%0 Book Section
%B Springer, Indam Series, Vol. 4, 2012
%D 2012
%T Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs
%A Toni Lassila
%A Andrea Manzoni
%A Alfio Quarteroni
%A Gianluigi Rozza
%K solution manifold
%X 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.
%B Springer, Indam Series, Vol. 4, 2012
%I Springer
%G en
%U http://hdl.handle.net/1963/6340
%1 6270
%2 Mathematics
%4 1
%# MAT/08 ANALISI NUMERICA
%$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T17:47:45Z\\nNo. of bitstreams: 1\\nqlmr-bumi_FINAL_SISSAreport.pdf: 377397 bytes, checksum: ecaf5713afa6a6f68992f6331631aff4 (MD5)