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REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2010
"... In this paper we present reduced basis approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a lowdimensional space associated with a smooth parametric manifold — to provide dimensi ..."
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Cited by 51 (8 self)
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In this paper we present reduced basis approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a lowdimensional space associated with a smooth parametric manifold — to provide dimension reduction; an efficient PODGreedy sampling method for identification of optimal and numerically stable approximations — to yield rapid convergence; accurate (Online) calculation of the solutiondependent stability factor by the Successive Constraint Method — to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the reduced basis approximation and associated outputs — to provide certainty in our predictions; and an OfflineOnline computational decomposition strategy for our reduced basis approximation and associated error bound — to minimize marginal cost and hence achieve high performance in the realtime and manyquery contexts. The method is applied to a transient natural convection problem in a twodimensional “complex” enclosure — a square with a small rectangle cutout — parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the reduced basis approximation con
Model Reduction of LargeScale Dynamical Systems
"... Abstract. Simulation and control are two critical elements of Dynamic DataDriven Application Systems (DDDAS). Simulation of dynamical systems such as weather phenomena, when augmented with realtime data, can yield precise forecasts. In other applications such as structural control, the presence of ..."
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Cited by 27 (16 self)
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Abstract. Simulation and control are two critical elements of Dynamic DataDriven Application Systems (DDDAS). Simulation of dynamical systems such as weather phenomena, when augmented with realtime data, can yield precise forecasts. In other applications such as structural control, the presence of realtime data relating to system state can enable robust active control. In each case, there is an ever increasing need for improved accuracy, which leads to models of higher complexity. The basic motivation for system approximation is the need, in many instances, for a simplified model of a dynamical system, which captures the main features of the original complex model. This need arises from limited computational capability, accuracy of measured data, and storage capacity. The simplified model may then be used in place of the original complex model, either for simulation and prediction, oractive control. As sensor networks and embedded processors proliferate our environment, technologies for such approximations and realtime control emerge as the next major technical challenge. This paper outlines the state of the art and outstanding challenges in the development of efficient and robust methods for producing reduced order models of large statespace systems. 1
Interpolatory Projection Methods for Parameterized Model Reduction
"... We provide a unifying projectionbased framework for structurepreserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reducedorder model. The parameter dependence may be linear or n ..."
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Cited by 16 (11 self)
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We provide a unifying projectionbased framework for structurepreserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reducedorder model. The parameter dependence may be linear or nonlinear and is retained in the reducedorder model. Moreover, we are able to give conditions under which the gradient and Hessian of the system response with respect to the system parameters is matched in the reducedorder model. We provide a systematic approach built on established interpolatory H2 optimal model reduction methods that will produce parameterized reducedorder models having high fidelity throughout a parameter range of interest. For single input/single output systems with parameters in the input/output maps, we provide reducedorder models that are optimal with respect to an H2 ⊗ L2 joint error measure. The capabilities of these approaches are illustrated by several numerical
Simulationbased optimal Bayesian experimental design for nonlinear systems
 Journal of Computational Physics
, 2012
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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
Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces. arXiv.org
, 2012
"... Abstract. Reduced order models, in particular the reduced basis method, rely on empirically built and problem dependent basis functions that are constructed during an offline stage. In the online stage, the precomputed problemdependent solution space, that is spanned by the basis functions, can t ..."
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Cited by 6 (2 self)
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Abstract. Reduced order models, in particular the reduced basis method, rely on empirically built and problem dependent basis functions that are constructed during an offline stage. In the online stage, the precomputed problemdependent solution space, that is spanned by the basis functions, can then be used in order to reduce the size of the computational problem. For complex problems, the number of basis functions required to guarantee a certain error tolerance can become too large in order to benefit computationally from the model reduction. To overcome this, the present work introduces a framework where local approximation spaces (in parameter space) are used to define the reduced order approximation in order to have explicit control over the online cost. This approach also adapts the local approximation spaces to local anisotropic behavior in the parameter space. We present the algorithm and numerous numerical tests. 1.
Efficient greedy algorithms for highdimensional parameter spaces with applications to empirical interpolation and reduced basis methods
, 2012
"... Abstract. We propose two new algorithms to improve greedy sampling of highdimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for highdime ..."
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Cited by 5 (1 self)
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Abstract. We propose two new algorithms to improve greedy sampling of highdimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for highdimensional parametrized functions. The first algorithm, based on a saturation assumption of the error in the greedy algorithm, is shown to result in a significant reduction of the workload over the standard greedy algorithm. In a further improved approach, this is combined with an algorithm in which the train set for the greedy approach is adaptively sparsified and enriched. A safety check step is added at the end of the algorithm to certify the quality of the sampling. Both these techniques are applicable to highdimensional problems and we shall demonstrate their performance on a number of numerical examples.
A.: Numerical methods for loworder modeling of fluid flows based on POD
 Int. J. Numer. Methods Fluids
, 2010
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Nonlinear parametric inversion using interpolatory model reduction. arXiv preprint arXiv:1311.0922
, 2013
"... Abstract. Nonlinear parametric inverse problems appear in several prominent applications; one such application is Diffuse Optical Tomography (DOT) in medical image reconstruction. Such inverse problems present huge computational challenges, mostly due to the need for solving a sequence of largesca ..."
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Cited by 4 (3 self)
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Abstract. Nonlinear parametric inverse problems appear in several prominent applications; one such application is Diffuse Optical Tomography (DOT) in medical image reconstruction. Such inverse problems present huge computational challenges, mostly due to the need for solving a sequence of largescale discretized, parametrized, partial differential equations (PDEs) in the forward model. In this paper, we show how interpolatory parametric model reduction can significantly reduce the cost of the inversion process in DOT by drastically reducing the computational cost of solving the forward problems. The key observation is that function evaluations for the underlying optimization problem may be viewed as transfer function evaluations along the imaginary axis; a similar observation holds for Jacobian evaluations as well. This motivates the use of systemtheoretic model order reduction methods. We discuss the construction and use of interpolatory parametric reduced models as surrogates for the full forward model. Within the DOT setting, these surrogate models can approximate both the cost functional and the associated Jacobian with very little loss of accuracy while significantly reducing the cost of the overall inversion process. Four numerical examples illustrate the efficiency of the proposed approach. Although we focus on DOT in this paper, we believe that our approach is applicable much more generally. Key words. DOT, PaLS, model reduction, rational interpolation. AMS subject classifications. 65F10, 65N22, 93A15, 93C05.
Comparison of some reduced representation approximations, submitted
, 2013
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