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:我们对反问题和数据同化及其相互关系的主题进行清晰和简洁的介绍。 我们的注释的第一部分涉及反问题; 这是指研究如何从数据中估计未知模型参数; 我们的注释的第二部分涉及数据同化; 这是指一个特定的反问题类别,其中未知参数是动态系统的初始条件(和/或状态),而数据包括部分和噪音的对州观测。 我们的注释的第三部分和最后一部分描述了数据同化方法的使用,通过引入人工算法时间来解决一般反问题。 我们的注释涉及的问题包括: 如何从数据同化中估算出未知模型参数, 包括(感化) 梯度下层、 变形巴、 蒙特卡洛、 重要取样和 Markov 链 Monte Carlo 的反向问题; 3DVAR、 4DVAR、 扩展和多言调 Kalman 过滤器, 数据同化的粒子过滤器。 第一部分和第二部分从Bayesian 配制成的一章开始, 其中, 问题由近义文献提供近距离的离子、 直径的直系、 直径、 直径、 直径、 直径、 直径、 直径、 直径、 直径、 直径、 分、直径、直径、直系、直系、直系、直系、直系、直系、 分、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系、直系