Model Reduction for Large Scale Systems

Abstract

Projection based model order reduction has become a mature technique for simulation of large classes of parameterized systems. However, several challenges remain for problems where the solution manifold of the parameterized system cannot be well approximated by linear subspaces. While the online efficiency of these model reduction methods is very convincing for problems with a rapid decay of the Kolmogorov n-width, there are still major drawbacks and limitations. Most importantly, the construction of the reduced system in the offline phase is extremely CPU-time and memory consuming for large scale and multi scale systems. For practical applications, it is thus necessary to derive model reduction techniques that do not rely on a classical offline/online splitting but allow for more flexibility in the usage of computational resources. A promising approach with this respect is model reduction with adaptive enrichment. In this contribution we investigate Petrov-Galerkin based model reduction with adaptive basis enrichment within a Trust Region approach for the solution of multi scale and large scale PDE constrained parameter optimization.

A relaxed localized trust-region reduced basis approach for optimization of multiscale problems

“…Moreover, the convergence is not affected by the choice of the reduction approach of the primal and dual equations, as long as a certified error control of the reduced functional is given. In the numerical experiments that were shown in [39,9,40], the TR-RB algorithm showed a remarkably robust behavior, including significant enhancements compared to the original version from [56]. Nevertheless, some hints for room for improvements concerning the overall cost were noticeable.…”

“…In this contribution, we will particularly build on a combination of the adaptive and certified trustregion reduced basis optimization method [39,9,40] with an underlying efficient discretization framework based on the Petrov-Galerkin LOD [24] and its recently introduced two-scale reduced basis approximation (TSRBLOD) [41]. The resulting variant of a trust-region localized RB method (TR-LRB) adaptively constructs local RB models in each of the TR subproblems and deviates from a classical globally resolved finite element approximation of the underlying multiscale state equation.…”

“…Before enrichment: check(12) for early termination; enrich the local RB corrector models at µ (k+1) if(40) including pre-assembly of the local estimators ∆ T loc ; construct the primal and dual two-scale models and enrich for all µ (k ) , k = 0, . .…”