SpringerBriefs on PDEs and Data Science

This book provides a comprehensive and systematic introduction to inverse problems for stochastic partial differential equations (SPDEs), with particular emphasis on stochastic parabolic and hyperbolic equations. It addresses both the unique challenges and new opportunities that arise in the stochastic setting. Key topics include inverse state problems (such as determining unknown initial conditions) and inverse source problems (identifying unknown source terms), with a focus on the mathematical tools essential for their analysis, especially global Carleman estimates tailored to SPDEs. The book explores fundamental issues of uniqueness, stability, and reconstruction under various measurement scenarios, including internal, boundary, and terminal observations. It highlights how stochasticity can fundamentally alter the nature of inverse problems, sometimes enabling solutions where deterministic approaches fail. Reconstruction methods such as Tikhonov regularization are also discussed in detail. This book is intended for graduate students and researchers in applied mathematics, stochastic analysis, and PDEs, as well as practitioners in fields like mathematical finance, physics, and engineering who require rigorous methods for uncertainty quantification. A moderate background in PDEs, functional analysis, and basic stochastic calculus is beneficial.