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Functional Data Approximation on Bounded Domains using Polygonal Finite Elements.

Juan Cao1,2, Yanyang Xiao3, Zhonggui Chen3

  • 1School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China.

Computer Aided Geometric Design
|June 13, 2018
PubMed
Summary
This summary is machine-generated.

We introduce novel quadratic serendipity finite elements for approximating functional data on 2D domains. These elements offer superior accuracy and efficiency on polygonal meshes compared to traditional methods.

Keywords:
Voronoi tessellationbarycentric coordinatesdata approximationpolygonal elements

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Computer Graphics

Background:

  • Functional data approximation on 2D domains is crucial for various scientific and engineering applications.
  • Traditional finite element methods often rely on simplicial meshes, which can be inefficient for complex geometries.
  • Generalized barycentric coordinates offer alternative frameworks for constructing finite elements.

Purpose of the Study:

  • To develop and analyze piecewise approximations of functional data on arbitrary 2D bounded domains using generalized barycentric finite elements.
  • To evaluate the approximation quality and convergence of quadratic serendipity elements on polygonal meshes.
  • To propose algorithms for generating adaptive meshes for functional data approximation.

Main Methods:

  • Construction and analysis of piecewise approximations using generalized barycentric finite elements, specifically quadratic serendipity elements.
  • Numerical experiments comparing approximation qualities using Wachspress, natural neighbor, Poisson, and mean value coordinates.
  • Development of two greedy algorithms for generating Voronoi meshes for adaptive approximations.
  • Refinement of polygonal meshes and parameter coefficients using L2-optimization.

Main Results:

  • Quadratic serendipity elements on polygonal domains exhibit space/accuracy advantages over traditional finite elements on simplicial meshes.
  • The proposed greedy algorithms effectively generate Voronoi meshes for adaptive functional data approximation.
  • L2-optimization further enhances the piecewise functional approximation accuracy.
  • Demonstrated efficacy in modeling features and discontinuities in functional data and image approximation.

Conclusions:

  • Quadratic serendipity finite elements provide an efficient and accurate method for functional data approximation on polygonal domains.
  • The developed greedy meshing algorithms and optimization techniques improve adaptive approximation capabilities.
  • This approach shows significant potential for applications in functional data analysis and image processing.