TY - CHAP T1 - Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs T2 - Springer, Indam Series, Vol. 4, 2012 Y1 - 2012 A1 - Toni Lassila A1 - Andrea Manzoni A1 - Alfio Quarteroni A1 - Gianluigi Rozza KW - solution manifold AB - The set of solutions of a parameter-dependent linear partial di fferential equation with smooth coe fficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affi ne parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affi ne expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold \\r\\nonly spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic\\r\\nequations con rming the predicted convergence rates. JF - Springer, Indam Series, Vol. 4, 2012 PB - Springer UR - http://hdl.handle.net/1963/6340 U1 - 6270 U2 - Mathematics U4 - 1 U5 - MAT/08 ANALISI NUMERICA ER -