Spectral Gap Optimization of Divergence Type Diffusion Operators

Shiba Biswal, Karthik Elamvazhuthi, Hans Mittelmann, Spring Berman

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this paper, we address the problem of maximizing the spectral gap of a divergence type diffusion operator. Our main application of interest is characterizing the distribution of a swarm of agents that evolve on a bounded domain in d according to a Markov process. A subclass of the divergence type operators that we introduce in this paper can describe the distribution of the swarm across the domain. We construct an operator that stabilizes target distributions that are bounded and strictly positive almost everywhere on the domain. Optimizing the spectral gap of the operator ensures fast convergence to this target distribution. The optimization problem is posed as the minimization of the second largest eigenvalue modulus (SLEM) of the operator (the largest eigenvalue is 0). We use the well-known Courant-Fisher min-max principle to characterize the SLEM. We also present a numerical scheme for solving the optimization problem, and we validate our optimization approach for two example target distributions.

Original languageEnglish (US)
Title of host publicationEuropean Control Conference 2020, ECC 2020
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages1268-1273
Number of pages6
ISBN (Electronic)9783907144015
StatePublished - May 2020
Event18th European Control Conference, ECC 2020 - Saint Petersburg, Russian Federation
Duration: May 12 2020May 15 2020

Publication series

NameEuropean Control Conference 2020, ECC 2020

Conference

Conference18th European Control Conference, ECC 2020
Country/TerritoryRussian Federation
CitySaint Petersburg
Period5/12/205/15/20

ASJC Scopus subject areas

  • Artificial Intelligence
  • Decision Sciences (miscellaneous)
  • Control and Systems Engineering
  • Mechanical Engineering
  • Computational Mathematics
  • Control and Optimization

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