Construction of energy-stable projection-based reduced order models

Abstract

a r t i c l e i n f o Keywords: Reduced order model (ROM) Proper orthogonal decomposition (POD)/ Galerkin projection Linear hyperbolic/incompletely parabolic systems Linear time-invariant (LTI) systems Numerical stability Lyapunov equation a b s t r a c tAn approach for building energy-stable Galerkin reduced order models (ROMs) for linear hyperbolic or incompletely parabolic systems of partial differential equations (PDEs) using continuous projection is developed. This method is an extension of earlier work by the authors specific to the equations of linearized compressible inviscid flow. The key idea is to apply to the PDEs a transformation induced by the Lyapunov function for the system, and to build the ROM in the transformed variables. For linear problems, the desired transformation is induced by a special inner product, termed the ''symmetry inner product'', which is derived herein for several systems of physical interest. Connections are established between the proposed approach and other stability-preserving model reduction methods, giving the paper a review flavor. More specifically, it is shown that a discrete counterpart of this inner product is a weighted L 2 inner product obtained by solving a Lyapunov equation, first proposed by Rowley et al. and 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.Published by Elsevier Inc.

Error analysis of an incremental proper orthogonal decomposition algorithm for PDE simulation data

“…The basis elements are called POD modes, and they are often used to create low order models of high-dimensional systems of ordinary differential equations or partial differential equations (PDEs) that can be simulated easily and even used for real-time applications. For more about the applications of POD in engineering and applied sciences and POD model order reduction, see, e.g., [1,4,8,10,11,13,17,26,28,30,31,36,40,51,53,55].…”

A note on incremental POD algorithms for continuous time data

“…Proper orthogonal decomposition (POD) is a data approximation technique that has been successfully used for many applications in various fields; see, e.g., [1,4,9,14,16,17,19,21,22,27,31,37,40,41,43]. The first part of any such application is to use POD to extract basis elements, called POD modes, from experimental or simulation data.…”

emgr—The Empirical Gramian Framework

“…Other choices are for example: covariance-weighted products for Gaussian-noise-driven systems yielding system covariances [76], reproducing kernel Hilbert spaces (RKHS) [77], such as the polynomial, Gaussian or Sigmoid kernels [78], or energy-stable inner products [79]. Also, weighted Gramians [36] and time-weighted system Gramians [80] can be computed using this interface, i.e., dp = @(x,y) mtimes([0:h:T].ˆk.…”