Adaptive Reduced Basis Trust Region Methods for Parameter Identification Problems

Kartmann, Michael; Keil, Tim; Ohlberger, Mario; Volkwein, Stephan; Kaltenbacher, Barbara

Abstract

In this contribution, we are concerned with model order reduction in the context of iterative regularization methods for the solution of inverse prob- lems arising from parameter identification in elliptic partial differential equations. Such methods typically require a large number of forward so- lutions, which makes the use of the reduced basis method attractive to re- duce computational complexity. However, the considered inverse problems are typically ill-posed due to their infinite-dimensional parameter space. Moreover, the infinite-dimensional parameter space makes it impossible to build and certify classical reduced-order models efficiently in a so-called ”offline phase”. We thus propose a new algorithm that adaptively builds a reduced parameter space in the online phase. The enrichment of the reduced parameter space is naturally inherited from the Tikhonov regu- larization within an iteratively regularized Gauß-Newton method. Finally, the adaptive parameter space reduction is combined with a certified re- duced basis state space reduction within an adaptive error-aware trust region framework. Numerical experiments are presented to show the ef- ficiency of the combined parameter and state space reduction for inverse parameter identification problems with distributed reaction or diffusion coefficients.