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42
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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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.
Control of the circular cylinder wake by TrustRegion methods and POD ReducedOrder Models
, 2008
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MPC for LargeScale Systems via Model Reduction and Multiparametric Quadratic Programming
, 2006
"... In this paper we present a methodology for achieving realtime control of systems modeled by partial differential equations. The methodology uses the explicit solution of the model predictive control (MPC) problem combined with model reduction. The explicit solution of the MPC problem leads to onli ..."
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In this paper we present a methodology for achieving realtime control of systems modeled by partial differential equations. The methodology uses the explicit solution of the model predictive control (MPC) problem combined with model reduction. The explicit solution of the MPC problem leads to online MPC functionality without having to solve an optimization problem at each time step. Reducedorder models are derived using a goaloriented, modelbased optimization formulation that yields efficient models tailored to the application at hand. The approach is demonstrated for reducedorder output feedback control of a largescale linear time invariant state space model of the discretized heat equation.
Improvement of reduced order modeling based on proper orthogonal decomposition. Research Report 6561
 INRIA
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POD/DEIM ReducedOrder Strategies for Efficient Four Dimensional Variational Data Assimilation
 Virginia Polytechnic Institute and State University
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Explicit Model Predictive Control for LargeScale Systems via Model Reduction
"... In this paper we present a framework for achieving constrained optimal realtime control for largescale systems with fast dynamics. The methodology uses the explicit solution of the model predictive control (MPC) problem combined with model reduction, in an outputfeedback implementation. The expli ..."
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In this paper we present a framework for achieving constrained optimal realtime control for largescale systems with fast dynamics. The methodology uses the explicit solution of the model predictive control (MPC) problem combined with model reduction, in an outputfeedback implementation. The explicit solution of the MPC problem leads to online MPC functionality without having to solve an optimization problem at each time step. Reducedorder models are derived using a goaloriented, modelconstrained optimization formulation that yields efficient models tailored to the control application at hand. The approach is illustrated on a challenging largescale flow problem that aims to control the shock position in a supersonic diffuser.
Model Reduction by Rational Interpolation
"... Abstract. The last two decades have seen major progress in interpolatory methods for model reduction of largescale dynamical systems have. The ability to produce optimal (at least locally) interpolatory reduced models at a modest cost for linear and bilinear systems, extensions to reducing paramet ..."
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Abstract. The last two decades have seen major progress in interpolatory methods for model reduction of largescale dynamical systems have. The ability to produce optimal (at least locally) interpolatory reduced models at a modest cost for linear and bilinear systems, extensions to reducing parametric systems, and the ability to produce reducedmodels directly from input/output measurements are some examples of these new developments. This chapter will give a survey of interpolatory model reduction methods including a detailed analysis of basic principles together with a presentation of the more recent developments. Discussion will be supported by numerical examples.
assimilation
"... reduced order approach to fourdimensional variational data ..."
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reduced order approach to fourdimensional variational data
Reduced order modeling based on POD of a parabolized NavierStokes equations model I: Forward Model
"... A Proper Orthogonal Decomposition (POD) based reduced order model of the parabolized NavierStokes (PNS) equations is derived in this article. A spacemarching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate ..."
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A Proper Orthogonal Decomposition (POD) based reduced order model of the parabolized NavierStokes (PNS) equations is derived in this article. A spacemarching finite difference method with time relaxation is used to obtain the solution of this problem, from which snapshots are obtained to generate the POD basis functions used to construct the reduced order model. In order to improve the accuracy and stability of the reduced order model in the presence of high Reynolds number, a Sobolev H1 norm calibration is applied to the POD construction process. Finally, some numerical tests with the high fidelity model as well as the POD reduced order model were carried out to demonstrate the efficiency and accuracy of the reduced order model for solving the PNS equations compared with the full PNS model. The efficiency of the H1 norm POD calibration is illustrated, along with the optimal dissipation coefficient derivation, yielding the best RMSE and correlation coefficient when comparing the PNS reduced order model with the full PNS model.
Reduced order models in PIDE constrained optimization
, 2010
"... Mathematical models for option pricing often result in partial differential equations originally starting with the BlackScholes model. In this context, recent enhancements are models driven by Levy processes, which lead to a partial differential equation with an additional integral term. If one s ..."
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Mathematical models for option pricing often result in partial differential equations originally starting with the BlackScholes model. In this context, recent enhancements are models driven by Levy processes, which lead to a partial differential equation with an additional integral term. If one solves the problems mentioned last numerically, this yields large linear systems of equations with dense matrices. However, by using the special structure and an iterative solver the problem can be handled efficiently. To further reduce the computational cost in the calibration phase we implement a reduced order model, like proper orthogonal decomposition (POD), which proves to be very efficient. In this paper we use a special multilevel trust region POD algorithm to calibrate the option pricing model and give numerical results supporting the gain in efficiency.