Bayesian Inverse Problems and Model Uncertainties

科学中传统的、非常自然的范式是建立从原因到结果的预测数学模型。然而，经常发生的情况是，对结果的观察是可用的，人们需要确定使观察成为可能的原因。后一类问题被称为逆问题。反问题的特点是对测量和所用模型的误差高度敏感，存在不止一种而是几种可能的解，以及它们的计算复杂性。这个项目专注于反问题的一个特殊但核心的方面：假设存在一个非常详细和复杂的预测模型，已知能够产生与观测结果很好匹配的预测。此外，假设该模型在计算上要求很高，并且它包含许多值未知或鲜为人知的参数。为了在规定的时间框架内解决逆问题，可能需要使用简化或减少的模型。鉴于逆问题对模型中的误差很敏感，通常会发生模型约简引入不受控制的误差，或者模型与现实之间的差异，这可能会使逆问题的解完全无用。研究者和他的同事们提出了一种通用的方法来处理统计框架中的建模误差问题，在这个项目中，目的是进一步发展这种方法，以便它能够以一种可靠的方式在有限的计算资源下找到有用的解决方案，并量化这种解决方案的可靠性。该项目的主要应用是在医学领域，包括绘制大脑活动图，使用便携式设备识别和定位中风，以及开发快速便携式计算工具来模拟血流，但其结果也有医疗应用之外的应用。反问题建模误差处理的技术难点在于它取决于反问题所要寻找的未知原因。然而，贝叶斯统计范式为这个问题提供了一个非常自然的解决方案。在贝叶斯上下文中，主要感兴趣的未知被描述为具有先验概率分布的随机变量，因此，可以估计建模误差的概率分布并将其作为似然模型的一部分。与忽略建模误差的结果相比，这一基本观察结果已被证明可以产生显著改善估计的算法。在本项目中，将进一步发展该方法，仔细跟踪建模误差分布的包含如何影响未知的贝叶斯后验分布，以及反过来，如何在使用数据更新未知的先验密度后更新建模误差分布。这样的跟踪将有望导致一种计算上有效的方法，在存在建模误差的情况下量化反解中的不确定性。该项目解决的一类问题是多尺度逆问题，其中主要感兴趣的未知数是描述系统的精细尺度行为，而观测值则代表宏观的粗尺度量。这些类型的问题经常出现在生物应用中，其中高保真微观模型在本质上通常是随机的，并且不能在标准贝叶斯框架中直接处理。

The traditional and very natural paradigm in science is to build predictive mathematical models that move from causes to consequences. However, it often happens that observations of consequences are available, and one needs to identify the causes that made the observations possible. The latter type of problems are known as inverse problems. Inverse problems are characterized by their high sensitivity to errors in the measurements and the models used, the existence of not just one but several possible solutions, and their computational complexity. This project focuses on one particular but central aspect in inverse problems: Assume that a very detailed and complex predictive model exists, known to be able to produce predictions that match well with observations. Furthermore, assume that the model is computationally very demanding, and it contains numerous parameters whose values are unknown or poorly known. To solve the inverse problem in the required time frame, it may be that a simplified, or reduced model, needs to be used. Given that inverse problems are sensitive to errors in the model, it typically happens that the model reduction introduces an uncontrolled error, or discrepancy between the model and reality, that may render the solution of the inverse problem completely useless. The investigator and his colleagues have proposed a general methodology to handle the modeling error problem in the statistical framework, and in this project, the aim is to develop the methodology further so that it allows a reliable way to find a useful solution with limited computational resources, and to quantify the reliability of such solution. The main applications in this project are in the field of medicine, including mapping of the brain activity, identification and localization of stroke using a portable equipment, and development of fast and portable computing tools to model blood flow, but the results also have applications beyond medical applications. The technical difficulty in handling the modeling error in an inverse problem is that it depends on the unknown cause that the inverse problem is seeking. However, the Bayesian statistical paradigm provides a very natural solution to this problem. In the Bayesian context, the unknown of primary interest is described as a random variable that has an a priori probability distribution, and therefore, it is possible to estimate a probability distribution of the modeling error and include it as part of the likelihood model. This basic observation has been shown to lead to algorithms that dramatically improve the estimates compared to results that ignore the modeling error. In this project, the methodology will be developed further, by carefully following how the inclusion of the modeling error distribution affects the Bayesian posterior distribution of the unknown, and conversely, how the modeling error distribution can be updated after the data is used to update the prior density of the unknown. Such tracking will hopefully lead to a computationally efficient way of quantifying uncertainties in the inverse solutions in the presence of modeling errors. One family of problems the project addresses is multi-scale inverse problems, in which the unknowns of primary interest are describing fine-scale behavior of the system, while the observation represents a macroscopic, coarse scale quantity. These types of problems often appear in biological applications, where the high-fidelity microscopic models are often stochastic in nature, and cannot be handled directly in the standard Bayesian framework.