Error Estimation for Reduced‐Order Models of Dynamical Systems

Abstract

Abstract. The use of reduced order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors, by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most importantly, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error while the derived bounds are within an order of magnitude. Key words. model reduction, proper orthogonal decomposition, small sample statistical condition estimation, adjoint method AMS subject classifications. 65L10, 65L991. Introduction. Model reduction of dynamical systems described by differential equations is ubiquitous in science and engineering [1]. Reduced models are used for efficient simulation [12,24] and control [13,22]. Moreover, the process of creating low-order models forces the researcher to isolate and quantify the dominant physical mechanisms, revealing effective design decisions that would not have been identified through numerical simulation, experiments or "black box" optimization methods [23].The Proper Orthogonal Decomposition (POD) method has been used extensively in a variety of fields including fluid dynamics [18], identification of coherent structures [8,16] , and study of the dynamic wind pressures acting on buildings [11], to name only a few. A detailed description [8] of the POD approach as a reduction method shows that, for a given number of modes, POD is the most efficient choice among all linear decompositions in the sense that it retains, on average, the greatest possible kinetic energy.As soon as one contemplates the use of a reduced model, questions concerning the quality of the approximation become paramount. To judge the quality of the reduced model, it is important to estimate its error. An algorithm for estimating the error of a class of reduction methods based on projection techniques was presented in [25]. In this approach, the original problem is linearized around the initial time. The resulting first-order error estimates are valid for only a small number of time steps (during which the Jacobian matrix can be considered constant). First-order estimates

Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models

“…However, we demonstrated that this control law is energetically inefficient. These numerical results agree to a large extent to those obtained previously by other researchers [38,40,69] using the twodimensional Navier-Stokes equations to solve the optimal control problem. Compared with those studies, the main advantage of our approach is that it leads to a significant reduction of the numerical costs because the optimization process itself is completely based on reduced-order models only.…”

Subspace fluid re-simulation

“…As expected, L2 error relative to a ground truth solution increases as the dynamics diverge from those of the training simulation. Characterizing the error of subspace simulations under arbitrary perturbations is an interesting and subtle topic [Homescu et al 2005], so we leave a more thorough analysis as future work.…”

Skipping steps in deformable simulation with online model reduction

“…Such backups are not acceptable in offline animation, since they would require coupled simulators (fluids, rigid bodies) to be backed up as well, which would invalidate any performance gains. The dual weighted residual (DWR) method [Meyer and Matthies 2003] as well as the method of Homescu et al [2006] take a different approach and solve the adjoint of a known simulation result backwards in time to obtain a priori reduced-order error bounds on perturbed versions of the simulation. These two methods still bootstrap off of existing simulation data, so they cannot be applied to the case where the simulation is only computed once.…”

“…This estimate is conservative -subspace error is usually characterized as a sum of projection error and integration error (for example, see Homescu et al [2006]), with integration error being the only component that compounds over time. We pessimistically assume that all of the error is integration error.…”