Carleman Estimates in Mean Field Games: Stability and Uniqueness for Nonlinear PDEs and Inverse Problems (Inverse and Ill-Posed Problems Series, 64)

This book provides a comprehensive exploration of Mean Field Games (MFG) theory, a mathematical framework for modeling the collective behavior of rational agents in complex systems. MFG theory can govern a range of societal phenomena, including finance, sociology, machine learning, and economics. The focus is on the system of two coupled nonlinear parabolic partial differential equations (PDEs) that define the Mean Field Games System. The book covers key theoretical topics such as solution stability and uniqueness, with a particular emphasis on Carleman estimates, which are used to estimate solution errors based on noise in the input data. It also introduces the theory of Ill-Posed and Inverse Problems within MFG theory. Both theoretical and numerical aspects of forward and inverse problems are explored through Carleman estimates, offering a rigorous foundation for researchers and practitioners in applied mathematics and related fields.

This book offers a rigorous approach to Carleman estimates, a key element of Mean Field Games theory, making it an essential resource for researchers, graduate students, and professionals looking to apply this powerful framework to complex, real-world systems.

Michael Victor Klibanov got his Masters degree in Mathematics in 1972 from Novosibirsk State University, Novosibirsk, Russian Federation. In 1977 he got PhD degree in Mathematics from Ural State University, Ekaterinburg, Russia. In 1986 He got his Doctor of Science degree in Mathematics from Computing Center of Russian Academy of Science, Novosibirsk, Russia.

In 1977-1989 Klibanov has held Associate Professor in Mathematics position in Samara State University, Samara, Russia. In 1990 he got Associate Professor position in the Department of Mathematics and Statistics at the University of North Carolina at Charlotte, Charlotte, USA.

Since 1994 Klibanov is now a tenured professor in the Department of Mathematics and Statistics at the University of North Carolina, where his research interests include Inverse Problems for Partial Differential Equations, Ill-Posed Problems and Mean Field Games. Klibanov is one of the very few World’s top experts in the field of Inverse and Ill-Posed Problems.

Klibanov has been recognized by the Stanford University Study as among the top 2% of the world’s most cited researchers. He was also awarded the Golden Medal for his "Distinguished Impact in Mathematics" from the Sobolev Institute of Mathematics ( Russia) in 2017. He has authored more than 175 papers and 3 books in the field of Inverse and Ill-Posed. Papers are published in the leading mathematical journals. He has made original and groundbreaking contributions in the area of coefficient inverse problems by introducing Carleman estimates for proofs of uniqueness theorems and constructions of globally convergent numerical methods for these problems. Klibanov’s idea of using Carleman estimates for inverse problems is currently World recognized as the one of the most foundational ideas in this field.

Jingzhi Li is a Professor in the Department of Mathematics, Southern University of Science and Technology. He has been engaged for many years in the research of numerical solution of partial differential equations related to inverse problems, and has achieved a series of research results in theoretical studies and numerical simulations in computational mathematics. Currently, his main research areas involve inverse problem theory and computational methods, shape optimization and the unified theory of differential forms, scientific computing, and finite element methods. In particular, he has developed the Stein extension theorem in differential forms and has made important contributions to the theory and algorithms of inverse imaging in mathematical physics inverse problems. In recent years, he has achieved some original results in the fundamental research on the theory and computation of mathematical physics and inverse problems, including several uniqueness theories in inverse problems, and Carleman estimation and convexification algorithms for inverse problems. He has published more than 80 articles in first rate mathematical journals including CMP, IP, SIAM series, JFA, JDE, Numer Math, JCP, etc., all of which are SCI indexed. He has presided over three National Natural Science Foundation of China (NSFC) projects, one Outstanding Youth Project of Shenzhen City, and participated in one key project of the National Natural Science Foundation of China.