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Thomas Hangelbroek
DSCF1595

Associate Professor

Department of Mathematics

University of Hawaiʻi at Mānoa

2565 McCarthy Mall

Honolulu, HI 96822

Email: hangelbr@math.hawaii.edu

My research lies at the interface of approximation theory, harmonic analysis and numerical analysis. I study kernels, radial basis functions, splines, and wavelets: i.e., basic tools for scattered data approximation, machine learning, and meshless solution of PDEs. I am interested in the approximation power of these tools, and in applying them to interesting problems. My work is supported by National Science Foundation grant DMS-2010051, New Directions in Mesh-Free Approximation with Localizable Kernels.

Recent Preprints

Extending Data to Improve Stability and Error Estimates Using Asymmetric Kansa-like Methods to Solve PDEs, with Francis J. Narcowich and Joseph D. Ward.

A Semi-Lagrangian scheme on embedded manifolds using generalized local polynomial reproductions, with Christian Rieger and Grady Wright
to appear in SIAM Journal on Numerical Analysis.

Generalized local polynomial reproductions, with Christian Rieger and Grady Wright
to appear in Journal of Foundations of Computational Mathematics.

Spectral stability and perturbation results for kernel differentiation matrices on the sphere, with Christian Rieger and Grady Wright
Numerische Mathematik 157, (2025), no. 2, pp. 505–538

Kernel Multigrid on Manifolds, with Christian Rieger.
J. Complexity 86 (2025), Paper No. 101900.

Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms , with Christian Rieger.
Math. Comp. 94, (2025), no. 351, pp. 381-407.

Highly Localized RBF Lagrange Functions for Finite Difference Methods on Spheres , with Wolfgang Erb, F. Narcowich, C. Rieger, and J. Ward.
BIT Numer Math 64 (2024), no. 2, Paper No. 16.

Most of my preprints can be found on the arXiv.

Recent Talks

Local Approximation with Kernels, AT15, San Antonio, May 2016

Kernel Approximation, Elliptic PDE and Boundary Effects, ICMA, Schloss Rauischholzhausen, March 2016

Beyond Quasi-unifomity: Kernel Approximation with a Local Mesh Ratio, CSE 15, Salt Lake City, March 2015

Recent progress on boundary effects in kernel approximation, FOCM 2014, Montevideo, December 2014

On boundaries in approximation by polyharmonic kernels, Curves and Surfaces 8, Paris, June 2014

Teaching

In Spring 2020 I am teaching Math 633 (Functional Analysis). If you are a student in one of my classes, course materials can be found on Laulima.

Previous Courses:

Fall 2019: Math 402 (Introduction to Partial Differential Equations)

Spring 2019: Math 100 (Survey of Mathematics), Math 242 (Calculus 2)

Fall 2018: Math 649 (Topics in Analysis: Abstract Harmonic Analysis)

Spring 2018: Math 633 (Functional Analysis)

Fall 2017: on leave

Spring 2017: Math 305 (Probabilistic Models)

Fall 2016: Math 241 (Calculus 1), Math 631 (Measure and Integration)

Spring 2016: Math 242 (Calculus 2)

Fall 2015: Math 321 (Intro to Advanced Mathematics), Math 649 (Harmonic Analysis)

Spring 2015: Math 633 (Functional Analysis)

Fall 2014: Math 252A (Accelerated Calculus 2), Math 431 (Principles of Analysis 1)