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MODEL REDUCTION FOR LARGESCALE SYSTEMS WITH HIGHDIMENSIONAL PARAMETRIC INPUT SPACE
, 2007
"... Abstract. A modelconstrained adaptive sampling methodology is proposed for reduction of largescale systems with highdimensional parametric input spaces. Our model reduction method uses a reduced basis approach, which requires the computation of highfidelity solutions at a number of sample points ..."
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Cited by 51 (11 self)
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Abstract. A modelconstrained adaptive sampling methodology is proposed for reduction of largescale systems with highdimensional parametric input spaces. Our model reduction method uses a reduced basis approach, which requires the computation of highfidelity 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 PDEconstrained 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 for examples with parametric input spaces of dimension 11 and 21, which describe thermal analysis and design of a heat conduction fin, and compared with statisticallybased sampling methods. For this example, the modelconstrained 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.
Optimal control of the cylinder wake in the laminar regime by TrustRegion methods and POD ReducedOrder Models
, 2008
"... In this paper, optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime (Re = 200). The control parameters are the amplitude and the frequency of the timeharmonic cylinder rotation. In order to reduce the size of the discretized optimali ..."
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Cited by 39 (10 self)
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In this paper, optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime (Re = 200). The control parameters are the amplitude and the frequency of the timeharmonic cylinder rotation. In order to reduce the size of the discretized optimality system, a Proper Orthogonal Decomposition (POD) ReducedOrder Model (ROM) is derived to be used as state equation. We then propose to employ the TrustRegion Proper Orthogonal Decomposition (TRPOD) approach, originally introduced by Fahl (2000), to update the reducedorder models during the optimization process. A lot of computational work is saved because the optimization process is now based only on lowfidelity models. A particular care was taken to derive a POD ROM for the pressure and velocity fields with an appropriate balance between model accuracy and robustness. The key enablers are the extension of the POD basis functions to the pressure data, the use of calibration methods for the POD ROM and the addition in the POD expansion of several nonequilibrium modes to describe various operating conditions. When the TRPOD algorithm is applied to the wake flow configuration, this approach converges to the minimum predicted by an openloop control approach and leads to a relative mean drag reduction of 30 % at reduced cost.
SurrogateAssisted Evolutionary Optimization Frameworks for HighFidelity Engineering Design Problems
 In Knowledge Incorporation in Evolutionary Computation
, 2004
"... Over the last decade, Evolutionary Algorithms (EAs) have emerged as a powerful paradigm for global optimization of multimodal functions. More recently, there has been significant interest in applying EAs to engineering design problems. However, in many complex engineering design problems where high ..."
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Cited by 26 (6 self)
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Over the last decade, Evolutionary Algorithms (EAs) have emerged as a powerful paradigm for global optimization of multimodal functions. More recently, there has been significant interest in applying EAs to engineering design problems. However, in many complex engineering design problems where highfidelity analysis models are used, each function evaluation may require a Computational Structural Mechanics (CSM), Computational Fluid Dynamics (CFD) or Computational ElectroMagnetics (CEM) simulation costing minutes to hours of supercomputer time. Since EAs typically require thousands of function evaluations to locate a near optimal solution, the use of EAs often becomes computationally prohibitive for this class of problems. In this paper, we present frameworks that employ surrogate models for solving computationally expensive optimization problems on a limited computational budget. In particular, the key factors responsible for the success of these frameworks are discussed. Experimental results obtained on benchmark test functions and realworld complex design problems are presented.
Recursive trustregion methods for multiscale nonlinear optimization
 SIAM J. OPTIM
, 2006
"... A class of trustregion methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a mean of speedi ..."
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Cited by 17 (3 self)
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A class of trustregion methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a mean of speeding up the computation of the step. This use is recursive, leading to true multilevel/multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partialdifferential equations. A simple algorithm of the class is then described and its numerical performance is shown to be numerically promising. This observation then motivates a proof of global convergence to firstorder stationary points on the fine grid that is valid for all algorithms in the class.
A Survey of Model Reduction Methods for Parametric Systems
, 2013
"... Numerical simulation of largescale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent largescale nature of the models leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational bu ..."
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Cited by 12 (4 self)
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Numerical simulation of largescale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent largescale nature of the models leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original largescale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey stateoftheart in parametric model reduction methods. Parametric model reduction targets the broad class of problems for which the equations governing the system behavior depend on a set of parameters. Examples include parameterized partial differential equations and largescale systems of parameterized ordinary differential
Centroidal Voronoi tessellation based proper orthogonal decomposition analysis
 in: Proc. 8th Conference on Control of Distributed Parameter Systems, Birkhauser
, 2002
"... Abstract. Proper orthogonal decompositions (POD) have been used to systematically extract the most energetic modes while centroidal Voronoi tessellations (CVT) have been used to systematically extract best representatives. We combine the ideas of CVT and POD into a hybrid method for model reduction. ..."
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Cited by 11 (8 self)
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Abstract. Proper orthogonal decompositions (POD) have been used to systematically extract the most energetic modes while centroidal Voronoi tessellations (CVT) have been used to systematically extract best representatives. We combine the ideas of CVT and POD into a hybrid method for model reduction. The optimality of such an approach and various practical implementation strategies are discussed. 1.
Parametric ReducedOrder Models for Probabilistic Analysis of Unsteady Aerodynamic Applications
"... Methodology is presented to derive reducedorder models for largescale parametric applications in unsteady aerodynamics. The specific case considered in this paper is a computational fluid dynamic (CFD) model with parametric dependence that arises from geometric shape variations. The first key cont ..."
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Cited by 10 (0 self)
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Methodology is presented to derive reducedorder models for largescale parametric applications in unsteady aerodynamics. The specific case considered in this paper is a computational fluid dynamic (CFD) model with parametric dependence that arises from geometric shape variations. The first key contribution of the methodology is the derivation of a linearized model that permits the effects of geometry variations to be represented with an explicit affine function. The second key contribution is an adaptive sampling method that utilizes an optimization formulation to derive a reduced basis that spans the space of geometric input parameters. The method is applied to derive efficient reducedorder models for probabilistic analysis of the effects of blade geometry variation for a twodimensional model problem governed by the Euler equations. Reducedorder models that achieve three orders of magnitude reduction in the number of states are shown to accurately reproduce CFD Monte Carlo simulation results at a fraction of the computational cost. I.