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 -