@article {2015,
	title = {Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations},
	journal = {Numerische Mathematik, (2015), 36 p. Article in Press},
	number = {Numerische Mathematik;},
	year = {2015},
	publisher = {Springer},
	abstract = {In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems.},
	doi = {10.1007/s00211-015-0743-4},
	url = {http://urania.sissa.it/xmlui/handle/1963/34491},
	author = {Gianluigi Rozza and Peng Chen and Alfio Quarteroni}
}
@article {2014,
	title = {Model Order Reduction in Fluid Dynamics: Challenges and Perspectives},
	number = {Modeling simulation and applications;volume 9; pages 235-273;},
	year = {2014},
	publisher = {Springer},
	abstract = {This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities - which are mainly related either to nonlinear convection terms and/or some geometric variability - that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and-in the unsteady case - long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.},
	doi = {10.1007/978-3-319-02090-7_9},
	author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza}
}
