Past Events

To construct localizable multiscale methods for nonlinear partial differential equations we consider a transfer operator that maps arbitrary admissible boundary data on the boundary of an oversampling domain to the respective (local) solution on the target subdomain; here the boundary of the latter must have a distance greater than zero from the boundary of the oversampling domain. Then, we try to approximate the set of all local solutions on the target subdomain. Interpreting the boundary data as some input parameter, we can view this set of local solutions as a set of solutions depending on a parameter. This motivates using methods from model order reduction such as the proper orthogonal decomposition (POD) or the Greedy algorithm to approximate this set. However, both the POD and the Greedy algorithm rely on a training set of finite cardinality that is chosen such that every point in the admissible parameter set is close to a point in the training set. Therefore, both algorithms suffer from the curse of dimensionality. We thus employ randomization and consider the parameter (here: boundary data) as a random variable with values in a Hilbert space. By choosing a suitable distribution we can then exploit the concentration of measure phenomenon, which is also sometimes called the "blessing of dimensionality" to break the curse.

 

In detail, we will present a randomized greedy algorithm that provides with high probability a certification for the whole parameter set rather than only for the parameters in the training set. Moreover, we will present a randomized POD and a corresponding error analysis that shows that for exponentially decaying eigenvalues of the randomized POD which uses the exact correlation operator (integral in the expectation) the approximation error between any solution corresponding to a parameter in the admissible parameter set and the approximation with the POD that uses a Monte-Carlo approximation converges exponentially as well.