Related papers: Localized linear polynomial operators and quadratu…
This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial…
Numerical simulations with rigid particles, drops or vesicles constitute some examples that involve 3D objects with spherical topology. When the numerical method is based on boundary integral equations, the error in using a regular…
We study a popular algorithm for fitting polynomial curves to scattered data based on the least squares with gradient weights. We show that sometimes this algorithm admits a substantial reduction of complexity, and, furthermore, find…
Many of the distributed localization algorithms are based on relaxed optimization formulations of the localization problem. These algorithms commonly rely on first-order optimization methods, and hence may require many iterations or…
In this paper we investigate multivariate integration in weighted unanchored Sobolev spaces of smoothness of arbitrarily high order. As quadrature points we employ higher order polynomial lattice point sets over $\mathbb{F}_{2}$ which are…
Locally refined spline surfaces (LRB) is a representation well suited for scattered data approximation. When a data set has local details in some areas and is largely smooth in other, LR B-splines allow the spatial distribution of degrees…
Highly localized kernels constructed by orthogonal polynomials have been fundamental in recent development of approximation and computational analysis on the unit sphere, unit ball and several other regular domains. In this work we first…
We introduce computable projection operators onto piecewise polynomial spaces, defined via sampling and discrete least-squares polynomial approximations. The resulting mappings exhibit (almost) optimal approximation properties in $L^2$ and…
Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate…
We consider a polynomial reconstruction of smooth functions from their noisy values at discrete nodes on the unit sphere by a variant of the regularized least-squares method of An et al., SIAM J. Numer. Anal. 50 (2012), 1513--1534. As nodes…
In this paper a local approximation method on the sphere is presented. As interpolation scheme we consider a partition of unity method, such as the modified spherical Shepard's method, which uses zonal basis functions (ZBFs) plus spherical…
In this article we propose building general-purpose function approximators on top of Haar Scattering Networks. We advocate that this architecture enables a better comprehension of feature extraction, in addition to its implementation…
We propose a new locally smeared operator product expansion to decompose nonlocal operators in terms of a basis of smeared operators. The smeared operator product expansion formally connects nonperturbative matrix elements determined…
This paper considers filtered polynomial approximations on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$, obtained by truncating smoothly the Fourier series of an integrable function $f$ with the help of a "filter" $h$, which is a…
Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer…
This paper discusses a methodology for determining a functional representation of a random process from a collection of scattered pointwise samples. The present work specifically focuses onto random quantities lying in a high dimensional…
Almost exponentially localized polynomial kernels are constructed on the unit ball $B^d$ in $\RR^d$ with weights %functions $W_\mu(x)= (1-|x|^2)^{\mu-1/2}$, $\mu \ge 0$, by smoothing out the coefficients of the corresponding orthogonal…
Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a usual compact space $\mathcal M$, Gr\"ochenig introduced the weighted least squares polynomials and the least squares quadrature from pointwise samples of a function, and…
Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer…
In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local variation of the number of quadrature points employed on each subinterval. In this…