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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 53 (9 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
An hp Certified Reduced Basis Method for Parametrized Parabolic Partial Differential Equations
, 2010
"... Abstract. We present a new “hp ” parameter multidomain certified reduced basis method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations. We propose, and provide theoretical justification for, a new procedure for adapt ..."
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Cited by 35 (5 self)
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Abstract. We present a new “hp ” parameter multidomain certified reduced basis method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations. We propose, and provide theoretical justification for, a new procedure for adaptive partition (“h”refinement) of the parameter domain into smaller parameter subdomains: we pursue a hierarchical splitting of the parameter (sub)domains based on proximity to judiciously chosen parameter anchor points within each subdomain. Subsequently, we construct individual standard RB approximation spaces (“p”refinement) over each subdomain. Greedy parameter sampling procedures and a posteriori error estimation play important roles in both the “h”type and “p”type stages of the new algorithm. We present illustrative numerical results for a convectiondiffusion problem: the new “hp”approach is considerably faster (respectively, more costly) than the standard “p”type reduced basis method in the online (respectively, offline) stage. Key words. reduced basis; a posteriori error estimation; Greedy sampling; htype; ptype; hp convergence; parameter domain decomposition
Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation
 SIAM J. Sci. Comput
"... Abstract. We present a new approach to treat nonlinear operators in reduced basis approximations of parametrized evolution equations. Our approach is based on empirical interpolation of nonlinear differential operators and their Fréchet derivatives. Efficient offline/online decomposition is obtain ..."
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Cited by 34 (17 self)
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Abstract. We present a new approach to treat nonlinear operators in reduced basis approximations of parametrized evolution equations. Our approach is based on empirical interpolation of nonlinear differential operators and their Fréchet derivatives. Efficient offline/online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals. An a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically. We introduce a new algorithm, the PODEIgreedy algorithm, which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronised way. The approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations. We show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions. In case of symmetries of the problem, the approach realizes an automatic and intuitive space–compression or even space–dimensionality reduction. We perform empirical investigations of the error convergence and run–times. In all cases we obtain a good run–time acceleration.
Reduced basis techniques for stochastic problems
, 2010
"... We report here on the recent application of a now classical general reduction technique, the ReducedBasis (RB) approach initiated in [PRV+02], to the specific context of differential equations with random coefficients. After an elementary presentation of the approach, we review two contributions of ..."
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Cited by 25 (6 self)
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We report here on the recent application of a now classical general reduction technique, the ReducedBasis (RB) approach initiated in [PRV+02], to the specific context of differential equations with random coefficients. After an elementary presentation of the approach, we review two contributions of the authors: [BBM+09], which presents the application of the RB approach for the discretization of a simple second order elliptic equation supplied with a random boundary condition, and [BL09], which uses a RB type approach to reduce the variance in the MonteCarlo simulation of a stochastic differential equation. We conclude the review with some general comments and also discuss possible tracks for further research in the direction.
Adaptive basis enrichment for the reduced basis method applied to finite volume schemes
 Proc. 5th International Symposium on Finite Volumes for Complex Applications
"... ABSTRACT. We derive an efficient reduced basis method for finite volume approximations of parameterized linear advectiondiffusion equations. An important step in deriving a reduced finite volume model with the reduced basis technology is the generation of a reduced basis space, on which the detaile ..."
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Cited by 22 (12 self)
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ABSTRACT. We derive an efficient reduced basis method for finite volume approximations of parameterized linear advectiondiffusion equations. An important step in deriving a reduced finite volume model with the reduced basis technology is the generation of a reduced basis space, on which the detailed numerical simulations are projected. We present a new strategy for this reduced basis generation. We apply an effective exploration of the parameter space by adaptive grids based on an a posteriori error estimate. The resulting method gives a considerable improvement concerning equal distribution of the model error over the parameter space compared to uniform parameter selections. It is computationally very efficient in terms of small ratio of trainingtime over modelerror.
A certified reduced basis method for the FokkerPlanck equation of dilute polymeric fluids: FENE Dumbbells in extensional flow
 SIAM Journal on Scientific Computing
"... Abstract. In this paper we present a reduced basis method for the parametrized Fokker–Planck equation associated with evolution of Finitely Extensible Nonlinear Elastic (FENE) dumbbells in a Newtonian solvent for a (prescribed) extensional macroscale flow. There are two new important ingredients: a ..."
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Cited by 20 (5 self)
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Abstract. In this paper we present a reduced basis method for the parametrized Fokker–Planck equation associated with evolution of Finitely Extensible Nonlinear Elastic (FENE) dumbbells in a Newtonian solvent for a (prescribed) extensional macroscale flow. There are two new important ingredients: a projection–based POD–Greedy sampling procedure for the stable identification of optimal reduced basis spaces; and a finite–time a posteriori bound for the error in the reduced basis prediction of the two outputs of interest — the optical anisotropy and the first normal stress difference. We present numerical results for stress–conformation hysteresis as a function of Weissenberg number and final time that demonstrate the rapid convergence of the reduced basis approximation and the effectiveness of the a posteriori error bounds. Key words. Reduced basis methods, POD, greedy algorithm, a posteriori error estimation, Fokker–Planck equation, FENE dumbbells, polymeric fluids, micromacro model. 1. Introduction. The
An Improved Error Bound for Reduced Basis Approximation of Linear Parabolic Problems
 Math. Comput., online first
"... Abstract. We consider a spacetime variational formulation for linear parabolic partial differential equations. We introduce an associated PetrovGalerkin truth finite element discretization with favorable discrete infsup constant βδ, the inverse of which enters into error estimates: βδ is unity f ..."
