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：本文研究了在椭圆型偏微分方程参数识别反问题的迭代正则化求解过程中，模型降阶方法的应用。此类方法通常需要大量正向求解计算，这使得采用缩减基方法降低计算复杂度具有吸引力。然而，由于参数空间具有无限维特性，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”高效构建并认证经典降阶模型。为此，我们提出一种新算法，可在在线阶段自适应构建缩减参数空间。该参数空间的扩充自然源于迭代正则化高斯-牛顿法中的吉洪诺夫正则化。最终，在自适应误差感知信赖域框架中，将自适应参数空间约简与经过认证的缩减基状态空间约简相结合。数值实验表明，针对具有分布式反应系数或扩散系数的反参数识别问题，所提出的参数与状态空间联合约简方法具有高效性。