Automated extension of fixed point PDE solvers for optimal design with bounded retardation

• Antragsteller: Professor Dr. Nicolas R. Gauger; Professor Dr. Andreas Griewank (†); Professor Dr. Thomas Slawig
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2006 bis 2015
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 25207711

This proposal concerns the development of mathematical methods, algorithmic techniques and software tools for the transition from simulation to optimization. We focus in particular on applications in aerodynamics to optimize wing shapes and in climate studies to identify parameters. The methodology is applicable to all areas of scientific computing, where large scale governing equations involving discretized PDEs are treated by custom made fixed point solvers. To exploit the domain specific experience and expertise invested in these simulation tools we propose to extend them in a semi-automated fashion. First they are augmented with an adjoint solvers to obtain (reduced) derivatives and then this sensitivity information is immediately used to determine optimization corrections. In other words, rather than applying an outer optimization loop we prefer the one-shot strategy of pursuing optimality simultaneously with the goals of primal and adjoint feasibility.To complete an optimization run in a small multiple of the number of iterations required for a simulation run it must be ensured that the contractivity factor of the combined one-shot scheme is only a certain fraction closer to 1 than that of the user supplied fixed point solver. The realization of this principle of bounded retardation of the convergence rate requires a significant amount of mathematical analysis, algorithmic experimentation, and software development. More specifically we propose to analyze and optimize the spectral properties of the combined iteration in a Hubert space setting, to constructively identify the required first and second derivatives, and to verify the efficacy of the scheme on full scale models in aerodynamics and climatology. The analysis and algorithm development will be carried out at the two Berlin Universities and the application specific work at the HU Berlin and the TU Berlin for aerodynamics and climatology, respectively.

• DFG-Verfahren: Schwerpunktprogramme
• Teilprojekt zu: SPP 1253: Optimierung mit partiellen Differentialgleichungen