Center for Advanced Intelligence Project

Speaker: Yuka Hashimoto
Title: Krylov Subspace Method for Nonlinear Dynamical Systems with Random Noise

Abstract:
Operator-theoretic analysis of nonlinear dynamical systems has attracted much attention in a variety of engineering and scientific fields. In this paper, we address a lifted representation of nonlinear dynamical systems with random noise based on transfer operators, and develop a novel Krylov subspace method for estimating it using finite data, with consideration of the unboundedness of operators. For this purpose, we first consider Perron-Frobenius operators with kernel-mean embeddings for such systems. Then, we extend the Arnoldi method, which is the most classical type of Kryov subspace methods, so that it can be applied to the current case. Meanwhile, the Arnoldi method requires the assumption that the operator is bounded, which is not necessarily satisfied for transfer operators on nonlinear systems. We accordingly develop the shift-invert Arnoldi method for the Perron-Frobenius operators to avoid this problem. By using estimated operators, we can evaluate the predictive accuracy, which is applicable, for example, to anomaly detection in complex systems.

This is a joint work with Isao Ishikawa, Masahiro Ikeda, Yoichi Matsuo and Yoshinobu Kawahara.

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