We provide a clear and concise introduction to the subjects of inverse problems and data assimilation, and their inter-relations. The first part of our notes covers inverse problems; this refers to the study of how to estimate unknown model parameters from data. The second part of our notes covers data assimilation; this refers to a particular class of inverse problems in which the unknown parameter is the initial condition (and/or state) of a dynamical system, and the data comprises partial and noisy observations of the state. The third and final part of our notes describes the use of data assimilation methods to solve generic inverse problems by introducing an artificial algorithmic time. Our notes cover, among other topics, maximum a posteriori estimation, (stochastic) gradient descent, variational Bayes, Monte Carlo, importance sampling and Markov chain Monte Carlo for inverse problems; and 3DVAR, 4DVAR, extended and ensemble Kalman filters, and particle filters for data assimilation. Each of parts one and two starts with a chapter on the Bayesian formulation, in which the problem solution is given by a posterior distribution on the unknown parameter. Then the following chapter specializes the Bayesian formulation to a linear-Gaussian setting where explicit characterization of the posterior is possible and insightful. The next two chapters explore methods to extract information from the posterior in nonlinear and non-Gaussian settings using optimization and Gaussian approximations. The final two chapters describe sampling methods that can reproduce the full posterior in the large sample limit. Each chapter closes with a bibliography containing citations to alternative pedagogical literature and to relevant research literature. We also include a set of exercises at the end of parts one and two. Our notes are thus useful for both classroom teaching and self-guided study.

翻译：本文对反问题、数据同化及其相互关系进行了清晰简洁的介绍。笔记第一部分涵盖反问题，即研究如何根据数据估计未知模型参数。第二部分涵盖数据同化，指一类特定的反问题，其中未知参数是动力系统的初始条件（和/或状态），数据包括对状态的部分有噪观测。第三部分也是最后一部分，描述如何通过引入人工算法时间，利用数据同化方法求解一般性反问题。我们的笔记涵盖以下内容（包括但不限于）：最大后验估计、（随机）梯度下降、变分贝叶斯、蒙特卡洛、重要性采样及马尔可夫链蒙特卡洛方法的应用于反问题；以及3DVAR、4DVAR、扩展卡尔曼滤波、集合卡尔曼滤波和粒子滤波方法的应用于数据同化。第一和第二部分均以贝叶斯公式化章节开篇，其中问题解由未知参数的后验分布给出。随后章节将贝叶斯公式化特化为线性-高斯设定，在该设定下后验可以显式刻画且具有深刻洞察力。接下来两章探讨在非线性和非高斯设定下，利用优化和高斯近似从后验中提取信息的方法。最后两章描述能在样本量足够大时再现完整后验的采样方法。每章末尾附有参考文献，引用替代性教学文献及相关研究文献。我们在第一和第二部分末还包含一组习题。因此，本笔记适用于课堂教学和自学。