Gaussian process regression for Bayesian inverse problems

基于偏微分方程的数学模型在科学和工程中无处不在。模型中的参数，例如在边界条件和系数中出现的参数，通常是不完全已知的，必须根据观测数据进行估计。参数的准确重建，以及对重建不确定性的估计，对于可靠的预测和风险评估至关重要。一个例子应用是地下碳储存，在那里，环境的精确组成，如不同岩层的位置和水力导电性，是不完全清楚的，必须通过测量来估计。出于安全考虑，对泄漏颗粒重新进入人类环境的风险进行量化是至关重要的。从数学上讲，从数据中学习未知模型参数的过程，也称为模型校准，可以表述为从有噪声的间接观测中恢复未知参数的逆问题。根据贝叶斯方法，我们得到了基于观测数据的未知参数的后验分布。由于现代应用中涉及的模型的复杂性，迫切需要开发有效的后验探索算法。本文的总体目标是设计、分析和实现基于高斯过程回归的计算方法，以准确、有效地求解偏微分方程中的逆问题。我们将汇集数值分析、近似理论、统计学和优化的思想，开发可在一系列领域应用和适应的复杂算法，包括在健康（医学成像）和环境（地下建模、地下碳储存）方面的巨大应用潜力。

Mathematical models based on partial differential equations appear everywhere in science and engineering. Parameters in the models, appearing for example in boundary conditions and coefficients, are typically not fully known and have to be estimated from observed data. Accurate reconstruction of the parameters, as well as an estimate of the uncertainty in the reconstruction, are crucial for reliable predictions and risk assessments. An example application is underground carbon storage, where the precise make-up of the environment, such as the location and hydraulic conductivity of different layers of rock, is not fully known and has to be estimated from measurements. For safety reasons, it is crucial to quantify the risk of leaked particles re-entering the human environment.Mathematically, the process of learning unknown model parameters from data, also known as model calibration, can be formulated as the inverse problem to recover unknown parameters from noisy, indirect observations. Following the Bayesian approach, we obtain a posterior distribution for the unknown parameters conditioned on the observed data. Because of the complexity of models involved in modern applications, there is a pressing need to develop efficient algorithms for exploring the posterior.The overall aim of this proposal is to design, analyse and implement computational methods based on Gaussian process regression to solve inverse problems in partial differential equations accurately and efficiently. We will bring together ideas from numerical analysis, approximation theory, statistics and optimisation, to develop sophisticated algorithms that can be applied and adapted across a range of sectors, including huge potential for applications in health (medical imaging) and environment (subsurface modelling, underground carbon storage).