arXiv subjects

Thomas Hangelbroek

Publications and source records attributed to Thomas Hangelbroek.

At least 19 recordsLinked to original sources

arXiv2026

Kernel approximation beyond the native space -- with applications to approximation on manifolds

This article treats kernel approximation and interpolation on embedded manifolds of $\mathbb{R}^N$using restrictions of positive and conditionally positive definite kernels. The main challenge is to develop an approximation theory that treats error measured in highly regular smoothness spaces relative to the kernel. This means that the order of smoothness is higher than that of the kernel's associated native space (in the positive definite case, the reproducing kernel Hilbert space generated by the kernel). This prevents the use of standard techniques for controlling error in this setting, especially RKHS space arguments like orthogonality of the interpolation projector, or bounds using the {\em power function}. To address this challenge, we extend methods for treating target functions given as potentials introduced by DeVore and Ron, and give conditions guaranteeing that restrictions of such functions span Sobolev smoothness spaces on the manifolds. Together with kernel-based Bernstein inequalities for embedded manifolds, these results give a comprehensive theory for interpolation error. As an application of the theory, we derive new error estimates for approximating the eigenspace of certain differential operators arising from kernel-based methods for partial differential equations on manifolds.

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arXiv2026

Smoothing by, and eccentric smoothing of, compactly supported RBFs

We consider compactly supported RBFs having algebraically decaying Fourier transforms. Here we focus especially on generalized Wendland RBFs and their modification by making them smoother away from zero, a process we call eccentric smoothing. Specifically, we consider mapping properties of the integral operators (sometimes known as covariance operator) for these novel type RBFs. Moreover, we show that eccentric smoothing makes wavelet-inspired compression technique for the kernel matrix feasible.

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arXiv2025

Generalized local polynomial reproductions

We present a general framework, treating Lipschitz domains in Riemannian manifolds, that provides conditions guaranteeing the existence of norming sets and generalized local polynomial reproduction - a powerful tool used in the analysis of various mesh-free methods and a mesh-free method in its own right. As a key application, we prove the existence of smooth local polynomial reproductions on compact subsets of algebraic manifolds in $\mathbb{R}^n$ with Lipschitz boundary. These results are then applied to derive new findings on the existence, stability, regularity, locality, and approximation properties of shape functions for a coordinate-free moving least squares approximation method on algebraic manifolds, which operates directly on point clouds without requiring tangent plane approximations. There are two appendices: the first derives high order Markov inequalities for polynomials on algebraic manifolds and the second gives instructions for calculating the dimension of the space of degree $m$ polynomials restricted to a real algebraic variety.

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arXiv2025

A Semi-Lagrangian scheme on embedded manifolds using generalized local polynomial reproductions

We analyze rates of uniform convergence for a class of high-order semi-Lagrangian schemes for first-order, time-dependent partial differential equations on embedded submanifolds of $\mathbb{R}^d$ (including advection equations on surfaces) by extending the error analysis of Falcone and Ferretti. A central requirement in our analysis is a remapping operator that achieves both high approximation orders and strong stability, a combination that is challenging to obtain and of independent interest. For this task, we propose a novel mesh-free remapping operator based on $\ell_1$ minimizing generalized polynomial reproduction, which uses only point values and requires no additional geometric information from the manifold (such as access to tangent spaces or curvature). Our framework also rigorously addresses the numerical solution of ordinary differential equations on manifolds via projection methods. We include numerical experiments that support the theoretical results and also suggest some new directions for future research.

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arXiv2025

Extending Data to Improve Stability and Error Estimates Using Asymmetric Kansa-like Methods to Solve PDEs

In this paper, a theoretical framework is presented for the use of a Kansa-like method to numerically solve elliptic partial differential equations on spheres and other manifolds. The theory addresses both the stability of the method and provides error estimates for two different approximation methods. A Kansa-like matrix is obtained by replacing the test point set $X$, used in the traditional Kansa method, by a larger set $Y$, which is a norming set for the underlying trial space. This gives rise to a rectangular matrix. In addition, if a basis of Lagrange (or local Lagrange) functions is used for the trial space, then it is shown that the stability of the matrix is comparable to the stability of the elliptic operator acting on the trial space. Finally, two different types of error estimates are given. Discrete least squares estimates of very high accuracy are obtained for solutions that are sufficiently smooth. The second method, giving similar error estimates, uses a rank revealing factorization to create a ``thinning algorithm'' that reduces $\#Y$ to $\#X$. In practice, this algorithm doesn't need $Y$ to be a norming set.

