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：本文关注在求解椭圆偏微分方程参数辨识反问题时，迭代正则化方法中的模型降阶技术。这类方法通常需要大量正问题求解，因此采用缩减基方法以降低计算复杂度具有显著优势。然而，由于无限维参数空间的存在，所考虑的反问题通常是不适定的。此外，无限维参数空间使得在所谓"离线阶段"高效构建并验证经典降阶模型变得不可行。为此，我们提出一种新算法，能够在在线阶段自适应地构建缩减参数空间。缩减参数空间的扩充机制自然地继承自迭代正则化高斯-牛顿方法中的Tikhonov正则化过程。最终，我们将自适应参数空间缩减与经过验证的缩减基状态空间缩减相结合，置于自适应误差感知信赖域框架中。数值实验表明，针对具有分布反应系数或扩散系数的反参数辨识问题，参数与状态空间的联合缩减策略具有显著效能。