Randomized Model Order Reduction

Description

In this talk we show how randomization as used say in randomized linear algebra or compressed sensing can be exploited both for constructing reduced order models and deriving bounds for the approximation error.

We propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods, where those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions [Buhr, Smetana 2018]. Extending methods from randomized linear algebra [Halko et al 2011] allows us to construct local spaces both for interfaces and subdomains that yield an approximation that converges
provably at a nearly optimal rate and can be generated at close to optimal computational complexity. To realize the latter, we build the reduced spaces adaptively, relying on a probabilistic a posteriori error estimator.

Moreover, we propose a constant-free, probabilistic a posteriori error estimator for reduced order approximations such as the reduced basis approximation for parametrized PDEs. This error estimator does not require to estimate any stability constants and is both reliable and efficient at (given) high probability.
Here, we rely on results similar to the restricted isometry property employed in compressed sensing [Vershynin 2012]. In order to obtain an a posteriori error estimator that is computationally feasible in the online stage we employ the solution of a reduced dual problem with random right-hand side, exploiting the typically fast convergence of reduced order models.

Work in collaboration with A. Buhr (University of Muenster), A. T. Patera (Massachusetts Institute of Technology), and O. Zahm (Massachusetts Institute of Technology)

Period	11 Apr 2018
Event title	4th Model Reduction of Parametrized Systems , MoRePaS 2018
Event type	Conference
Conference number	4
Location	Nantes, France
Degree of Recognition	International