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arXiv2024

Kernel Multi-Grid on Manifolds

Kernel methods for solving partial differential equations on surfaces have the advantage that those methods work intrinsically on the surface and yield high approximation rates if the solution to the partial differential equation is smooth enough. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods to some extent, although the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a $τ\ge 2$-cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting linear algebra can be accelerated by using the Lagrange function decay, with convergence rates which are obtained by a rigorous analysis. In particular, we can show that the computational cost to solve the linear system scales log-linear in the degrees of freedom.

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arXiv2023

Spectral stability and perturbation results for kernel differentiation matrices on the sphere

We investigate the spectrum of differentiation matrices for certain operators on the sphere that are generated from collocation at a set of scattered points $X$ with positive definite and conditionally positive definite kernels. We focus on the cases where these matrices are constructed from collocation using all the points in $X$ and from local subsets of points (or stencils) in $X$. The former case are called global methods (e.g., the Kansa or radial basis function (RBF) pseudospectral method), while the latter are referred to as local methods (e.g., the RBF finite difference (RBF-FD) method). Both techniques are used extensively for numerically solving certain partial differential equations on spheres, as well as other domains. For time-dependent PDEs like the diffusion equation, the spectrum of the differentiation matrices and their stability under perturbations are central to understanding the temporal stability of the underlying numerical schemes. In the global case, we present a perturbation estimate for differentiation matrices which discretize operators that commute with the Laplace-Beltrami operator. In doing so, we demonstrate that if such an operator has negative (non-positive) spectrum, then the differentiation matrix does, too. For conditionally positive definite kernels this is particularly challenging since the differentiation matrices are not necessarily diagonalizable. This perturbation theory is then used to obtain bounds on the spectra of the local RBF-FD differentiation matrices based on the conditionally positive definite surface spline kernels. Numerical results are presented to confirm the theoretical estimates.

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arXiv2023

Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms

Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick, which shows that functions having double the smoothness required by the RKHS (along with complicated, albeit complicated boundary behavior) can be approximated with higher convergence rates than the optimal rates for the whole space. Other advances allowed interpolation of target functions which are less smooth, and different norms which measure interpolation error. The current state of the art of error analysis for RBF interpolation treats target functions having smoothness up to twice that of the native space, but error measured in norms which are weaker than that required for membership in the RKHS. Motivated by the fact that the kernels and the approximants they generate are smoother than required by the native space, this article extends the doubling trick to error which measures higher smoothness. This extension holds for a family of kernels satisfying easily checked hypotheses which we describe in this article, and includes many prominent RBFs. In the course of the proof, new convergence rates are obtained for the abstract operator considered by Devore and Ron, and new Bernstein estimates are obtained relating high order smoothness norms to the native space norm.

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arXiv2023

Highly Localized RBF Lagrange Functions for Finite Difference Methods on Spheres

The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the spheres can be used to create effective, stable finite difference methods based on radial basis functions (RBF-FD). For certain classes of PDEs this approach leads to precise convergence estimates for stencils which grow moderately with increasing discretization fineness.

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arXiv2019

Anisotropic Gaussian approximation in $L_2(\mathbb{R}^2)$

Let $\mathcal{D}$ be the dictionary of Gaussian mixtures: the functions created by affine change of variables of a single Gaussian in $n$ dimensions. $\mathcal{D}$ is used pervasively in scientific applications to a degree that practitioners often employ it as their default choice for representing their scientific object. Its use in applications hinges on the perception that this dictionary is large enough, and its members are local enough in space and frequency, to provide efficient approximation to "almost all objects of interest". However, and perhaps surprisingly, only a handful of concrete theoretical results are actually known on the ability to use Gaussian mixtures in lieu of mainstream representation systems. The present paper shows that, in 2D, Gaussian mixtures are effective in resolving anisotropic structures, too. In this setup, the "smoothness class" is comprised of 2D functions that are sparsely represented using curvelets. An algorithm for $N$-term approximation from (a small subset of) $\mathcal{D}$ is presented, and the error bounds are then shown to be on par with the errors of $N$-term curvelet approximation. The latter are optimal, essentially by definition. Our approach is based on providing effective approximation from $\mathcal{D}$ to the members of the curvelet system, mimicking the approach in arXiv:0802.2517 and arXiv:0911.2803 where the mother wavelets are approximated. When the error is measured in the $1$-norm, this adaptation of the prior approach, combined with standard tools, yields the desired results. However, handling the $2$-norm case is much more subtle and requires substantial new machinery: in this case, the error analysis cannot be solely done on the space domain: some of it has to be carried out on frequency. Since, on frequency, all members of $\mathcal{D}$ are centered at the origin, a delicate analysis for controlling the error there is needed.

