Large-scale dynamical systems are an intrinsic part of many areas of science and engineering. Frequently, they are simulated using high-dimensional discretized equations whose accurate solutions often can only be obtained at very high computational cost. For this reason, there has been a lot of research on the development of reduced order models (ROM's) in the last few decades. Ideally, these models are low-dimensional but still manage to replicate the important characteristics of the system. One popular model is POD which provides the optimally ordered, orthonormal basis from a set of data. A reduced order model is then constructed by representing the solution using a subset of this basis, e.g. projecting the problem onto a lower dimensional subspace in order to obtain a lower dimensional model. In this thesis, we explore both how to evaluate the response of a POD model to perturbations and how to improve its ability to handle perturbations. We also investigate the use of sensitivity analysis on POD reduced order models. Additionally, we look at the POD method's dependence on the type of system being modeled, and ways to improve computational efficiency.
We first present results comparing the outcomes of using essentially the same POD basis for different types of PDE's that demonstrate the accuracy of the POD model depends not only on the ability of the POD basis to represent the solution as set by the tolerance used in the POD selection procedure, but also on the type of equation. Additionally, the results demonstrate that the way in which inaccuracies arise in POD reduced order models are dependent on the type of equation being modeled.
Next, we introduce an error indicator that detects when the POD reduced order model stops performing well on a perturbed system as the numerical solution evolves. This has the advantage that the POD basis can be adjusted as needed during the evolution of the ROM. The error indicator is shown to effectively detect the error as it arises during the evolution of the model on four different types of PDE's. Additionally, we found that sensitivity analysis applied to POD models with significantly reduced dimension compared to their full dimensional counterparts still generated accurate sensitivities.
We address the related issue of how to improve the ability of a POD model to handle perturbations in its parameters. To do this, we introduce a number of basis augmentation strategies. Our tests of these strategies demonstrate that using augmented bases can substantially improve the accuracy of POD reduced order models on perturbed systems.
Lastly, we consider how to improve the computational efficiency of POD. We propose that the implicit filtering associated with POD enables the use of larger timesteps in the POD model. Our computational experiments support this idea that POD based ROM's will generally have reduced stability timestep restrictions. The consequence of this observation is that the ability to use a larger timestep will typically lead to greater computational efficiency, and thus help to offset the additional cost per timestep that evolving a POD based ROM requires for problems with nonlinear terms that cannot be precomputed.
