Adaptive methods for problems with infinitely many parameters and their computational complexity

Markus Bachmayr (RWTH Aachen)

________________________________________________________________________________

In sparse polynomial approximations of elliptic PDEs that depend on infinitely 
many parameters - as, for example, via a series expansion of a coefficient given 
by a random field  - certain structural features of the parameter dependence 
turn out to have an impact on the achievable convergence rate. In particular, 
suitable multilevel representations of random fields lead to optimal fully 
discrete convergence rates that are naturally related to the smoothness of 
random field realizations. However, this requires an independently adapted 
spatial discretization for each coefficient of a polynomial in the parametric 
expansion. This leads to the question whether such approximations can also be 
constructed by numerical methods at optimal computational costs. We show that 
this is the case for adaptive finite element spatial discretizations based on 
standard newest vertex bisection.

________________________________________________________________________________