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Proper Orthogonal Decomposition Methods for Partial Differential Equations

Zhendong Luo and Goong Chen
Academic Press
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
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Numerical methods for solving partial differential equations (PDEs) are often extremely demanding of computational resources. The primary methods require the division of the computational domain into discrete meshes. The domains are generally multidimensional and often irregularly shaped. The corresponding meshes have many degrees of freedom related to the number of nodes in the mesh partition. For even routine problems the number of unknowns can be in the hundreds of thousands to millions. Even large fast computers can’t find solutions to such problems in reasonable times.
The theme of the current book is the development of effective methods that generate numerical solutions for time-dependent PDEs using a relatively small amount of data but produce solutions that are accurate and suitable for applications. The advantages of these methods are reduced data storage, computer time and computational complexity, often by orders of magnitude. The key to the authors’ approach is proper orthogonal decomposition (POD). Using singular value decomposition, POD methods generate a set of modes that optimally represent either simulation or model data and significantly reduce the number of modes needed to model the behavior of the PDEs.
A critical piece in the authors’ application of POD techniques is the method of snapshots. Two options exist for identifying the optimal basis modes from a given complex system represented by the PDE. One can either collect data directly from an experiment or simulate the system as it evolves according to its dynamics. In either case snapshots of the dynamics are taken at prescribed time intervals from which optimal modes can be identified. 
The authors develop specific POD algorithms adapted and applied to conventional finite difference, finite element, and finite volume computational schemes as well as the theoretical basis to support them. Each of these is described in a separate chapter. The finite difference POD algorithm is applied to two-dimensional equations with examples that include the stationary Stokes equation and the shallow water equation. After a review of the theory of Sobolev spaces and related elliptic theory, the next chapter develops the finite element POD algorithm and applies the algorithm to the viscoelastic wave equation and versions of Burger’s and Navier-Stokes equations. Finally, the finite volume version of the algorithm is presented and applied to the Sobolev and Boussinesq equations.
This book is intended for readers very comfortable with partial differential equations at the graduate level. It has more the character of a research monograph than a textbook, and there are no exercises. Although POD methods and applications are described in some detail, this is not an introduction to proper orthogonal decomposition such as one can find in Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control by Brunton and Kutz.


Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and material science. He did his PhD work in dynamical systems and celestial mechanics.