SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A3878-A3906, January 2020.
The problem of estimating numerically a distributed parameter from indirect measurements arises in many applications, and in that context the choice of the discretization plays an important role. In fact, guaranteeing a certain level of accuracy of the forward model that maps the unknown to the observations may require a fine discretization, adding to the complexity of the problem and to the computational cost. On the other hand, reducing the complexity of the problem by adopting a coarser discretization may increase the modeling error and can be very detrimental for ill-posed inverse problems. To balance accuracy and complexity, we propose an adaptive algorithm for adjusting the discretization level automatically and dynamically while estimating the unknown distributed parameter by an iterative scheme. In the Bayesian paradigm, all unknowns, including the metric that defines the discretization, are modeled as random variables. Our approach couples the discretization with a Bayesian hierarchical hyperparameter that is estimated simultaneously with the unknown parameter of primary interest. The viability of the proposed algorithm, the Bayesian mesh adaptation (BMA) is assessed on two test cases: a fan-beam X-ray tomography problem and an inverse source problem for a Darcy flow model.

中文翻译：

---

贝叶斯网格自适应估计分布参数

SIAM科学计算杂志，第42卷，第6期，第A3878-A3906页，2020年1月。
在许多应用中都出现了用间接测量方法从数值上估算分布参数的问题，在这种情况下，离散化的选择起着重要的作用。实际上，要保证将未知数映射到观测值的正向模型具有一定水平的准确性，可能需要进行精细离散化，从而增加了问题的复杂性和计算成本。另一方面，通过采用较粗糙的离散化来降低问题的复杂性可能会增加建模误差，并且可能会对不适定的逆问题产生极大的不利影响。为了平衡精度和复杂度，我们提出了一种自适应算法，该算法可自动动态地调整离散化级别，同时通过迭代方案估计未知的分布参数。在贝叶斯范式中，所有未知数，包括定义离散化的度量的模型都被建模为随机变量。我们的方法将离散化与贝叶斯分层超参数结合在一起，贝叶斯分层超参数与主要关注的未知参数同时估计。在两个测试案例上评估了所提出算法贝叶斯网格自适应（BMA）的可行性：扇形束X射线断层扫描问题和Darcy流模型的反源问题。