We consider inverse problems estimating distributed parameters from indirect noisy observations through discretization of continuum models described by partial differential or integral equations. It is well understood that the errors arising from the discretization can be detrimental for ill-posed inverse problems, as discretization error behaves as correlated noise. While this problem can be avoided with a discretization fine enough to suppress the modeling error level below that of the exogenous noise that is addressed, e.g., by regularization, the computational resources needed to deal with the additional degrees of freedom may require high performance computing environment. Following an earlier idea, we advocate the notion that the discretization is one of the unknowns of the inverse problem, and is updated iteratively together with the solution. In this approach, the discretization, defined in terms of an underlying metric, is refined selectively only where the representation power of the current mesh is insufficient. In this paper we allow the metrics and meshes to be anisotropic, and we show that this leads to significant reduction of memory allocation and computing time.

翻译：我们考虑一类反问题，即通过偏微分方程或积分方程描述的连续体模型的离散化，从间接含噪观测中估计分布参数。普遍认为，离散化产生的误差对不适定反问题具有破坏性影响，因为离散化误差表现为相关噪声。尽管通过足够精细的离散化将建模误差水平抑制到外部噪声水平以下（例如通过正则化处理）可避免此问题，但处理额外自由度所需的计算资源可能需要高性能计算环境。基于先前思路，我们主张离散化本身是反问题的未知量之一，应与解同步迭代更新。在该方法中，以底层度规定义的离散化仅在当前网格表示能力不足时被选择性细化。本文允许度规与网格具有各向异性，并证明此举能显著减少内存分配与计算时间。