Jonas Kusch, RWTH Aachen/TU Kaiserslautern
Frequency Based Preconditioning And Smoothing for Shape Optimization
In this talk, a preconditioner for shape optimization in the case of small Reynolds numbers will be derived. The preconditioner mimics the local Hessian behavior, which is derived analytically for the Stokes equations and investigated numerically for the Navier-Stokes equations. The derivation will show that the Hessian symbol has smoothing behavior, which is why we can think of the method as a local smoothing technique that will accelerate the optimization process. This acceleration is crucial as the raw steepest descent method will need a large number of computationally expensive optimization steps. In order to ensure a computationally cheap approximation of the Hessian symbol, the pseudo-differential Hessian behavior of order one is approximated by even order differential operators. To obtain a good approximation of the linear scaling behavior caused by the pseudo-differential Hessian behavior, we will make use of the windowed Fourier transform. We will use the knowledge of local frequencies to locally adapt the preconditioner, such that we have linear scaling for relevant frequencies.