相关论文: Fast expansion into harmonics on the ball

We present algorithms for solving spatially nonlocal diffusion models on the unit sphere with spectral accuracy in space. Our algorithms are based on the diagonalizability of nonlocal diffusion operators in the basis of spherical harmonics,…

For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion…

The vector space $V^k$ of the eigenfunctions of the Laplacian on the three sphere $S^3$, corresponding to the same eigenvalue $lambda_k = -k (k +2)$, has dimension $(k + 1)^2$. After recalling the standard bases for $V^k$, we introduce a…

We present an indirect higher order boundary element method utilising NURBS mappings for exact geometry representation and an interpolation-based fast multipole method for compression and reduction of computational complexity, to counteract…

In this paper, we consider the Dirichlet problem of the three-dimensional Laplace equation in the unit ball with a shrinking hole. The problem typically arises from homogenization problems in domains perforated with tiny holes. We give an…

Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $\epsilon$. If $\epsilon$ is…

We consider the propagation of acoustic time-harmonic waves in a homogeneous media containing periodic lattices of spherical or cylindrical inclusions. It is assumed that the wavelength has the order of the periods of the lattice while the…

An explicit expression for the general bivariate Krawtchouk polynomials is obtained in terms of the standard Krawtchouk and dual Hahn polynomials. The bivariate Krawtchouk polynomials occur as matrix elements of the unitary reducible…

We show that there are harmonic functions on a ball ${\mathbb{B}_n}$ of $\mathbb{R}^n$, $n\ge 2$, that are continuous up to the boundary (and even H\"older continuous) but not in the Sobolev space $H^s(\mathbb{B}_n)$ for any $s$…

We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the $d$-dimensional Euclidean space. When the radius $r$ of the balls is large, this volume can be approximated by a polynomial of…

We report on a new 3D numerical code designed to solve the Einstein equations for general vacuum spacetimes. This code is based on the standard 3+1 approach using cartesian coordinates. We discuss the numerical techniques used in developing…

We prove dispersive estimates for the wave equation in the exterior of a torus. Because no separation of variables into a basis of eigenfunctions and eigenvalues exists for the time harmonic problem, we introduce a related approximate…

This paper is devoted to the efficient numerical solution of the Helmholtz equation in a two- or three-dimensional rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and…

In this paper we address some problems concerning an approximate Dirichlet domain. We show that under some assumptions the approximate Dirichlet domain can work equally well as an exact Dirichlet domain. In particular, we consider a problem…

We present an efficient and highly scalable geometric method for two-dimensional ideal fluid dynamics on the sphere. The starting point is Zeitlin's finite-dimensional model of hydrodynamics. The efficiency stems from exploiting a…

We use lookup tables to design faster algorithms for important algebraic problems over finite fields. These faster algorithms, which only use arithmetic operations and lookup table operations, may help to explain the difficulty of…

In this paper, we establish hardness and approximation results for various $L_p$-ball constrained homogeneous polynomial optimization problems, where $p \in [2,\infty]$. Specifically, we prove that for any given $d \ge 3$ and $p \in…

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…

We develop a spectral method for solving the incompressible generalized Navier--Stokes equations in the ball with no-flux and prescribed slip boundary conditions. The algorithm achieves an optimal complexity per time step of…

Consider the Laplacian in a bounded domain in R^d with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are `quasi-orthogonal' on the boundary with respect to a certain norm. Boundary orthogonality is proved…