This monograph presents a comprehensive exploration of model order reduction techniques rooted in orthogonal polynomials, addressing diverse dynamical systems such as linear, coupled, bilinear, time-delay, and nonlinear systems. By integrating general and specific polynomials—such as Laguerre and Chebyshev—as well as discrete orthogonal polynomials, the monograph introduces unified frameworks and innovative approaches, including structure-preserving reduction and hybrid methods. Organized thematically, the monograph combines theory with practical algorithms, emphasizing efficiency and accuracy. It features dedicated chapters on continuous-time and discrete-time systems, step-by-step derivations, and computational case studies. Readers will gain cutting-edge tools to simplify complex systems, accelerate simulations, and enhance design processes in engineering and applied mathematics.

Yao-Lin Jiang received the B.S. degree from Sichuan University, Chengdu, China, in 1985, and the M.S. and Ph.D. degrees from Xi'an Jiaotong University, Shaanxi, China, in 1988 and 1992, respectively, all in mathematics. From 1992 to 1994, he was a Postdoctoral Fellow with the Institute of Engineering Mechanics, Xi'an Jiaotong University, and from 1996 to 1997, a Postdoctoral Fellow with the Faculty of Engineering, the Chinese University of Hong Kong, Hong Kong. From 1999 to 2000, he was a Research Fellow with the Department of Electronic Engineering at the City University of Hong Kong, Hong Kong, and from 2004 to 2005, he was a Senior Research Fellow with the Research Council, Department of Computer Science at the Katholieke Universiteit Leuven (KU Leuven), Belgium. Since 1998, he has been a Full Professor with the School of Mathematical and Statistics, Xi'an Jiaotong University, and since 2011, he has been the Chang Jiang Professor of China. His research interests include numerical solutions of partial differential equations (i.e., finite element methods), model order reduction, waveform relaxation, space-time parallel (time methods), geometric numerical integration methods, numerical algorithms by neural networks, integrated circuit simulation, building environment simulation, fast simulation of power systems, control theory and applications, Riemannian optimization, Lie group methods, dynamics of nonlinear systems, etc.