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204
Adaptive PetrovGalerkin methods for first order transport equations
, 2011
"... Abstract. We propose a general framework for well posed variational formulations of linear unsymmetric operators, taking first order transport and evolution equations in bounded domains as primary orientation. We outline a general variational framework for stable discretizations of boundary value pr ..."
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Cited by 79 (8 self)
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Abstract. We propose a general framework for well posed variational formulations of linear unsymmetric operators, taking first order transport and evolution equations in bounded domains as primary orientation. We outline a general variational framework for stable discretizations of boundary value problems for these operators. To adaptively resolve anisotropic solution features such as propagating singularities the variational formulations should allow one, in particular, to employ as trial spaces directional representation systems. Since such systems are known to be stable in L2 special emphasis is placed on L2stable formulations. The proposed stability concept is based on perturbations of certain “ideal ” test spaces in PetrovGalerkin formulations. We develop a general strategy for realizing corresponding schemes without actually computing excessively expensive test basis functions. Moreover, we develop adaptive solution concepts with provable error reduction. The results are illustrated by first numerical experiments.
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
A General Multipurpose Interpolation Procedure: The Magic Points
 Communications on Pure and Applied Analysis
"... Abstract. Lagrangian interpolation is a classical way to approximate general functions by finite sums of well chosen, predefined, linearly independent interpolating functions; it is much simpler to implement than determining the best fits with respect to some Banach (or even Hilbert) norms. In add ..."
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Cited by 45 (11 self)
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Abstract. Lagrangian interpolation is a classical way to approximate general functions by finite sums of well chosen, predefined, linearly independent interpolating functions; it is much simpler to implement than determining the best fits with respect to some Banach (or even Hilbert) norms. In addition, only partial knowledge is required (here values on some set of points). The problem of defining the best sample of points is nevertheless rather complex and is in general open. In this paper we propose a way to derive such sets of points. We do not claim that the points resulting from the construction explained here are optimal in any sense. Nevertheless, the resulting interpolation method is proven to work under certain hypothesis, the process is very general and simple to implement, and compared to situations where the best behavior is known, it is relatively competitive.
Convergence rates of best Nterm Galerkin approximations for a class of elliptic sPDEs ∗
, 2010
"... Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D ⊂ R d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2 (D)orthogonal bases, a ..."
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Cited by 36 (9 self)
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Deterministic Galerkin approximations of a class of second order elliptic PDEs with random coefficients on a bounded domain D ⊂ R d are introduced and their convergence rates are estimated. The approximations are based on expansions of the random diffusion coefficients in L 2 (D)orthogonal bases, and on viewing the coefficients of these expansions as random parameters y = y(ω) = (yi(ω)). This yields an equivalent parametric deterministic PDE whose solution u(x, y) is a function of both the space variable x ∈ D and the in general countably many parameters y. We establish new regularity theorems decribing the smoothness properties of the solution u as a map from y ∈ U = (−1, 1) ∞ to V = H 1 0(D). These results lead to analytic estimates on the V norms of the coefficients (which are functions of x) in a socalled “generalized polynomial chaos”(gpc) expansion of u. Convergence estimates of approximations of u by best Nterm truncated Vvalued polynomials in the variable y ∈ U are established. These estimates are of the form N −r, where the rate of convergence r depends only on the decay of the random input expansion. It
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.
Generalized Multiscale Finite Element Methods (GMsFEM)
, 2013
"... In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach i ..."
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Cited by 30 (10 self)
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In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be reused for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarsegrid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method. 1
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.