Nonlinear estimation with Perron-Frobenius operator and Karhunen-Loève expansion

Uncertainty Quantification for Stochastic Nonlinear Systems using Perron-Frobenius Operator and Karhunen-Loève Expansion

In this paper, a methodology for propagation of uncertainty in stochastic nonlinear dynamical systems is investigated. The process noise is approximated using KarhunenLoève (KL) expansion. Perron-Frobenius (PF) operator is used to predict the evolution of uncertainty. A multivariate Kolmogorov-Smirnov test is used to verify the proposed framework. The method is applied to predict uncertainty ev...

Compact weighted Frobenius-Perron operators and their spectra

In this note we characterize the compact weighted Frobenius-Perron operator $p$ on $L^1(Sigma)$ and determine their spectra. We also show that every weakly compact weighted Frobenius-Perron operator on $L^1(Sigma)$ is compact.

Weighted Frobenius-Perron and Koopman operators

Performance Bounds for Dispersion Analysis: A Comparison Between Monte Carlo and Perron-Frobenius Operator Approach

We compare computational cost and accuracy between two different approaches of dispersion analysis. One is the conventional Monte Carlo method and the other being the Perron-Frobenius operator approach, that directly propagates the joint probability density function using Liouville equation. It is shown that with same computational budget, Perron-Frobenius operator approach rewards better accur...

A Posteriori Error and Optimal Reduced Basis for Stochastic Processes Defined by a Finite Set of Realizations

The use of reduced basis has spread to many scientific fields for the last fifty years to condense the statistical properties of stochastic processes. Among these basis, the classical Karhunen-Loève basis corresponds to the Hilbertian basis that is constructed as the eigenfunctions of the covariance operator of the stochastic process of interest. The importance of this basis stems from its opti...