# On thin plate spline interpolation

## TL;DR

This paper provides a PDE-based proof for improved error estimates in thin plate spline interpolation and demonstrates the effectiveness of ${
m f H}$-matrix techniques for large-scale problems.

## Contribution

It introduces a new PDE-based proof for error estimate improvements and applies ${
m f H}$-matrix methods to efficiently solve large interpolation problems.

## Key findings

- Error estimates can be improved by $h^{1/2}$
- ${
m f H}$-matrix techniques are effective for large problems
- The proof simplifies understanding of error bounds in spline interpolation

## Abstract

We present a simple, PDE-based proof of the result [M. Johnson, 2001] that the error estimates of [J. Duchon, 1978] for thin plate spline interpolation can be improved by $h^{1/2}$. We illustrate that ${\mathcal H}$-matrix techniques can successfully be employed to solve very large thin plate spline interpolation problems