Data Driven Nonlinear Model Reduction and Metric Complexity Measures for Large Scale Systems
This project addresses the challenges of modeling, simulation, and analysis for infinite dimensional nonlinear systems governed by nonlinear partial differential equations (PDEs). These systems arise in many applications such as aerodynamics, heat transfer, solid-state circuits, devices, composite materials, energy efficient buildings, combustion, and process control. The objective of this research is to develop nonlinear model reduction (MR) methods for control design of systems governed by nonlinear PDEs.
The proposed MR algorithms are based on a global geometric framework which preserve the intrinsic geometry of the nonlinear PDEs and capture the geodesic distances between pairs of solution snapshots. This contrasts with the current MR algorithms which fail to capture the nonlinear degrees of freedom. The proposed reduced models apply to nonlinear PDEs by computing the corresponding geodesics. The latter is achieved by characterizing the PDE solutions as extremals of energy functionals. This idea is motivated by Arnold’s variational characterization of the Euler equation in terms of geodesics of diffeomorphisms. Specifically, data from simulations are used to compute new geodesic based proper orthogonal decomposition reduced order models (ROMs). Sensitivity analysis with respect to key parameters will be carried out to study their effect on the numerical solutions and the resulting ROMs. Stability issues of the latter are addressed through so-called closure models.
MR is divided in two stages consisting of information acquisition and information processing. In the first stage an input-output experiment/simulation is carried out and data collected. In the second stage a representation of the available information is obtained. MR is then formulated as an optimal input design and model selection problems. The latter are characterized in terms of measures of metric complexity, the epsilon-entropy and various n-widths. The latter quantify the uncertainty and inherent error generated in the information collecting stage due to lack of data or inaccurate measurements, and the representation error due to loss of information. The proposed methods will be applied to three prototype problems including nonlinear convection, unsteady airflow in office buildings, unsteady and turbulent flows in agile micro-air vehicles.
This proposal addresses the challenge of designing reduced order models for systems described by nonlinear PDEs. These systems arise in many applications such as in control of vehicular platoons, microelectromechanical systems, smart structures, aerodynamics, energy efficient buildings, combustion control in gas turbines and rockets, and process control of distillation processes. For example, in aerodynamic possible benefits include separation control, drag reduction, and lift enhancement resulting in significant fuel savings.
The results will also apply to spectral imaging which typically generates a large amount of high-dimensional data that are acquired in different sub-bands for each spatial location of interest. The high dimensionality of spectral data imposes limitations on numerical analysis. As such, there is an emerging demand for robust data compression techniques with loss of less relevant information to manage real spectral data. The proposed reduced-order data modeling techniques can be applied in order to compute low-dimensional models by projecting high-dimensional clusters onto subspaces spanned by local reduced-order bases.