Today, mathematical methods, models, and computational algorithms are playing increasingly significant roles in addressing major challenges arising from scientific research and technological development. Although many novel methods and algorithms, such as deep learning and artificial intelligence, are emerging and reshaping various areas at an unprecedented pace, their core ideas and working mechanisms are inherently related to and deeply rooted in some essential mathematical foundations and principles. By performing an in-depth survey on the underlying foundations, principles, and algorithms, this book aims to navigate the vast landscape of mathematical methods widely used in diverse domains.  

This book starts with a survey of mathematical foundations, including essential concepts and theorems in real analysis, linear algebra, etc. Then it covers a broad spectrum of applied mathematical methods, ranging from traditional ones such as optimizations and dynamics modeling, to state-of-the-art such as machine learning, deep learning, and reinforcement learning. The emphasis is placed on methods regarding statistical modeling, stochastic and dynamical system modeling, optimal decision making, and statistical learning. For each part, this book organizes fundamental definitions, theorems, methods, and algorithms in a logical, self-explanatory way.