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Reduced Basis Method for Parametrized Elliptic Optimal Control Problems
We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique. Copyright \ua9 by SIAM. Unauthorized reproduction of this article is prohibited
Reduced basis method for parametrized optimal control problems governed by PDEs
This master thesis aims at the development, analysis and computer implementation of effcient numerical methods for the solution of optimal control problems based on parametrized partial differential equations. Our goal isfto develop a new approach based on suitable model reduction paradigm --the reduced basis method (RB)-- for the rapid and reliable solution of control problems which may occur in several engineering contexts. In particular, we develop the methodology for parametrized quadratic optimization problem with either coercive elliptic equations or Stokes equations as constraints. Firstly, we recast the optimal control problem in the framework of mixed variational problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then the usual ingredients of the RB methodology are provided: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling to perform competitive Offine-Online splitting in the computational procedure; an efficient and rigorous a posteriori error estimation on the state, control and adjoint variables as well as on the cost functional. The reduction scheme is applied to several numerical tests conrming the theoretical results and demonstrating the efficiency of the proposed technique. Moreover an application to an (idealized) inverse problem in haemodynamics is discussed, showing the versatility and potentiality of the method in tackling parametrized optimal control problems that could arise in a a broad variety of application contexts
A fast Monte-Carlo method with a Reduced Basis of Control Variates applied to Uncertainty Propagation and Bayesian Estimation
The Reduced-Basis Control-Variate Monte-Carlo method was introduced recently
in [S. Boyaval and T. Leli\`evre, CMS, 8 2010] as an improved Monte-Carlo
method, for the fast estimation of many parametrized expected values at many
parameter values. We provide here a more complete analysis of the method
including precise error estimates and convergence results. We also numerically
demonstrate that it can be useful to some parametrized frameworks in
Uncertainty Quantification, in particular (i) the case where the parametrized
expectation is a scalar output of the solution to a Partial Differential
Equation (PDE) with stochastic coefficients (an Uncertainty Propagation
problem), and (ii) the case where the parametrized expectation is the Bayesian
estimator of a scalar output in a similar PDE context. Moreover, in each case,
a PDE has to be solved many times for many values of its coefficients. This is
costly and we also use a reduced basis of PDE solutions like in [S. Boyaval, C.
Le Bris, Nguyen C., Y. Maday and T. Patera, CMAME, 198 2009]. This is the first
combination of various Reduced-Basis ideas to our knowledge, here with a view
to reducing as much as possible the computational cost of a simple approach to
Uncertainty Quantification
A reduced basis localized orthogonal decomposition
In this work we combine the framework of the Reduced Basis method (RB) with
the framework of the Localized Orthogonal Decomposition (LOD) in order to solve
parametrized elliptic multiscale problems. The idea of the LOD is to split a
high dimensional Finite Element space into a low dimensional space with
comparably good approximation properties and a remainder space with negligible
information. The low dimensional space is spanned by locally supported basis
functions associated with the node of a coarse mesh obtained by solving
decoupled local problems. However, for parameter dependent multiscale problems,
the local basis has to be computed repeatedly for each choice of the parameter.
To overcome this issue, we propose an RB approach to compute in an "offline"
stage LOD for suitable representative parameters. The online solution of the
multiscale problems can then be obtained in a coarse space (thanks to the LOD
decomposition) and for an arbitrary value of the parameters (thanks to a
suitable "interpolation" of the selected RB). The online RB-LOD has a basis
with local support and leads to sparse systems. Applications of the strategy to
both linear and nonlinear problems are given
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