Title: Localized Model Order Reduction
Abstract: In the last decades numerical simulations based on partial differential equations (PDEs) have significantly gained importance in engineering applications. However, both the geometric complexity of the considered structures, as say ships, aircrafts, and turbines, and the intricacy of the simulated physical phenomena often make a straightforward application of say the Finite Element (FE) method prohibitive. This is particularly true if multiple simulation requests or real-time simulation response is desired as in engineering design and optimization. One way to tackle such complex problems is to exploit the natural decomposition of the structures into components and apply model order reduction within domain decomposition methods.
In this talk we first introduce local approximation spaces for some domain decomposition methods, which are optimal in the sense that they minimize the approximation error among all spaces of the same dimension. As the approximation of those optimal local approximation spaces say with the FE method can be very expensive we propose next to use methods from randomized linear algebra to construct local spaces which yield an approximation that converges with a nearly optimal rate, but can be generated at nearly optimal computational complexity. Finally, we generalize the procedure to parameter-dependent PDEs facilitating a real-time simulation response.
We provide several numerical experiments that demonstrate an exponential convergence of the reduced approximation also for irregular geometries and show that the approximation based on the local spaces generated by random sampling converges indeed with a nearly optimal rate.
