This is a concise mathematical introduction to Monte Carlo methods, a rich family of algorithms with far-reaching applications in science and engineering. Monte Carlo methods are an exciting subject for mathematical statisticians and computational and applied mathematicians: the design and analysis of modern algorithms are rooted in a broad mathematical toolbox that includes ergodic theory of Markov chains, Hamiltonian dynamical systems, transport maps, stochastic differential equations, information theory, optimization, Riemannian geometry, and gradient flows, among many others. These lecture notes celebrate the breadth of mathematical ideas that have led to tangible advancements in Monte Carlo methods and their applications. To accommodate a diverse audience, the level of mathematical rigor varies from chapter to chapter, giving only an intuitive treatment to the most technically demanding subjects. The aim is not to be comprehensive or encyclopedic, but rather to illustrate some key principles in the design and analysis of Monte Carlo methods through a carefully-crafted choice of topics that emphasizes timeless over timely ideas. Algorithms are presented in a way that is conducive to conceptual understanding and mathematical analysis -- clarity and intuition are favored over state-of-the-art implementations that are harder to comprehend or rely on ad-hoc heuristics. To help readers navigate the expansive landscape of Monte Carlo methods, each algorithm is accompanied by a summary of its pros and cons, and by a discussion of the type of problems for which they are most useful. The presentation is self-contained, and therefore adequate for self-guided learning or as a teaching resource. Each chapter contains a section with bibliographic remarks that will be useful for those interested in conducting research on Monte Carlo methods and their applications.

翻译：本文是对蒙特卡洛方法的一个简明数学导论，这类丰富的算法族在科学与工程领域具有深远应用。蒙特卡洛方法对于数理统计学家、计算数学与应用数学家而言是一个充满活力的研究主题：现代算法的设计与分析植根于广泛的数学工具箱，包括马尔可夫链的遍历理论、哈密顿动力系统、传输映射、随机微分方程、信息论、优化、黎曼几何以及梯度流等诸多领域。本讲义旨在展现推动蒙特卡洛方法及其应用取得实质性进展的广阔数学思想。为适应不同背景的读者，各章节的数学严谨程度有所差异，对技术难度最高的主题仅作直观性阐述。其目标并非追求全面性或百科全书式的覆盖，而是通过精心选取的专题——强调永恒性思想而非时效性内容——来阐释蒙特卡洛方法设计与分析中的若干核心原理。算法的呈现方式侧重于概念理解与数学分析，优先追求清晰度与直观性，而非难以理解或依赖特定启发式策略的最新实现。为帮助读者在广阔的蒙特卡洛方法领域中建立认知脉络，每种算法均附有其优缺点总结及适用问题类型的讨论。内容阐述自成体系，既适合自主学习，也可作为教学资源。每章均设有文献评注部分，对有志于从事蒙特卡洛方法及其应用研究的读者具有重要参考价值。