SIAM Journal on Scientific Computing

We study a novel variation on the Ulam--von Neumann Monte Carlo method for solving a linear system. This is an old randomized procedure that results from using a random walk to stochastically evaluate terms in the Neumann series. In order to apply this ...

In [SIAM J. Sci. Comput., 36 (2014), pp. A693--A713] the authors present a new coarse propagator for the parareal method applied to oscillatory PDEs that exhibit time-scale separation and show, under certain regularity constraints, superlinear convergence ...

The identification of the interface of an inclusion in a diffusion process is considered. This task is viewed as a parameter identification problem in which the parameter space bears the structure of a shape manifold. A corresponding optimum experimental ...

We present a framework for derivative-based global sensitivity analysis (GSA) for models with high-dimensional input parameters and functional outputs. We combine ideas from derivative-based GSA, random field representation via Karhunen--Loève expansions,...

Fractional Laplace equations are becoming important tools for mathematical modeling and prediction. Recent years have shown much progress in developing accurate and robust algorithms to numerically solve such problems, yet most solvers for fractional ...

We present a high-index optimization-based shrinking dimer (HiOSD) method to compute index-$k$ saddle points as a generalization of the optimization-based shrinking dimer method for index-1 saddle points [L. Zhang, Q. Du, and Z. Zheng, SIAM J. Sci. ...

For a scalar function conserved by an unsteady flow, its flux through a simple curve is usually expressed as an Eulerian flux, a double integral both in time and in space. We show that this Eulerian flux equals a Lagrangian flux, a line integral in space ...

A novel approach for nonintrusive uncertainty propagation is proposed. Our approach overcomes the limitation of many traditional methods, such as generalized polynomial chaos methods, which may lack sufficient accuracy when the quantity of interest ...

In this paper, we study the ratio of the $L_1 $ and $L_2 $ norms, denoted as $L_1/L_2$, to promote sparsity. Due to the nonconvexity and nonlinearity, there has been little attention to this scale-invariant model. Compared to popular models in the ...

This paper is concerned with direct and inverse scattering by a locally perturbed infinite plane (called a locally rough surface in this paper) on which a Neumann boundary condition is imposed. A novel integral equation formulation is proposed for the ...

We construct and analyze a class of extrapolated and linearized Runge--Kutta (RK) methods, which can be of arbitrarily high order, for the time discretization of the Allen--Cahn and Cahn--Hilliard phase field equations, based on the scalar auxiliary ...

In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the triangle, including Clenshaw's algorithm and sparse differentiation operators. ...

For the time-fractional phase-field models, the corresponding energy dissipation law has not been well studied on both the continuous and the discrete levels. In this work, we address this open issue. More precisely, we prove for the first time that the ...

In this paper, a novel pre- and post-processing algorithm is presented that can double the convergence rate of finite element approximations for linear wave problems. In particular, it is shown that a $q$-step pre- and post-processing algorithm can ...

We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [Appl. Numer. Math., 122 (2017), pp. 14--38]), which we hereby refer to as the surrogate matrix methodology. This methodology ...

Gaussian kernels can be an efficient and accurate tool for multivariate interpolation. For smooth functions, high accuracies are often achieved near the flat limit where the interpolation matrix becomes increasingly ill-conditioned. Stable evaluation ...

Domain decomposition methods are robust and parallel scalable, preconditioned iterative algorithms for the solution of the large linear systems arising in the discretization of elliptic partial differential equations by finite elements. The convergence ...

An efficient $hp$-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We show that ...

In this paper we present benchmark problems for non-self-adjoint elliptic eigenvalue problems with large defect and ascent. We describe the derivation of the benchmark problem with a discontinuous coefficient and mixed boundary conditions. Numerical ...

In this paper, a fast multipole method (FMM) is proposed to compute long-range interactions of wave sources embedded in 3-dimensional (3-D) layered media. The layered media Green's function for the Helmholtz equation, which satisfies the transmission ...

A parareal algorithm based on an exponential $\theta$-scheme is proposed for the stochastic Schrödinger equation with weak damping and additive noise. It proceeds as a two-level temporal parallelizable integrator with the exponential $\theta$-scheme as the ...

In this paper, a mass-conservative temporal second order and spatial fourth order characteristic finite volume method (MC-T2S4-CFVM) for solving atmospheric pollution advection diffusion problems is developed. While the characteristics tracking is ...

This paper concerns the numerical approximation of low-energy eigenstates of the linear random Schrödinger operator. Under oscillatory high-amplitude potentials with a sufficient degree of disorder it is known that these eigenstates localize in the sense of ...

In this paper, we consider the fast numerical solution of an optimal control formulation of the Keller--Segel model for bacterial chemotaxis. Upon discretization, this problem requires the solution of huge-scale saddle point systems to guarantee accurate ...

We propose and analyze a new numerical method for computing the ground state of the modified Gross--Pitaevskii equation for modeling the Bose--Einstein condensate with a higher order interaction by adapting the density function formulation and the ...

The piecewise constant Mumford--Shah (PCMS) model and the Rudin--Osher--Fatemi (ROF) model are two important variational models in image segmentation and image restoration, respectively. In this paper, we explore a linkage between these models. We prove ...

Orthogonal maps are the solutions of the mathematical model of paper-folding, also called the origami problem. They consist of a system of first-order fully nonlinear equations involving the gradient of the solution. The Dirichlet problem for orthogonal ...

This paper is concerned with uniqueness, phase retrieval, and shape reconstruction methods for solving inverse electromagnetic source scattering problems with multifrequency sparse phased or phaseless far field data. With the phased data, we show that the ...

Many problems in solid state physics and quantum chemistry require the solution of the Schrödinger equation in the presence of an electromagnetic field. In this paper, we construct, analyze, and numerically validate conservative discontinuous Galerkin (...