 2021. Finding optimal (or low energy) centroidal Voronoi tessellations (CVTs) on a two-dimensional domain is a challenging problem. One must navigate an energy landscape whose desirable critical points have sufficiently small basins of attractions that are inaccessible with Monte Carlo initialized gradient descent methods

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1497-A1526, January 2021. We develop interior penalty discontinuous Galerkin difference methods for the wave equation in second-order form. The new schemes are energy conserving or energy dissipating depending on the simple choice of centered or upwind fluxes and are superconvergent away from boundaries. Unlike analogous methods using

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1472-A1496, January 2021. We develop a fourth order accurate finite difference method for the three dimensional elastic wave equation in isotropic media with the piecewise smooth material property. In our model, the material property can be discontinuous at curved interfaces. The governing equations are discretized in second order form

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1441-A1471, January 2021. Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be nonrobust in the presence of Neumann boundary conditions. In this paper, we overcome this issue by formulating the RBF-generated finite difference method in a discrete least squares setting instead. This allows

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B508-B537, January 2021. This paper proposes a new, conservative fully discrete scheme for the numerical solution of the regularized shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1412-A1440, January 2021. We introduce a numerical method, multilevel designed quadrature for computing the statistical solution of partial differential equations with random input data. Similar to multilevel Monte Carlo methods, our method relies on hierarchical spatial approximations in addition to a parametric/stochastic sampling strategy

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1389-A1411, January 2021. In this paper, we propose a meshfree method based on the Gaussian radial basis function (RBF) to solve both classical and fractional PDEs. The proposed method takes advantage of the analytical Laplacian of Gaussian functions so as to accommodate the discretization of the classical and fractional Laplacians in

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1362-A1388, January 2021. The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead of the standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in geophysical electromagnetics. We establish the well-posedness

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1336-A1361, January 2021. We propose a class of adaptive enriched Galerkin-characteristics finite element methods for efficient numerical solution of the incompressible Navier--Stokes equations in primitive variables. The proposed approach combines the modified method of characteristics to deal with convection terms, the finite element

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1305-A1335, January 2021. We propose a recurrent neural network for a “model-free” simulation of a dynamical system with unknown parameters without prior knowledge. The deep learning model aims to jointly learn the nonlinear time marching operator and the effects of the unknown parameters from a time series data. We assume that the time

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1273-A1304, January 2021. We numerically investigate and improve upon a computational approach originally introduced in [R. Cottereau, Internat. J. Numer. Methods Engrg., 95 (2013), pp. 71--90] which aims at evaluating the effective coefficient of a medium modeled by a highly oscillatory coefficient. This computational approach is based

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1248-A1272, January 2021. In this paper, two finite difference numerical schemes are proposed and analyzed for the droplet liquid film model with a singular Lennard--Jones energy potential involved. Both first and second order accurate temporal algorithms are considered. In the first order scheme, the convex potential and the surface diffusion

SIAM Journal on Scientific Computing, Ahead of Print. We develop a multigrid solver steered by an a posteriori estimator of the algebraic error. We adopt the context of a second-order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree $p \ge 1$. Our solver employs zero pre- and one postsmoothing by the overlapping Schwarz (block-Jacobi) method and features

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1221-A1247, January 2021. Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton--Jacobi--Bellman (HJB) equations, which are notoriously difficult when the state dimension is large. Existing strategies for high-dimensional problems often rely on specific, restrictive problem structures or are valid

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1194-A1220, January 2021. We consider a linear kinetic transport equation under a diffusive scaling that converges to a diffusion equation as the Knudsen number $\varepsilon\rightarrow 0$. In [S. Boscarino, L. Pareschi, and G. Russo, SIAM J. Sci. Comput., 35 (2013), pp. A22--A51; Z. Peng et al., J. Comput. Phys., 415 (2020), 109485], to

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B479-B507, January 2021. The binary fluid surfactant phase-field model, coupled with two Cahn--Hilliard equations and Navier--Stokes equations, is a very complex nonlinear system, which poses many challenges to the design of numerical schemes. As far as the author knows, due to the highly nonlinear coupling nature, there is no fully decoupled

