Adaptive Reduced Basis Trust Region Methods for Parameter Identification Problems

TL;DR

This paper introduces an adaptive reduced basis trust region method that efficiently solves inverse parameter identification problems in elliptic PDEs by dynamically reducing the parameter and state spaces during the iterative process.

Contribution

The paper presents a novel algorithm that adaptively constructs a reduced parameter space during the online phase, integrating it with a certified reduced basis state space within a trust region framework.

Findings

Significant reduction in computational cost demonstrated.

Effective handling of ill-posed inverse problems shown.

Numerical experiments confirm efficiency and accuracy.

Abstract

In this contribution, we are concerned with model order reduction in the context of iterative regularization methods for the solution of inverse problems arising from parameter identification in elliptic partial differential equations. Such methods typically require a large number of forward solutions, which makes the use of the reduced basis method attractive to reduce 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 regularization within an iteratively regularized Gau{\ss}-Newton method. Finally, the adaptive parameter space reduction is combined with a certified reduced basis state space reduction within an adaptive error-aware trust region framework. Numerical experiments are presented to show the efficiency of the combined parameter and state space reduction for inverse parameter identification problems with distributed reaction or diffusion coefficients.