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Cited by 13 (3 self)
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Abstract. We consider a spacetime variational formulation for linear parabolic partial differential equations. We introduce an associated PetrovGalerkin truth finite element discretization with favorable discrete infsup constant βδ, the inverse of which enters into error estimates: βδ is unity for the heat equation; βδ decreases only linearly in time for noncoercive (but asymptotically stable) convection operators. The latter in turn permits effective longtime a posteriori error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. The paper contains a full analysis and various extensions for the formulation introduced briefly by Urban and Patera (2012) as well as numerical results for a model reactionconvectiondiffusion equation. 1.
REDUCED BASIS METHOD FOR FINITE VOLUME APPROXIMATION OF EVOLUTION EQUATIONS ON PARAMETRIZED GEOMETRIES
"... parameters that describe the geometry of the underlying problem. One can think of applications in control theory and optimization which depend on timeconsuming parameterstudies of such problems. Therefore, we want to reduce the order of complexity of the numerical simulations for such P2DEs. Reduc ..."
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Cited by 10 (7 self)
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parameters that describe the geometry of the underlying problem. One can think of applications in control theory and optimization which depend on timeconsuming parameterstudies of such problems. Therefore, we want to reduce the order of complexity of the numerical simulations for such P2DEs. Reduced Basis (RB) methods are a means to achieve this goal. These methods have gained popularity over the last few years for model reduction of finite element approximations of elliptic and instationary parabolic equations. We present a RB method for parabolic problems with general geometry parameterization and finite volume (FV) approximations. After a mapping on a reference domain, the parabolic equation leads to a convectiondiffusionreaction equation with anisotropic diffusion tensor. Suitable FV schemes with gradient reconstruction allow to discretize such problems. A model reduction of the resulting numerical scheme can be obtained by an RB technique. We present experimental results, that demonstrate the applicability of the RB method, in particular the computational acceleration. Key words. Reduced basis methods, model reduction, geometry transformation, heat equation AMS subject classifications. 76M12, 76R50, 35K05
Adaptive reduced basis methods for nonlinear convectiondiffusion equations
 in Finite Volumes for Complex Applications VI Problems & Perspectives
, 2011
"... evolution equations and depend on timeconsuming parameter studies or need to ensure critical constraints on the simulation time. For both settings, model order reduction by the reduced basis methods is a suitable means to reduce computational time. In this proceedings, we show the applicability of ..."
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Cited by 9 (1 self)
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evolution equations and depend on timeconsuming parameter studies or need to ensure critical constraints on the simulation time. For both settings, model order reduction by the reduced basis methods is a suitable means to reduce computational time. In this proceedings, we show the applicability of the reduced basis framework to a finite volume scheme of a parametrized and highly nonlinear convectiondiffusion problem with discontinuous solutions. The complexity of the problem setting requires the use of several new techniques like parametrized empirical operator interpolation, efficient a posteriori error estimation and adaptive generation of reduced data. The latter is usually realized by an adaptive search for base functions in the parameter space. Common methods and effects are shortly revised in this presentation and supplemented by the analysis of a new strategy to adaptively search in the time domain for empirical interpolation data. Key words: finite volume methods, model reduction, reduced basis methods, empirical interpolation MSC2010: 65M08, 65J15, 65Y20 1
Efficient Reduced Models and APosteriori Error Estimation for Parametrized Dynamical Systems by Offline/Online Decomposition
, 2009
"... Reduced basis (RB) methods are an effective approach for model reduction of parametrized partial differential equations. In the field of dynamical systems ’ order reduction, these methods are not very established, but the interest in reduction of parametrized systems is increasing. In the current p ..."
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Cited by 8 (3 self)
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Reduced basis (RB) methods are an effective approach for model reduction of parametrized partial differential equations. In the field of dynamical systems ’ order reduction, these methods are not very established, but the interest in reduction of parametrized systems is increasing. In the current presentation, we show that some characteristic components of RBmethods can be transfered to model reduction of parametrized linear dynamical systems. We assume an affine parameter dependence of the system components, which allows an offline/online decomposition and is the basis for efficient reduced simulation. Additionally, error control is possible by aposteriori error estimators for the state and output, based on residual analysis and primaldual techniques. Experiments demonstrate the applicability of the reduced parametrized systems, the reliability of the error estimators and the runtime gain by the reduction technique. The aposteriori error estimation technique can straightforwardly be applied to all traditional projectionbased reduction techniques of nonparametric linear systems, such