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 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 decrease 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 increase so much as to require high performance computing environments. Following an earlier idea, we advocate the notion of the discretization as one of the unknowns of the inverse problem, which 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.

中文翻译：

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逆问题的自适应各向异性贝叶斯网格划分

我们考虑逆问题，通过偏微分或积分方程描述的连续介质模型的离散化，从间接噪声观测中估计分布参数。众所周知，离散化产生的误差对于不适定反问题可能是有害的，因为离散化误差表现为相关噪声。虽然这个问题可以通过足够精细的离散化来将建模误差水平降低到所解决的外源噪声的水平以下（例如通过正则化）来避免，但是处理附加自由度所需的计算资源可能会增加太多，以至于需要高性能计算环境。遵循早期的想法，我们提倡将离散化的概念作为逆问题的未知数之一，它与解决方案一起迭代更新。在这种方法中，仅当当前网格的表示能力不足时，才选择性地细化根据基础度量定义的离散化。在本文中，我们允许度量和网格是各向异性的，并且我们表明这会显着减少内存分配和计算时间。