Karen Willcox

Balanced Model Reduction via the Proper Orthogonal Decomposition

A new method for performing a balanced reduction of a high-order linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshotsisused to obtainlow-rank,reduced-rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain.Theapproximationsarethenusedtoobtainabalancedreducedorder model. The method is particularly effective when a small number of outputs is of interest. It is demonstrated for a linearized high-order system that models unsteady motion of a two-dimensional airfoil. Computation of the exact grammians would be impractical for such a large system. For this problem, very accurate reducedorder models are obtained that capture the required dynamics with just three states. The new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition. Although further development is necessary, the concept also extends to nonlinear systems.

Aerodynamic Data Reconstruction and Inverse Design Using Proper Orthogonal Decomposition

The application of proper orthogonal decomposition for incomplete (gappy) data for compressible external aerodynamic problems has been demonstrated successfully in this paper for the first time. Using this approach, it is possible to construct entire aerodynamic flowfields from the knowledge of computed aerodynamic flow data or measured flow data specified on the aerodynamic surface, thereby demonstrating a means to effectively combine experimental and computational data. The sensitivity of flow reconstruction results to available measurements and to experimental error is analyzed. Another new extension of this approach allows one to cast the problem of inverse airfoil design as a gappy data problem. The gappy methodology demonstrates a great simplification for the inverse airfoil design problem and is found to work well on a range of examples, including both subsonic and transonic cases.

Missing point estimation in models described by proper orthogonal decomposition

This paper presents a new method of missing point estimation (MPE) to derive efficient reduced-order models for large-scale parameter-varying systems. Such systems often result from the discretization of nonlinear partial differential equations. A projection-based model reduction framework is used where projection spaces are inferred from proper orthogonal decompositions of data-dependent correlation operators. The key contribution of the MPE method is to perform online computations efficiently by computing Galerkin projections over a restricted subset of the spatial domain. Quantitative criteria for optimally selecting such a spatial subset are proposed and the resulting optimization problem is solved using an efficient heuristic method. The effectiveness of the MPE method is demonstrated by applying it to a nonlinear computational fluid dynamic model of an industrial glass furnace. For this example, the Galerkin projection can be computed using only 25% of the spatial grid points without compromising the accuracy of the reduced model.

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

United States. Air Force Office of Scientific Research (Computational Mathematics Grant FA9550-12-1-0420)

Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space

A model-constrained adaptive sampling methodology is proposed for the reduction of large-scale systems with high-dimensional parametric input spaces. Our model reduction method uses a reduced basis approach, which requires the computation of high-fidelity solutions at a number of sample points throughout the parametric input space. A key challenge that must be addressed in the optimization, control, and probabilistic settings is the need for the reduced models to capture variation over this parametric input space, which, for many applications, will be of high dimension. We pose the task of determining appropriate sample points as a PDE-constrained optimization problem, which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters. The methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21, which describe thermal analysis and design of a heat conduction fin, and compared with statistically based sampling methods. For these examples, the model-constrained adaptive sampling leads to reduced models that, for a given basis size, have error several orders of magnitude smaller than that obtained using the other methods.

Surrogate-Based Optimization Using Multifidelity Models with Variable Parameterization and Corrected Space Mapping

Engineers are increasingly using high-fidelity models for numerical optimization. However, the computational cost of these models, combined with the large number of objective function and constraint evaluations required by optimization methods, can render such optimization computationally intractable. Surrogate-based optimization (SBO) optimization using a lower-fidelity model most of the time, with occasional recourse to the high-fidelity model is a proven method for reducing the cost of optimization. One branch of SBO uses lower-fidelity physics models of the same system as the surrogate. Until now however, surrogates using a different set of design variables from that of the high-fidelity model have not been available to use in a provably convergent numerical optimization. New methods are herein developed and demonstrated to reduce the computational cost of numerical optimization of variableparameterization problems, that is, problems for which the low-fidelity model uses a different set of design variables from the high-fidelity model. Four methods are presented to perform mapping between variable-parameterization spaces, the last three of which are new: space mapping, corrected space mapping, a mapping based on proper orthogonal decomposition (POD), and a hybrid between POD mapping and space mapping. These mapping methods provide links between different models of the same system and have further applications beyond formal optimization strategies. On an unconstrained airfoil design problem, it achieved up to 40% savings in highfidelity function evaluations. On a constrained wing design problem it achieved 76% time savings, and on a bat flight design problem, it achieved 45% time savings. On a large-scale practical aerospace application, such time savings could represent weeks. Thesis Supervisor: Karen Willcox Title: Associate Professor of Aeronautics and Astronautics Thesis Supervisor: Robert Haimes Title: Principal Research Engineer