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arXiv2017

Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds

This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are spatially highly localized. The construction of such functions is computationally efficient and generalizes the construction given by the authors for restricted surface splines on $\mathbb{R}^d$. The kernels for which the theory applies includes the Sobolev-Matérn kernels for closed, compact, connected, $C^\infty$ Riemannian manifolds.

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arXiv2016

An inverse theorem for compact Lipschitz regions in $R^d$ using localized kernel bases

While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important and difficult setting of bounded domains have been notably absent. This article develops inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods. The inverse estimates we consider control Sobolev norms of linear combinations of a localized basis by the $L_p$ norm over a bounded domain. The localized basis is generated by forming local Lagrange functions for certain types of RBFs (namely Matérn and surface spline RBFs). In this way it extends the boundary-free construction of Fuselier, Hangelbroek, Narcowich, Ward and Wright.

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arXiv2013

The Polyharmonic Dirichlet Problem and Path Counting

The purpose of this article is to provide a solution to the $m$-fold Laplace equation in the half space $R_+^d$ under certain Dirichlet conditions. The solutions we present are a series of $m$ boundary layer potentials. We give explicit formulas for these layer potentials as linear combinations of powers of the Laplacian applied to the Dirichlet data, with coefficients determined by certain path counting problems.

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arXiv2012

On the density of polyharmonic splines

This article treats the question of fundamentality of the translates of a polyharmonic spline kernel (also known as a surface spline) in the space of continuous functions on a compact set $Ω\subset \RR^d$ when the translates are restricted to $Ω$. Fundamentality is not hard to demonstrate when a low degree polynomial may be added or when translates are permitted to lie outside of $Ω$; the challenge of this problem stems from the presence of the boundary, for which all successful approximation schemes require an added polynomial. When $Ω$ is the unit ball, we demonstrate that translates of polyharmonic splines are fundamental by considering two related problems: the fundamentality in the space of functions vanishing at the boundary and fundamentality of the restricted kernel in the space of continuous function on the sphere. This gives rise to a new approximation scheme composed of two parts: one which approximates purely on $\partial Ω$, and a second part involving a shift invariant approximant of a function vanishing outside of a neighborhood $Ω$.

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arXiv2011

On Local RBF Approximation

The purpose of this paper is to investigate RBF approximation with highly nonuniform centers. Recently, DeVore and Ron have developed a notion of the local density of a set of centers -- a notion that permits precise pointwise error estimates for surface spline approximation. We give an equivalent, alternative characterization of local density, one that allows effective placement of centers at different resolutions. We compare, also, the pointwise results of DeVore--Ron to previously works of Wu and Schaback and of Duchon.

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arXiv2010

Kernel Approximation on Manifolds II: The $L_{\infty}$-norm of the $L_2$-projector

This article addresses two topics of significant mathematical and practical interest in the theory of kernel approximation: the existence of local and stable bases and the L_p--boundedness of the least squares operator. The latter is an analogue of the classical problem in univariate spline theory, known there as the "de Boor conjecture". A corollary of this work is that for appropriate kernels the least squares projector provides universal near-best approximations for functions f\in L_p, 1\le p\le \infty.

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arXiv2010

Surface Spline Approximation on SO(3)

The purpose of this article is to introduce a new class of kernels on SO(3) for approximation and interpolation, and to estimate the approximation power of the associated spaces. The kernels we consider arise as linear combinations of Green's functions of certain differential operators on the rotation group. They are conditionally positive definite and have a simple closed-form expression, lending themselves to direct implementation via, e.g., interpolation, or least-squares approximation. To gauge the approximation power of the underlying spaces, we introduce an approximation scheme providing precise L_p error estimates for linear schemes, namely with L_p approximation order conforming to the L_p smoothness of the target function.

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arXiv2010

The Penalized Lebesgue Constant for Surface Spline Interpolation

Problems involving approximation from scattered data where data is arranged quasi-uniformly have been treated by RBF methods for decades. Treating data with spatially varying density has not been investigated with the same intensity, and is far less well understood. In this article we consider the stability of surface spline interpolation (a popular type of RBF interpolation) for data with nonuniform arrangements. Using techniques similar to those recently employed by Hangelbroek, Narcowich and Ward to demonstrate the stability of interpolation from quasi-uniform data on manifolds, we show that surface spline interpolation on R^d is stable, but in a stronger, local sense. We also obtain pointwise estimates showing that the Lagrange function decays very rapidly, and at a rate determined by the local spacing of datasites. These results, in conjunction with a Lebesgue lemma, show that surface spline interpolation enjoys the same rates of convergence as those of the local approximation schemes recently developed by DeVore and Ron.

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