Related papers: A fast matrix-free algorithm for spectral approxim…
Orthogonal systems in $\mathrm{L}_2(\mathbb{R})$, once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation…
This work establishes new results on spectral theory and time evolution for matrix-valued discrete Schr\"odinger operators on the space of square-summable matrix sequences. The matrix-valued formalism is employed to streamline notation,…
We show that the method of splitting the operator ${\rm e}^{\epsilon(T+V)}$ to fourth order with purely positive coefficients produces excellent algorithms for solving the time-dependent Schr\"odinger equation. These algorithms require…
This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many…
We construct quasi-Monte Carlo methods to approximate the expected values of linear functionals of Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in…
We review and further develop the recently introduced numerical approach for scattering calculations based on a so called pseudo-time Schroedinger equation, which is in turn a modification of the damped Chebyshev polynomial expansion…
In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional…
A Discontinuous Galerkin (DG) Finite Element Method (FEM) approach for the 3D time dependent Maxwell equations in unbounded domains is presented. The method (implemented in the FEM library NGsolve) is based on the covariant transformation…
This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations [1] with the problem coefficient depending on only one…
Spacetime discontinuous Galerkin (SDG) finite element methods are used to solve such PDEs involving space and time variables arising from wave propagation phenomena in important applications in science and engineering. To support an…
We present a spectrally accurate fast algorithm for evaluating the solution to the scalar wave equation in free space driven by a large collection of point sources in a bounded domain. With $M$ sources temporally discretized by $N_t$ time…
We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the…
Many scientific and engineering challenges can be formulated as optimization problems which are constrained by partial differential equations (PDEs). These include inverse problems, control problems, and design problems. As a major…
This article presents a new high-order accurate algorithm for finding a particular solution to a linear, constant-coefficient partial differential equation (PDE) by means of a convolution of the volumetric source function with the Green's…
We give a stochastic optimization algorithm that solves a dense $n\times n$ real-valued linear system $Ax=b$, returning $\tilde x$ such that $\|A\tilde x-b\|\leq \epsilon\|b\|$ in time: $$\tilde O((n^2+nk^{\omega-1})\log1/\epsilon),$$ where…
Given an arbitrary matrix $A\in\mathbb{R}^{n\times n}$, we consider the fundamental problem of computing $Ax$ for any $x\in\mathbb{R}^n$ such that $Ax$ is $s$-sparse. While fast algorithms exist for particular choices of $A$, such as the…
The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation…
Achieving a substantial part of peak performance on todays and future high-performance computing systems is a major challenge for simulation codes. In this paper we address this question in the context of the numerical solution of partial…
We present a family of algorithms for the numerical approximation of the Schr\"odinger equation with potential concentrated at a finite set of points. Our methods belong to the so-called fast and oblivious convolution quadrature algorithms.…
In order to inherit numerically the ergodicity of the damped stochastic nonlinear Schr\"odinger equation with additive noise, we propose a fully discrete scheme, whose spatial direction is based on spectral Galerkin method and temporal…