Related papers: Numerical Integration over the Unit Sphere by usin…
Inverse scattering involving microwave and ultrasound waves require numerical solution of nonlinear optimization problem. To alleviate the computational burden of a full three-dimensional (3-D) inverse problem, it is a common practice to…
For a distribution function $F$ on $\mathbb{R}^d$ and a point $q\in \mathbb{R}^d$, the \emph{spherical depth} $\SphD(q;F)$ is defined to be the probability that a point $q$ is contained inside a random closed hyper-ball obtained from a pair…
We mimic the conventional explicit Total Variation Diminishing Runge-Kutta (TVDRK) schemes and propose a class of numerical integrators to solve differential equations on a unit sphere. Our approach utilizes the exponential map inherent to…
We developed a numerical code which evolves the semiclassical Einstein's equation (with the quantum stress-energy contribution added as a source term) for the spherically symmetric metric inside an evaporating semiclassical charged black…
We develop a test for spherical symmetry of a multivariate distribution $\Pr$ that works well even when the dimension of the data $d$ is larger than the sample size $n$. We propose a non-negative measure of spherical asymmetry $\zeta(\Pr)$…
A finite set $X \seq \RR^n$ with a weight function $w : X \longrightarrow \RR_{>0}$ is called \emph{Euclidean $t$-design} in $\RR^n$ (supported by $p$ concentric spheres) if the following condition holds: \[ \sum_{i=1}^p…
We apply boundary integral equations for the first time to the two-dimensional scattering of time-harmonic waves from a smooth obstacle embedded in a continuously-graded unbounded medium. In the case we solve the square of the wavenumber…
Numerical relativity codes that do not make assumptions on spatial symmetries most commonly adopt Cartesian coordinates. While these coordinates have many attractive features, spherical coordinates are much better suited to take advantage…
This short paper is concerned with the use of spherical t-designs as optimal designs for the spherical harmonic regression model in three dimensions over a range of specified criteria. The nature of the designs is explored and their…
We present an efficient numerical code based on spectral methods to integrate the field equations of general Robinson-Trautmann spacetimes. The most natural basis functions for the spectral expansion of the metric functions are spherical…
We present an $\epsilon$-bounded compression method for unit-norm embeddings that achieves 1.5$\times$ compression, 25% better than the best prior lossless method. The method exploits that spherical coordinates of high-dimensional unit…
We propose a sampling scheme on the sphere and develop a corresponding spherical harmonic transform (SHT) for the accurate reconstruction of the diffusion signal in diffusion magnetic resonance imaging (dMRI). By exploiting the antipodal…
T-SNE is a well-known approach to embedding high-dimensional data and has been widely used in data visualization. The basic assumption of t-SNE is that the data are non-constrained in the Euclidean space and the local proximity can be…
Sine-square deformation (SSD) is a treatment proposed in quantum systems, which spatially modifies a Hamiltonian, gradually decreasing the local energy scale from the center of the system toward the edges by a sine-squared envelope…
This paper proposes a sketching strategy based on spherical designs, which is applied to the classical spherical basis function approach for massive spherical data fitting. We conduct theoretical analysis and numerical verifications to…
This paper develops a systematic and geometric theory of optimal quantization on the unit sphere $\mathbb S^2$, focusing on finite uniform probability distributions supported on the spherical surface - rather than on lower-dimensional…
At its core a $t$-design is a method for sampling from a set of unitaries in a way which mimics sampling randomly from the Haar measure on the unitary group, with applications across quantum information processing and physics. We construct…
This expository paper provides a unified and pedagogical introduction to optimal quantization for probability measures supported on spherical curves and discrete subsets of the sphere, emphasizing both continuous and discrete settings. We…
A universal numerical method is developed for the investigation of magnetic neutron scattering. By applying the pseudospectral-time-domain (PSTD) algorithm to the spinor version of the Schr\"odinger equation, the evolution of the spin-state…
This paper introduces a novel boundary integral approach of shape uncertainty quantification for the Helmholtz scattering problem in the framework of the so-called parametric method. The key idea is to construct an integration grid whose…