Projection-based Reduced Order Models Research Articles

Construction of energy-stable projection-based reduced order models

An 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 L2 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.

Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment

A new approach for stabilizing unstable reduced order models (ROMs) for linear time-invariant (LTI) systems through an a posteriori post-processing step applied to the algebraic ROM system is developed. The key idea is to modify the unstable eigenvalues of the ROM system by moving these eigenvalues into the stable half of the complex plane. It is demonstrated that this modification to the ROM system eigenvalues can be accomplished using full state feedback (a.k.a. pole placement) algorithms from control theory. This approach ensures that the modified ROM is stable provided the system’s unstable poles are controllable and observable; however, the accuracy of the stabilized ROM is not guaranteed. To remedy this difficulty and guarantee an accurate stabilized ROM, a constrained nonlinear least-squares optimization problem for the stabilized ROM eigenvalues in which the error in the ROM output is minimized is formulated. This optimization problem is small and therefore computationally inexpensive to solve. Performance of the proposed algorithms is evaluated on two test cases for which ROMs constructed via the proper orthogonal decomposition (POD)/Galerkin method suffer from instabilities.

Parametric Reduced Order Models Using Adaptive Sampling and Interpolation

Over the past decade, a number of approaches have been put forth to improve the accuracy of projection-based reduced order models over parameter ranges. These can be classified as either i.) building a global basis that is suitable for a large parameter set by applying sampling strategies, ii.) identifying parameter dependent coefficient functions in the reduced order model, or iii.) changing the basis as parameters change. We propose a strategy that combines sampling with basis interpolation. We apply sampling strategies that identify suitable parameter values from which associated basis functions are interpolated at any parameter value in a region. While our approach has practical limits to roughly a handful of parameters, it has the advantage of achieving a desired level of accuracy in parametric reduced-order models of relatively small size. We present this method using a proper orthogonal decomposition model of a nonlinear partial differential equation with variable coefficients and initial conditions.

Reduced-order model tracking and interpolation to solve PDE-based Bayesian inverse problems

This work presents a computationally efficient probabilistic framework that enables the identification of model parameters from noisy measurements of the response. We consider transient PDE-based models, where the parameters correspond to physical properties. An efficient and reliable procedure for estimation of those unknown parameters is pursued. The proposed framework uses a Bayesian approach, an efficient sequential Monte Carlo sampling scheme, and adaptive reduced-order models (ROMs). The Bayesian approach has several advantages including the ability to provide not only point estimates of the quantities of interest, but also measures of credibility and robustness concerning those estimates. The associated sequential Monte Carlo method sampling scheme is embarrassingly parallelizable, as well as efficient in terms of the number of calls to the forward solver (e.g. a finite element code) used in evaluating the likelihood function. We propose to use a ROM adaptation procedure where projection-based ROMs are seen as points on a certain Riemannian manifold and are ‘tracked’ and interpolated during the sampling process using a database of precomputed ROMs. This approach ensures that an appropriate ROM is used for the likelihood evaluation in every region of the parameter space, thus leading to computational savings while conserving sufficient accuracy. Using numerical examples, we illustrate the capabilities of the proposed framework, and show that it leads to quality estimates with a quantified predictive uncertainty.

Stabilization of projection-based reduced order models of the Navier–Stokes

A new method of stabilizing low-order, proper orthogonal decomposition based reduced-order models of the Navier–Stokes equations is proposed. Unlike traditional approaches, this method does not rely on empirical turbulence modeling or modification of the Navier–Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the solution, the new proposed basis functions also provide stable reduced-order models. The proposed approach is illustrated with two test cases: two-dimensional flow inside a square lid-driven cavity and a two-dimensional mixing layer.