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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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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
Identifying structural variability using Bayesian inference
"... A stochastic approach is proposed for estimating the variability in structural parameters present in a large set of metalframe structures, given only measurements of modal frequency performed on a subset of the structures. The key step is a new statistical model relating simulation and experiment, ..."
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A stochastic approach is proposed for estimating the variability in structural parameters present in a large set of metalframe structures, given only measurements of modal frequency performed on a subset of the structures. The key step is a new statistical model relating simulation and experiment, including terms representing not only the measurement noise, but also the unknown structural variability. This latter is modelled by random variables whose hyperparameters are themselves stochastic, and these hyperparameters are estimated by Bayes ’ theorem. The evaluation of the posterior distribution is efficiently performed by combining a number of modern numerical tools: kriging surrogates for the finiteelement analysis, probabilistic collocation uncertainty quantification, and Markov chain MonteCarlo. The method is demonstrated for a metalframe model with two uncertain parameters, using data from specially designed experiments with controlled variability. The output probability densities on the structural parameters are useful for input to subsequent uncertainty quantification. 1
18.337 Parallel Computing Parallel PDEconstrained Optimization
, 2008
"... The efficient solution of largescale systems resulting from the discretization of partial differential equations (PDE) are of interest to engineers in many fields. Over the past several decades, as processors became faster, the timetosolution for a given largescale system has decreased thereby p ..."
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The efficient solution of largescale systems resulting from the discretization of partial differential equations (PDE) are of interest to engineers in many fields. Over the past several decades, as processors became faster, the timetosolution for a given largescale system has decreased thereby permitting the solution of larger and larger systems. It is human nature to desire the solution of systems larger than those we
A SURVEY OF PROJECTIONBASED MODEL REDUCTION METHODS FOR PARAMETRIC DYNAMICAL SYSTEMS ∗
"... Abstract. 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 often leads to unmanageable demands on computational resources. Model reduction aims to reduce this ..."
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Abstract. 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 often 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 state of the art 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 equations. The goal of parametric model reduction is to generate low cost but accurate models that characterize system response for different values of the parameters. This paper surveys stateoftheart meth
t DataDriven Combined State and Parameter Reduction for ExtremeScale Inverse Problems
"... In this contribution we present an accelerated optimizationbased approach for combined state and parameter reduction of a parametrized linear control system which is then used as a surrogate model in a Bayesian inverse setting. Following the basic ideas presented in [Lieberman, Willcox, Ghattas. ..."
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In this contribution we present an accelerated optimizationbased approach for combined state and parameter reduction of a parametrized linear control system which is then used as a surrogate model in a Bayesian inverse setting. Following the basic ideas presented in [Lieberman, Willcox, Ghattas. Parameter and state model reduction for largescale statistical inverse settings, SIAM J. Sci. Comput., 32(5):25232542, 2010], our approach is based on a generalized datadriven optimization functional in the construction process of the surrogate model and the usage of a trustregiontype solution strategy that results in an additional speedup of the overall method. In principal, the model reduction procedure is based on the offline construction of appropriate lowdimensional state and parameter spaces and an online inversion step based on the resulting surrogate model that is obtained through projection of the underlying control system onto the reduced spaces. The generalization and enhancements presented in this work are shown to decrease overall computational time and increase accuracy of the reduced order model and thus allow an application to extremescale problems. Numerical experiments for a generic model and a fMRI connectivity model are presented in order to compare the computational efficiency of our improved method with the original approach. 1
scheme using PETSc for a Parabolic Optimal Control Problem
"... Abstract—This work presents a parallel implementation of the ..."
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Politecnico di MilanoMIT. We acknowledge many helpful
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Parameter and State Model Reduction for Bayesian Statistical Inverse Problems
, 2009
"... Decisions based on singlepoint estimates of uncertain parameters neglect regions of significant probability. We consider a paradigm based on decisionmaking under uncertainty including three steps: identification of parametric probability by solution of the statistical inverse problem, propagation ..."
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Decisions based on singlepoint estimates of uncertain parameters neglect regions of significant probability. We consider a paradigm based on decisionmaking under uncertainty including three steps: identification of parametric probability by solution of the statistical inverse problem, propagation of that uncertainty through complex models, and solution of the resulting stochastic or robust mathematical programs. In this thesis we consider the first of these steps, solution of the statistical inverse problem, for partial differential equations (PDEs) parameterized by field quantities. When these field variables and forward models are discretized, the resulting system is highdimensional in both parameter and state space. The system is therefore expensive to solve. The statistical inverse problem is one of Bayesian inference. With assumption on prior belief about the form of the parameter and an assignment of normal error in sensor measurements, we derive the solution to the statistical inverse problem analytically, up to a constant of proportionality. The parametric probability