Goal-oriented, model-constrained optimization for reduction of large-scale systems

Optimization-oriented reduced-order models should target a particular output functional, span an applicable range of dynamic and parametric inputs, and respect the underlying governing equations of the system. To achieve this goal, we present an approach for determining a projection basis that uses a goal-oriented, model-constrained optimization framework. The mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems. Results for a simple linear model problem of the two-dimensional heat equation demonstrate the ability of the goal-oriented approach to target a particular output functional of interest. Application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady Euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition.

Non-linear model reduction for uncertainty quantification in large-scale inverse problems

SUMMARY We present a model reduction approach to the solution of large-scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non-linear terms in the reduced model. To achieve this, we present a formulation that employs masked projection of the discrete equations; that is, we compute an approximation of the non-linear term using a select subset of interpolation points. Further, through this formulation we show similarities among the existing techniques of gappy proper orthogonal decomposition, missing point estimation, and empirical interpolation via coefficient-function approximation. The resulting model reduction methodology is applied to a highly non-linear combustion problem governed by an advection‐diffusion-reaction partial differential equation (PDE). Our reduced model is used as a surrogate for a finite element discretization of the non-linear PDE within the Markov chain Monte Carlo sampling employed by the Bayesian inference approach. In two spatial dimensions, we show that this approach yields accurate results while reducing the computational cost by several orders of magnitude. For the full three-dimensional problem, a forward solve using a reduced model that has high fidelity over the input parameter space is more than two million times faster than the full-order finite element model, making tractable the solution of the statistical inverse problem that would otherwise require many years of CPU time. Copyright q 2009 John Wiley & Sons, Ltd.

Simultaneous Optimization of a Multiple-Aircraft Family

Multidisciplinary design optimization is considered in the context of designing a family of aircraft. Aframework is developed in which multiple aircraft, each with different missions but sharing common parts, can be optimized simultaneously.Thenewframeworkisusedtogaininsighttotheeffectofdesignvariablescalingontheoptimization algorithm. Results are presented for a two-member family whose individual missions differ signie cantly. Both missions can be satise ed with common designs. Moreover, optimizing both airplanes simultaneously rather than following the traditional baseline plus derivative approach vastly improves the common solution. A cost modeling framework is outlined that allows the value of commonality to be quantie ed for design and manufacturing costs. A notional example is presented to show the cost benee t that may be achieved by designing a common family of aircraft.

Parametric Reduced-Order Models for Probabilistic Analysis of Unsteady Aerodynamic Applications

DOI: 10.2514/1.35850 We address the problem of propagating input uncertainties through a computational fluid dynamics model. Methods such as Monte Carlo simulation can require many thousands (or more) of computational fluid dynamics solves, rendering them prohibitively expensive for practical applications. This expense can be overcome with reduced-order models that preserve the essential flow dynamics. The specific contributions of this paper are as follows: first, to derive a linearized computational fluid dynamics model that permits the effects of geometry variations to be represented with an explicit affine function; second, to propose an adaptive sampling method to derive a reduced basis that is effective over the joint probability density of the geometry input parameters. The method is applied to derive efficient reduced models for probabilistic analysis of a two-dimensional problem governedbythelinearized Eulerequations.Reduced-order modelsthatachieve 3-orders-of-magnitude reduction in the number of states are shown to accurately reproduce computational fluid dynamics Monte Carlo simulation results at a fraction of the computational cost.