SIAM Journal on Scientific Computing

In this work, we design, analyze, and numerically test an invariant preserving discontinuous Galerkin method for solving the nonlinear Camassa--Holm equation. This model is integrable and admits peakon solitons. The proposed numerical method is high order ...

Summation-by-parts (SBP) finite-difference discretizations share many attractive properties with Galerkin finite-element methods (FEMs), including time stability and superconvergent functionals; however, unlike FEMs, SBP operators are not completely ...

Randomized sampling has recently been proven a highly efficient technique for computing approximate factorizations of matrices that have low numerical rank. This paper describes an extension of such techniques to a wider class of matrices that are not ...

The aim of this study is to develop a novel sixth-order weighted essentially nonoscillatory (WENO) finite difference scheme. To design new WENO weights, we present two important measurements: a discontinuity detector (at the cell boundary) and a ...

The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially with the number ...

We propose a novel combination of the reduced basis method with low-rank tensor techniques for the efficient solution of parameter-dependent linear systems in the case of several parameters. This combination, called rbTensor, consists of three ingredients. ...

This paper presents a fast and powerful method for the computation of eigenvalue bounds for Hessian matrices $\nabla^2{\varphi(x)}$ of nonlinear twice continuously differentiable functions $\varphi:\mathcal{U}\subseteq{\mathbb R}^n\rightarrow{\mathbb R}$ ...

In this work, we present a novel multilevel Monte Carlo method for kinetic simulation of stochastic reaction networks characterized by having simultaneously fast and slow reaction channels. To produce efficient simulations, our method adaptively classifies ...

This paper is concerned with the design and implementation of efficient solution algorithms for elliptic PDE problems with correlated random data. The energy orthogonality that is built into stochastic Galerkin approximations is cleverly exploited to give an ...

We present a forward semi-Lagrangian numerical method for systems of transport equations able to advect smooth and discontinuous fields with high-order accuracy. The numerical scheme is composed of an integration of the transport equations along the ...

We introduce a Bernstein--Bezier basis for the pyramid, whose restriction to the face reduces to the Bernstein--Bezier basis on the triangle or quadrilateral. The basis satisfies the standard positivity and partition of unity properties common to Bernstein ...

We present and analyze a new space-time parallel multigrid method for parabolic equations. The method is based on arbitrarily high order discontinuous Galerkin discretizations in time and a finite element discretization in space. The key ingredient of the ...

In this paper hyperbolic partial differential equations (PDEs) with random coefficients are discussed. We consider the challenging problem of flux functions with coefficients modeled by spatiotemporal random fields. Those fields are given by correlated ...

In this paper we are concerned with the plane wave methods for Helmholtz equations with large wave numbers. We extend the plane wave weighted least-squares (PWLS) method and the plane wave discontinuous Galerkin (PWDG) method to Helmholtz equations in ...

Similarly to polynomials, smooth radial basis function (RBF) interpolants converge exponentially fast to analytic functions on a one dimensional bounded domain but are also vulnerable to the Runge phenomenon [R. Platte, IMA J. Numer. Anal. 31 (2014), pp. ...

We introduce a time-integrator to sample with high order of accuracy the invariant distribution for a class of semilinear stochastic PDEs (SPDEs) driven by an additive space-time noise. Combined with a postprocessor, the new method is a modification with ...

The nonmonotonic propagation of fronts is considered. When the speed function $F:\mathbb{R}^{n} \times [0,T]\rightarrow \mathbb{R}$ is prescribed, the nonlinear advection equation $\phi_{t}+F|\nabla \phi|=0$ is a Hamilton--Jacobi equation known as the ...

In this paper, we propose an algorithm for estimating parameters of a source term of a linear advection-diffusion equation with an uncertain advection-velocity field. First, we apply a minimax state estimation technique in order to reduce uncertainty ...

Solutions for many problems of interest exhibit singular behaviors at domain corners or points where the boundary condition changes type. For these types of problems, direct spectral methods with the usual polynomial basis functions do not lead to a ...

A bilinear quadrature numerically evaluates a continuous bilinear map, such as the $L^2$ inner product, on continuous $f$ and $g$ belonging to known finite-dimensional function spaces. Such maps arise in Galerkin methods for differential and integral ...

The accurate approximation of high-dimensional functions is an essential task in uncertainty quantification and many other fields. We propose a new function approximation scheme based on a spectral extension of the tensor-train (TT) decomposition. We ...

We develop a gauge-invariant frozen Gaussian approximation (GIFGA) method for the Schrödinger equation (LSE) with periodic potentials in the semiclassical regime. The method generalizes the Herman--Kluk propagator for LSE to the case with periodic media. It ...

In this work, which is the continuation of [SIAM J. Sci. Comput., 38 (2016), pp. A737--A764], we propose numerical schemes for linear kinetic equations which are able to deal with the fractional diffusion limit. When the collision frequency degenerates for ...

We discuss the problem of constructing an accurate function approximation when data are corrupted by unexpected errors. The unexpected corruption errors are different from the standard observational noise in the sense that they can have much larger ...

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this ...

The paper investigates a variant of semi-implicit spectral deferred corrections (SISDC) in which the stiff, fast dynamics correspond to fast propagating waves (``fast-wave slow-wave problem''). We show that for a scalar test problem with two imaginary ...

Forward-backward methods are a very useful tool for the minimization of a functional given by the sum of a differentiable term and a nondifferentiable one, and their investigation has comprised several efforts from many researchers in the last decade. In ...

Tensor network formats are an efficient tool for numerical computations in many dimensions, yet even this tool often becomes too time- and memory-consuming for a single compute node when applied to problems of scientific interest. Intending to overcome ...

As a phase space language for quantum mechanics, the Wigner function approach bears a close analogy to classical mechanics and has been drawing growing attention, especially in simulating quantum many-body systems. However, deterministic numerical solutions ...

We develop a framework for multifidelity information fusion and predictive inference in high-dimensional input spaces and in the presence of massive data sets. Hence, we tackle simultaneously the “big N" problem for big data and the curse of dimensionality ...