  In this paper, we consider the model reduction problem of large-scale
systems, such as systems obtained through the discretization of partial
differential equations. We propose a computationally optimal randomized proper
orthogonal decomposition (RPOD*) technique to obtain the reduced order model by
perturbing the primal and adjoint system using Gaussian white noise. We show
that the computations required by the RPOD* algorithm is orders of magnitude
cheaper when compared to the balanced proper orthogonal decomposition (BPOD)
algorithm and BPOD output projection algorithm while the performance of the
RPOD* algorithm is much better than BPOD output projection algorithm. It is
optimal in the sense that a minimal number of snapshots is needed. We also
relate the RPOD* algorithm to random projection algorithms. The method is
tested on two advection-diffusion equations.
