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.

翻译：本文关注于在求解椭圆偏微分方程参数识别反问题的迭代正则化方法中，模型降阶的应用。此类方法通常需要大量正问题求解，这使得使用简化基方法降低计算复杂度具有吸引力。然而，由于参数空间是无限维的，所考虑的反问题通常是病态的。此外，无限维参数空间使得无法在所谓的“离线阶段”高效地构建和验证经典降阶模型。因此，我们提出一种新算法，该算法在线阶段自适应地构建简化参数空间。简化参数空间的扩充自然继承自迭代正则化高斯-牛顿方法中的吉洪诺夫正则化。最终，自适应参数空间约简与基于认证简化基的状态空间约简相结合，集成于自适应误差感知信赖域框架中。数值实验表明，对于具有分布反应或扩散系数的反参数识别问题，这种参数与状态空间联合约简方法具有高效性。