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B455-B478, January 2021. Nowadays the Potts model works as the key model in a broad spectrum of applications in image processing and computer vision, which can be mathematically formulated in the form of min-cuts and, meanwhile, solved in terms of flow maximizing under the perspective of primal and dual. It is of great interest in developing

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1163-A1193, January 2021. We propose and analyze a Stein variational reduced basis method (SVRB) to solve large-scale PDE-constrained Bayesian inverse problems. To address the computational challenge of drawing numerous samples requiring expensive PDE solves from the posterior distribution, we integrate an adaptive and goal-oriented model

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1134-A1162, January 2021. This work develops a new multifidelity ensemble Kalman filter (MFEnKF) algorithm based on a linear control variate framework. The approach allows for rigorous multifidelity extensions of the EnKF, where the uncertainty in coarser fidelities in the hierarchy of models represents control variates for the uncertainty

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B431-B454, January 2021. The shallow water equations provide the basic modeling equations for a number of coastal flooding hazards, such as tsunamis and storm surge. In realistic scenarios, there are often structures important to these flows that have a large extent but small width, including sea walls, berms, and harbor barriers. Explicit

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1108-A1133, January 2021. In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the theoretical framework is nonstandard, and the development of suitable numerical

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1081-A1107, January 2021. We present a reduced basis method for parametrized linear elliptic partial differential equations (PDEs) in a least-squares finite element framework. A rigorous and reliable error estimate is developed, and is shown to bound the error with respect to the exact solution of the PDE, in contrast to estimates that

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1062-A1080, January 2021. A quality-Bayesian approach, combining the direct sampling method and the Bayesian inversion, is proposed to reconstruct the locations and intensities of the unknown acoustic sources using partial data. First, we extend the direct sampling method by constructing new indicator functions to obtain the approximate

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B405-B430, January 2021. We propose a novel algorithm for the approximation of surface quasi-geostrophic (SQG) flows modeled by a nonlinear partial differential equation coupling transport and fractional diffusion phenomena. The time discretization consists of an explicit strong-stability-preserving three-stage Runge--Kutta method while

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B381-B404, January 2021. A new numerical scheme is proposed for flow computation in complex discrete fracture networks. The method is based on a three-field domain decomposition framework in which independent variables are introduced at the interfaces generated in the process of decoupling the original problem on the whole network into

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1027-A1061, January 2021. In this paper, we design a truly exact perfect absorbing layer (PAL) for domain truncation of the two-dimensional Helmholtz equation in an unbounded domain with bounded scatterers. This technique is based on a complex compression coordinate transformation in polar coordinates and a judicious substitution of the

SIAM Journal on Scientific Computing, Ahead of Print. The stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations with parametric or random inputs. However, computing stochastic Galerkin approximations for a given problem requires the solution of large coupled systems

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page C154-C176, January 2021. We are interested in parallelizing the least angle regression (LARS) algorithm for fitting linear regression models to high-dimensional data. We consider two parallel and communication avoiding versions of the basic LARS algorithm. The two algorithms have different asymptotic costs and practical performance. One

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B354-B380, January 2021. Resistive magnetohydrodynamics (MHD) is a continuum base-level model for conducting fluids (e.g., plasmas and liquid metals) subject to external magnetic fields. The efficient and robust solution of the MHD system poses many challenges due to the strongly nonlinear, non--self-adjoint, and highly coupled nature of

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1001-A1026, January 2021. We consider the iterative solution of regularized saddle-point systems. When the leading block is symmetric and positive semidefinite on an appropriate subspace, Dollar et al. [SIAM J. Matrix Anal. Appl., 28 (2006), pp. 170--189] describe how to apply the conjugate gradient (CG) method coupled with a constraint

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A980-A1000, January 2021. Domain discretization is considered a dominant part of solution procedures for solving partial differential equations. It is widely accepted that mesh generation is among the most cumbersome parts of the finite element method analysis and often requires human assistance, especially in complex three-dimensional

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A953-A979, January 2021. In this study, we present a new class of quasi-interpolation schemes for the approximation of multivariate functions on sparse grids. Each scheme in this class is based on shifts of kernels constructed from one-dimensional radial basis functions such as multiquadrics. The kernels are modified near the boundaries

