EU Regional School - Heinkenschloss Seminar - Part 1
Prof. Dr. Heinkenschloss - Model Reduction in PDE-Constrained Optimization
The numerical solution of optimization problems governed by partial differential equations (PDEs) requires the repeated solution of coupled systems of PDEs. Model reduction can be used to substantially lower the computational cost by using reduced order models (ROMs) as surrogates for the expensive original objective and constraint functions, or to use ROMs to accelerate subproblem solves in traditional Newton-type methods. In these lectures I will present approaches for the integration of projection based ROMs into PDE-constrained optimization, discuss their computational costs and convergence properties, and demonstrate their performance on example problems. I will review the generation of projection based ROMs, as well as Newton-type optimization algorithms. The integration of projection based ROMs and optimization will first be discussed for relatively simple PDE constrained optimization problems that allow for precomputations of ROMs and computations of global error bounds, and then for nonlinear PDE constrained optimization problems where such precomputations and generations of global error bounds are typically impossible.