Parameter identification problems in partial differential equations (PDEs) consist in determining one or more unknown functional parameters in a PDE. Here, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity function in an elliptic PDE from noisy observations of the PDE solution, the performance of Bayesian procedures based on Gaussian process priors is investigated. Recent asymptotic theoretical guarantees establishing posterior consistency and convergence rates are reviewed and expanded upon. An implementation of the associated posterior-based inference is provided, and illustrated via a numerical simulation study where two different discretisation strategies are devised. The reproducible code is available at: https://github.com/MattGiord.

翻译：偏微分方程中的参数识别问题旨在确定方程中一个或多个未知函数参数。本文考虑此类问题的贝叶斯非参数方法。以椭圆型偏微分方程中基于含噪解观测值推断扩散系数函数的代表性案例为切入点，系统研究了基于高斯过程先验的贝叶斯方法的性能。回顾并拓展了近期关于后验一致性与收敛速度的渐近理论保证，提供了相应后验推断的实现方案，并通过数值模拟研究展示了两种不同离散化策略的适用性。可复现代码见：https://github.com/MattGiord。