Abstract:
Numerous model reduction methods, such as reduced basis
and its generalization, proper orthogonal decomposition, generalized
empirical interpolation, rely on the implicit assumption that the solution
manifold that gathers the solution to a PDE as parameters vary can
be well approximated by low dimensional spaces. This is made more
precise by assuming that the Kolmogorov n-width of the manifold has
certain decay. In this talk, I shall discuss strategies that allow to
rigorously establish rates of decay for the Kolmogorov n-width
of solution manifolds associated with relevant parametric PDE’s.
One key result is that the rate of decay of n-width of sets in Banach
space is almost preserved under the action of holomorphic maps.
Joint work with Ronald DeVore |