Localized Reduced Basis Methods for PDE-constrained Parameter Optimization

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This projects is concerned with model reduction for parameter optimization of nonlinear elliptic partial differential equations (PDEs). The goal is to develop a new paradigm for PDE-constrained optimization based on adaptive online enrichment. The essential idea is to design a localized version of the reduced basis (RB) method which is called Localized Reduced Basis Method (LRBM). This allows us to tighten the quality of the reduced order approximation online within each iteration of the applied optimization algorithms. A localized a posteriori error analysis ensures convergence of the reduced basis solution to the solution of the underlying infinite dimensional parameter optimization problem. In the case of a locally inaccurate approximation quality the RB discretization is improved only locally in a very efficient way. The approach is designed for numerical multiscale methods, trust region based optimization methods and for iteratively regularized Gauß-Newton algorithms.