Reduced Order Modeling for Prediction and Control of Large-Scale Systems.
This report describes work performed from June 2012 through May 2014 as a part of a Sandia Early Career Laboratory Directed Research and Development (LDRD) project led by the first author. The objective of the project is to investigate methods for building stable and efficient proper orthogonal decomposition (POD)/Galerkin reduced order models (ROMs): models derived from a sequence of high-fidelity simulations but having a much lower computational cost. Since they are, by construction, small and fast, ROMs can enable real-time simulations of complex systems for onthe- spot analysis, control and decision-making in the presence of uncertainty. Of particular interest to Sandia is the use of ROMs for the quantification of the compressible captive-carry environment, simulated for the design and qualification of nuclear weapons systems. It is an unfortunate reality that many ROM techniques are computationally intractable or lack an a priori stability guarantee for compressible flows. For this reason, this LDRD project focuses on the development of techniques for building provably stable projection-based ROMs. Model reduction approaches based on continuous as well as discrete projection are considered. In the first part of this report, an approach for building energy-stable Galerkin ROMs for linear hyperbolic or incompletely parabolic systems of partial differential equations (PDEs) using continuous projection is developed. The key idea is to apply a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables. It is shown that, for many PDE systems including the linearized compressible Euler and linearized compressible Navier-Stokes equations, the desired transformation is induced by a special inner product, termed the “symmetry inner product”. Attention is then turned to nonlinear conservation laws. A new transformation and corresponding energy-based inner product for the full nonlinear compressible Navier-Stokes equations is derived, and it is demonstrated that if a Galerkin ROM is constructed in this inner product, the ROM system energy will be bounded in a way that is consistent with the behavior of the exact solution to these PDEs, i.e., the ROM will be energy-stable. The viability of the linear as well as nonlinear continuous projection model reduction approaches developed as a part of this project is evaluated on several test cases, including the cavity configuration of interest in the targeted application area. In the second part of this report, some POD/Galerkin approaches for building stable ROMs using discrete projection are explored. It is shown that, for generic linear time-invariant (LTI) systems, a discrete counterpart of the continuous symmetry inner product is a weighted L2 inner product obtained by solving a Lyapunov equation. This inner product was first proposed by Rowley et al., and is termed herein the “Lyapunov inner product“. Comparisons between the symmetry inner product and the Lyapunov inner product are made, and the performance of ROMs constructed using these inner products is evaluated on several benchmark test cases. Also in the second part of this report, a new ROM stabilization approach, termed “ROM stabilization via optimization-based eigenvalue reassignment“, is developed for generic LTI systems. At the heart of this method is a constrained nonlinear least-squares optimization problem that is formulated and solved numerically to ensure accuracy of the stabilized ROM. Numerical studies reveal that the optimization problem is computationally inexpensive to solve, and that the new stabilization approach delivers ROMs that are stable as well as accurate. Summaries of “lessons learned“ and perspectives for future work motivated by this LDRD project are provided at the end of each of the two main chapters.