A certified reduced basis method for parametrized elliptic optimal control problems
Mark Kärcher,Martin A. Grepl +1 more
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TLDR
The reduced basis method is employed as a surrogate model for linear-quadratic optimal control problems governed by parametrized elliptic partial dierential equations and it is shown that, based on the as- sumption of parameter dependence, the reduced order optimal control problem and the proposed bounds can be evaluated in an online computational procedure.Abstract:
In this paper, we employ the reduced basis method as a surrogate model for the solu- tion of linear-quadratic optimal control problems governed by parametrized elliptic partial dierential equations. We present a posteriori error estimation and dual procedures that provide rigorous bounds for the error in several quantities of interest: the optimal control, the cost functional, and general linear output functionals of the control, state, and adjoint variables. We show that, based on the as- sumption of ane parameter dependence, the reduced order optimal control problem and the proposed bounds can be eciently evaluated in an oine-online computational procedure. Numerical results are presented to conrm the validity of our approach.read more
Citations
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Numerical Approximation Of Partial Differential Equations
TL;DR: In this article, a numerical approximation of partial differential equations was used to detect harmful downloads of books on the Internet, where people have search hundreds of times for their chosen novels like this numerical approximation, but end up in harmful downloads.
Book ChapterDOI
Chapter 2: Reduced Basis Methods for Parametrized PDEs—A Tutorial Introduction for Stationary and Instationary Problems
TL;DR: A class of model reduction techniques for parametric partial differential equations, the so-called Reduced Basis (RB) methods, allow to obtain low-dimensional parametric models for various complex applications, enabling accurate and rapid numerical simulations.
Journal ArticleDOI
Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations
TL;DR: The reduced basis method is extended to the case of noncoercive (elliptic) equations, such as the Stokes equations, and applied to solve a benchmark vorticity minimization problem for a parametrized bluff body immersed in a two or a three-dimensional flow through boundary control, demonstrating the effectivity of the methodology.
Journal ArticleDOI
Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations
TL;DR: A multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations is developed and it is proved the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data is proved.
Journal ArticleDOI
Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls
TL;DR: This paper introduces reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable and proposes two different error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional.
References
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Book
Optimal Control of Systems Governed by Partial Differential Equations
TL;DR: In this paper, the authors consider the problem of minimizing the sum of a differentiable and non-differentiable function in the context of a system governed by a Dirichlet problem.
Book
Numerical Approximation of Partial Differential Equations
Alfio Quarteroni,Alberto Valli +1 more
TL;DR: In this article, the authors provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation.
Journal ArticleDOI
Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations
TL;DR: (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations are considered.
Journal ArticleDOI
Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods
Christophe Prud'Homme,Dimitrios V. Rovas,Karen Veroy,Luc Machiels,Yvon Maday,Anthony T. Patera,Gabriel Turinici +6 more
TL;DR: The method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
Journal ArticleDOI
Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations
TL;DR: In this paper, the authors extended the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in volving.
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