skip to main content


Title: Construction of energy-stable projection-based reduced order models

Our paper aims to unify and extend several approaches for building stable projection-based reduced order models (ROMs) using the energy method and the concept of “energy-stability”. Attention is focused on linear time-invariant (LTI) systems. First, an approach for building energy stable Galerkin ROMs for linear hyperbolic or incompletely parabolic systems of partial differential equations (PDEs) using continuous projection is proposed. The key idea is to apply to the system a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables. The result of this procedure will be a ROM that is energy-stable for any choice of reduced basis. It is shown that, for many PDE systems, the desired transformation is induced by a special inner product, termed the “symmetry inner product”. Next, attention is turned to building energy-stable ROMs via discrete projection. A discrete counterpart of the continuous symmetry inner product, termed the “Lyapunov inner product”, is derived. Moreover, it is shown that the Lyapunov inner product can be computed in a black-box fashion for a stable LTI system ari sing from the discretization of a system of PDEs in space. Projection in this inner product guarantees a ROM that is energy-stable,more » again for any choice of reduced basis. Connections between the Lyapunov inner product and the inner product induced by the balanced truncation algorithm are made. We also made comparisons between the symmetry inner product and the Lyapunov inner product. Performance of ROMs constructed using these inner products is evaluated on several benchmark test cases.« less
 [1] ;  [2] ;  [2] ;  [2]
  1. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Report Number(s):
Journal ID: ISSN 0096-3003; PII: S0096300314014489
Grant/Contract Number:
Accepted Manuscript
Journal Name:
Applied Mathematics and Computation
Additional Journal Information:
Journal Volume: 249; Journal Issue: C; Journal ID: ISSN 0096-3003
Research Org:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org:
USDOE National Nuclear Security Administration (NNSA)
Country of Publication:
United States
97 MATHEMATICS AND COMPUTING; reduced order model (ROM); proper orthogonal decomposition (POD)/Galerkin projection; linear hyperbolic/incompletely parabolic systems; linear time-invariant (LTI) systems; numerical stability; Lyapunov equation
OSTI Identifier: