SIAM Review, Volume 65, Issue 4, Page 1074-1105, November 2023.
Bayesian methods have been widely used in the last two decades to infer statistical properties of spatially variable coefficients in partial differential equations from measurements of the solutions of these equations. Yet, in many cases the number of variables used to parameterize these coefficients is large, and oobtaining meaningful statistics of their probability distributions is difficult using simple sampling methods such as the basic Metropolis--Hastings algorithm---in particular, if the inverse problem is ill-conditioned or ill-posed. As a consequence, many advanced sampling methods have been described in the literature that converge faster than Metropolis--Hastings, for example, by exploiting hierarchies of statistical models or hierarchies of discretizations of the underlying differential equation. At the same time, it remains difficult for the reader of the literature to quantify the advantages of these algorithms because there is no commonly used benchmark. This paper presents a benchmark Bayesian inverse problem---namely, the determination of a spatially variable coefficient, discretized by 64 values, in a Poisson equation, based on point measurements of the solution---that fills the gap between widely used simple test cases (such as superpositions of Gaussians) and real applications that are difficult to replicate for developers of sampling algorithms. We provide a complete description of the test case and provide an open-source implementation that can serve as the basis for further experiments. We have also computed $2\times 10^{11}$ samples, at a cost of some 30 CPU years, of the posterior probability distribution from which we have generated detailed and accurate statistics against which other sampling algorithms can be tested.

中文翻译：

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偏微分方程系数贝叶斯反演的基准

SIAM Review，第 65 卷，第 4 期，第 1074-1105 页，2023 年 11 月。
贝叶斯方法在过去二十年中被广泛使用，通过测量偏微分方程的解来推断偏微分方程中空间变量系数的统计特性。然而，在许多情况下，用于参数化这些系数的变量数量很大，并且使用简单的采样方法（例如基本的 Metropolis-Hastings 算法）很难获得其概率分布的有意义的统计数据，特别是如果逆问题是病态的或不适定的。因此，文献中描述了许多先进的采样方法，它们比 Metropolis-Hastings 收敛得更快，例如，通过利用统计模型的层次结构或基础微分方程的离散化层次结构。与此同时，由于没有通用的基准，文献读者仍然很难量化这些算法的优势。本文提出了一个基准贝叶斯逆问题，即根据解的点测量确定泊松方程中由 64 个值离散化的空间可变系数，该问题填补了广泛使用的简单检验之间的空白采样算法开发人员难以复制的案例（例如高斯叠加）和实际应用。我们提供了测试用例的完整描述，并提供了可以作为进一步实验基础的开源实现。我们还花费了大约 30 个 CPU 年的时间计算了 $2\times 10^{11}$ 个样本的后验概率分布，从中我们生成了详细而准确的统计数据，可以根据这些统计数据来测试其他采样算法。