  In this paper we propose a novel approach to compute the Koopman operator
from sparse time series data. In recent years there has been considerable
interests in operator theoretic methods for data-driven analysis of dynamical
systems. Existing techniques for the approximation of the Koopman operator
require sufficiently large data sets, but in many applications, the data set
may not be large enough to approximate the operators to acceptable limits. In
this paper, using ideas from robust optimization, we propose an algorithm to
compute the Koopman operator from sparse data. We enrich the sparse data set
with artificial data points, generated by adding bounded artificial noise and
and formulate the noisy robust learning problem as a robust optimization
problem and show that the optimal solution is the Koopman operator with
smallest error. We illustrate the efficiency of our proposed approach in three
different dynamical systems, namely, a linear system, a nonlinear system and a
dynamical system governed by a partial differential equation.
