Foreword
In memoriam David Gottlieb 1944–2008
A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations
In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space spanned by solutions of the...
David Gottlieb 1944–2008
error bounds for reduced-basis approximations of parametrized parabolic partial differential equations
In this paper, we extend the reduced-basis methods and associated error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space spanned by solutions...
A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation
We propose a new reduced basis element-cum-component mode synthesis approach for parametrized elliptic coercive partial differential equations. In the Offline stage we construct a Library of interoperable parametrized reference relevant to some family of problems; in the Online stage we instantiate and connect reference components (at ports) to rapidly form and query parametric . The method is based on static condensation at the interdomain level, a conforming eigenfunction “port” representation...
A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations
We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the rapid and reliable evaluation prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are (i) (provably) rapidly convergent global reduced–basis approximations — Galerkin projection onto a space spanned...
A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : “convex inverse” bound conditioners
We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space spanned by solutions of the governing partial differential equation at selected points in parameter space; (ii) a posteriori error estimation – relaxations of the error-residual equation...
Error Estimation for Reduced-Basis Approximation of Parametrized Elliptic Coercive Partial Differential Equations: “Convex Inverse” Bound Conditioners
We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence. The essential components are () (provably) rapidly convergent global reduced-basis approximations – Galerkin projection onto a space spanned by solutions of the governing partial differential equation at selected points in parameter space; () error estimation – relaxations of the error-residual equation...
A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations
We present in this article two components: these components can in fact serve various goals independently, though we consider them here as an ensemble. The first component is a technique for the prediction of linear functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential features are () (provably) rapidly convergent global reduced–basis approximations — Galerkin projection onto a space spanned by solutions...
convergence of the Greedy algorithm for the parametrized reduced basis method
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that...
A priori convergence of the greedy algorithm for the parametrized reduced basis method
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that...
Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations
In this paper, we extend the reduced-basis approximations developed earlier for elliptic and parabolic partial differential equations with parameter dependence to problems involving (a) dependence on the parameter, and (b) dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure:...
convergence of the Greedy algorithm for the parametrized reduced basis method
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that...
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