Particle systems and kinetic equations modeling interacting agents in high dimension

M. Fornasier*, Jan Haskovec, J. Vybíral

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

In this paper we explore how concepts of high-dimensional data compression via random projections onto lower-dimensional spaces can be applied for tractable simulation of certain dynamical systems modeling complex interactions. In such systems, one has to deal with a large number of agents (typically millions) in spaces of parameters describing each agent of high dimension (thousands or more). Even with today's powerful computers, numerical simulations of such systems are prohibitively expensive. We propose an approach for the simulation of dynamical systems governed by functions of adjacency matrices in high dimension, by random projections via Johnson-Lindenstrauss embeddings, and recovery by compressed sensing techniques. We show how these concepts can be generalized to work for associated kinetic equations by addressing the phenomenon of the delayed curse of dimension, known in information-based complexity for optimal numerical integration problems and measure quantization in high dimensions.

Original languageEnglish (US)
Pages (from-to)1727-1764
Number of pages38
JournalMultiscale Modeling and Simulation
Volume9
Issue number4
DOIs
StatePublished - Dec 1 2011

Keywords

  • Compressed sensing
  • Delayed curse of dimension
  • Dimensionality reduction
  • Dynamical systems
  • Flocking and swarming
  • High-dimensional kinetic equations
  • Johnson-Lindenstrauss embedding
  • Optimal integration of measures in high dimension

ASJC Scopus subject areas

  • Chemistry(all)
  • Modeling and Simulation
  • Ecological Modeling
  • Physics and Astronomy(all)
  • Computer Science Applications

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