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Geometric diffusions as a tool for harmonic analysis and structure definition of data: multiscale methods.

R R Coifman1, S Lafon, A B Lee

  • 1Department of Mathematics, Program in Applied Mathematics, Yale University, New Haven, CT 06510, USA. coifman-ronald@yale.edu

Proceedings of the National Academy of Sciences of the United States of America
|May 19, 2005
PubMed
Summary

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This study introduces a novel framework using diffusion semigroups and Markov matrices for multiscale analysis. It enables efficient data representation and homogenization of complex structures, generalizing Newtonian principles.

Area of Science:

  • Multiscale geometric organization
  • Data analysis and representation
  • Computational mathematics

Background:

  • Previous work established a framework for structural multiscale geometric organization.
  • Complex structures require advanced methods for organization and representation.
  • Existing methods may not efficiently handle the multiscale nature of heterogeneous structures.

Purpose of the Study:

  • To generate multiscale analyses using diffusion semigroups for organizing complex structures.
  • To develop scaling functions of Markov matrices for macroscopic descriptions.
  • To construct fast-order N algorithms for data representation and homogenization.

Main Methods:

  • Application of diffusion semigroups to generate multiscale analyses.

Related Experiment Videos

  • Construction of scaling functions based on Markov matrices.
  • Development of fast-order N algorithms.

    • Main Results:

    • A method for organizing and representing complex structures across multiple scales.
    • Macroscopic descriptions derived from local transitions via Markov matrix diffusion.
    • Efficient algorithms for data representation and homogenization of heterogeneous structures.

    Conclusions:

    • The diffusion semigroup approach provides a robust method for multiscale analysis.
    • The developed algorithms offer efficient solutions for complex data organization and homogenization.
    • This work generalizes aspects of the Newtonian paradigm for system analysis.