Reduced Order Modeling for Prediction and Control of Large-Scale Systems.

Abstract

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.

The Adjoint Petrov–Galerkin method for non-linear model reduction

“…The development of stable and accurate reduced-order modeling techniques for complex non-linear systems is the motivation for the current work.A significant body of research aimed at producing accurate and stable ROMs for complex non-linear problems exists in the literature. These efforts include, but are not limited to, "energy-based" inner products [9,11], symmetry transformations [12], basis adaptation [13,14], L 1 -norm minimization [15], projection subspace rotations [16], and least-squares residual minimization approaches [17,18,19,20,21,22,23,24]. The Least-Squares Petrov-Galerkin (LSPG) [22] method comprises a particularly popular leastsquares residual minimization approach and has been proven to be an effective tool for non-linear model reduction.…”

Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier–Stokes equations

“…The conservative form contains rational functions of the unknowns and it is therefore not possible to pre-compute ROMs using standard Galerkin projection; to attain any computational speed-up a hyper-reduction step is necessary. Hyper-reduction is not always desirable, as it can destroy energy conservation properties and/or symplectic time-evolution maps [11,22]. On the other hand, if the equations are expressed in primitive variables, hyperreduction can be avoided because all nonlinearities that appear are polynomial.…”

“…Rowley et al [37] show that Galerkin projection preserves the stability of an equilibrium point at the origin if the ROM is constructed in an energy-based inner product. Barone et al [7], Kalashnikova et al [22] demonstrate that a symmetry transformation leads to a stable formulation for a Galerkin ROM for the linearized compressible Euler equations and nonlinear compressible Navier-Stokes equations with solid wall and far-field boundary conditions. Serre et al [42] propose applying the stabilizing projection developed by Barone et al [7], Kalashnikova et al [22] to a skew-symmetric system constructed by augmenting a given linear system with its adjoint system.…”

“…Barone et al [7], Kalashnikova et al [22] demonstrate that a symmetry transformation leads to a stable formulation for a Galerkin ROM for the linearized compressible Euler equations and nonlinear compressible Navier-Stokes equations with solid wall and far-field boundary conditions. Serre et al [42] propose applying the stabilizing projection developed by Barone et al [7], Kalashnikova et al [22] to a skew-symmetric system constructed by augmenting a given linear system with its adjoint system. The downside to these methods is that they are inherently embedded methods: access to the governing PDEs and/or the code that discretizes these PDEs is required.…”

Construction of energy-stable projection-based reduced order models

“…The online time-integration of the ROM system (4) (with the ROM coefficient matrix computed within Spirit and written to disk) is then performed using a fourth-order Runge-Kutta scheme in MATLAB. For more information on the Spirit code, the reader is referred to [55,54].…”