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page C127-C153, January 2021. Large-scale PDE simulations using high-order finite-element methods on unstructured meshes are an indispensable tool in science and engineering. The widely used open-source PETSc library offers an efficient representation of generic unstructured meshes within its DMPlex module. This paper details our recent implementation

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B323-B353, January 2021. We present a numerical method which accurately computes the discrete spectrum and associated bound states of semi-infinite Hamiltonians which model electronic “edge” states localized at boundaries of one- and two-dimensional crystalline materials. The problem is nontrivial since arbitrarily large finite “hard” (Dirichlet)

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A929-A952, January 2021. A stochastic Galerkin formulation for a stochastic system of balanced or conservation laws may fail to preserve the hyperbolicity of the original system. In this work, we develop a hyperbolicity-preserving stochastic Galerkin formulation for the one-dimensional shallow water equations by carefully selecting the

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B298-B322, January 2021. The design of freeform optical surfaces is an inverse problem in illumination optics. Combining the laws of geometrical optics and energy conservation gives rise to a generalized Monge--Ampère equation. The underlying mathematical structure of some optical systems allows for an optimal-transport formulation of the

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B271-B297, January 2021. We consider the problem of imaging a crack network embedded in some homogeneous background from measured multistatic far field data generated by acoustic plane waves. We propose two novel approaches that can be seen as extensions of linear sampling-type methods and that provide indicator functions which are sensitive

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A908-A928, January 2021. This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear PDEs. The main idea of these algorithms is to utilize the solutions on the $k$th-level adaptive meshes to find the solutions on the $(k+1)$th-level adaptive meshes which are constructed by performing

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A883-A907, January 2021. This paper presents an adaptive randomized algorithm for computing the butterfly factorization of an $m\times n$ matrix with $m\approx n$ provided that both the matrix and its transpose can be rapidly applied to arbitrary vectors. The resulting factorization is composed of $\mathcal{O}(\log n)$ sparse factors, each

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A859-A882, January 2021. We design and analyze a Hybrid High-Order (HHO) method on unfitted meshes to approximate elliptic interface problems by means of a consistent penalty method à la Nitsche. The curved interface can cut through the mesh cells in a rather general fashion. Robustness with respect to the cuts is achieved by using a cell

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A828-A858, January 2021. We propose high-order well-balanced finite-volume schemes for a broad class of hydrodynamic systems with attractive-repulsive interaction forces and linear and nonlinear damping. Our schemes are suitable for free energies containing convolutions of an interaction potential with the density, which are essential for

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A800-A827, January 2021. The aim of this paper is to propose a robust numerical solver, which is capable of efficiently solving a three-dimensional elliptic problem in a data-sparse quantized tensor format. In particular, we use the combined Tucker and quantized tensor train format (TQTT), which allows us to use astronomically large grid

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A772-A799, January 2021. Solutions to the linear complementarity problem (LCP) are naturally sparse in many applications such as bimatrix games and portfolio section problems. Despite that it gives rise to the hardness, sparsity makes optimization faster and enables relatively large scale computation. Motivated by this, we take the sparse

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A744-A771, January 2021. This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babuška's theorem, we show well-posedness of the first-order optimality systems for a typical model problem with linear state

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B243-B270, January 2021. Iron loss determination in the magnetic core of an electrical machine, such as a motor or a transformer, is formulated as an inverse heat source problem. The sensor positions inside the object are optimized in order to minimize the uncertainty in the reconstruction in the sense of the A-optimality of Bayesian experimental

SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A721-A743, January 2021. Explicit stabilized methods are an efficient alternative to implicit schemes for the time integration of stiff systems of differential equations in large dimension. In this paper we derive explicit stabilized integrators of orders one and two for the optimal control of stiff systems. We analyze their favorable stability

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A685-A719, January 2021. The complex Langevin (CL) method is a classical numerical strategy to alleviate the numerical sign problem in the computation of lattice field theories. Mathematically, it is a simple numerical tool to compute a wide class of high-dimensional and oscillatory integrals. However, it is often observed that the CL method

SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A663-A684, January 2021. A widely used form of test matrix is the randsvd matrix constructed as the product $A = U \Sigma V^*$, where $U$ and $V$ are random orthogonal or unitary matrices from the Haar distribution and $\Sigma$ is a diagonal matrix of singular values. Such matrices are random but have a specified singular value distribution