TY - GEN
T1 - Spectral Gap Optimization of Divergence Type Diffusion Operators
AU - Biswal, Shiba
AU - Elamvazhuthi, Karthik
AU - Mittelmann, Hans
AU - Berman, Spring
N1 - Funding Information:
This work was supported by the Arizona State University Global Security Initiative. The work of Hans Mittelmann was supported in part by the Air Force Office of Scientific Research under grant FA9550-19-1-0070.
PY - 2020/5
Y1 - 2020/5
N2 - 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.
AB - 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.
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M3 - Conference contribution
AN - SCOPUS:85090162957
T3 - European Control Conference 2020, ECC 2020
SP - 1268
EP - 1273
BT - European Control Conference 2020, ECC 2020
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 18th European Control Conference, ECC 2020
Y2 - 12 May 2020 through 15 May 2020
ER -