Algebraic methods in approximation theory

Hal Schenck

Research output: Contribution to journalShort survey

Abstract

This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis; their properties depend on combinatorics, topology, and geometry of a simplicial or polyhedral subdivision of a region in Rk, and are often quite subtle. We describe four algebraic techniques which are useful in the study of splines: homology, graded algebra, localization, and inverse systems. Our goal is to give a hands-on introduction to the methods, and illustrate them with concrete examples in the context of splines. We highlight progress made with these methods, such as a formula for the third coefficient of the polynomial giving the dimension of the spline space in high degree. The objects appearing here may be computed using the spline package of the Macaulay2 software system.

Original languageEnglish (US)
Pages (from-to)14-31
Number of pages18
JournalComputer Aided Geometric Design
Volume45
DOIs
StatePublished - 2016

Keywords

  • Homology
  • Inverse system
  • Localization
  • Polyhedral complex
  • Spline

ASJC Scopus subject areas

  • Modeling and Simulation
  • Automotive Engineering
  • Aerospace Engineering
  • Computer Graphics and Computer-Aided Design

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