Principles of Multiscale Modeling,

Weinan
E
,
Cambridge U. Press
,
New York
, 2011. $75.00 (466 pp.). ISBN 978-1-107-09654-7Buy at Amazon

During the past 30 years, researchers have rapidly developed multiscale modeling. Nowadays it is ubiquitous throughout science and engineering, in subjects as varied as epidemiology and materials science. As a sign that the field has reached maturity, many publishers offer at least one journal with “multiscale” in the title.

Physicists familiar with multiscale modeling naturally associate it with the multigrid algorithm for elliptic partial differential equations developed by Achi Brandt and others starting in the late 1970s. However, in Principles of Multiscale Modeling, Princeton University professor Weinan E takes an unusually broad view that encompasses everything from Fourier analysis to matched asymptotics, from moving contact lines to the deformation of carbon nanotubes, and from fast multipole methods to domain decomposition.

Written by a leader in modern applied mathematics, Principles of Multiscale Modeling is a unified and well-organized synthesis of the physical ideas and mathematical techniques behind the multiscale approach to understanding physical phenomena. Other books on the subject tend to focus on a single aspect. For example, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives (Cambridge University Press, 2011), by Jean-Pierre Fouque and coauthors, focuses on specific applications; Multiscale Methods: Averaging and Homogenization (Springer, 2008), by Grigorios Pavliotis and Andrew Stuart, centers on mathematical techniques; and Multiscale Finite Element Methods: Theory and Applications (Springer, 2009), by Yalchin Efendiev and Thomas Hou, focuses on computational algorithms.

However, Principles of Multiscale Modeling reflects its author’s breadth of experience and interest in physics, mathematical analysis, and scientific computation. It is ambitious in scope and in its insistence on taking seriously all stages in multiscale modeling, from fundamental physical models to efficient computational algorithms by way of rigorous mathematical analysis. I am not aware of any other work that covers all those topics with equal attention and rigor.

Chapter 1 explains that traditional macroscopic models based on constitutive relations derived using entropy considerations, symmetry, or linearization sometimes fail. In that case one needs a hierarchy of models to properly describe the macroscopic behavior: Coarse-graining a microscopic model is insufficient. The next chapter is a crash course on analytical methods; it’s reminiscent of the classic text Advanced Mathematical Methods for Scientists and Engineers (Springer, 1978) by Carl Bender and Steven Orszag, but it is supplemented with more recent material on renormalization-group analysis and homogenization. Chapter 3 is a review of what Weinan E calls classical multiscale algorithms—for example, multigrid methods, fast multipole methods, and adaptive mesh refinement. Chapters 4 and 5 derive and analyze a hierarchy of physical models—quantum mechanical and continuum mechanical—and then use that hierarchy to derive particular “multi-physics” models. The next five chapters are each devoted to numerical algorithms for a particular problem: macroscopic behavior, singularities, elliptic equations, multiple time scales, and rare events.

Principles of Multiscale Modeling is really a guide, not a textbook. It is highly compressed and lacks exercises and weblinks to numerical codes. It also assumes a depth of cultural immersion in numerical analysis and physics beyond that of the average graduate student. Indeed, several chapters include the caveat that what is covered is not “an exhaustive treatment” but “we hope to convey the basic ideas.”

Although the book is comprehensive, it inevitably has some gaps. It does not mention the adaptive wavelet method as an alternative to adaptive mesh refinement. A brief discussion of that and related methods, along with other applications of multiresolution analysis, would have fit naturally in the chapter 3 section on multiresolution analysis and wavelet bases. I would also have liked to see a discussion of reduced-order models for complex macroscopic physical phenomena such as turbulent solutions of the Navier–Stokes equations.

I do have a quibble with this book. It seems to have been produced cheaply using standard LaTeX macros and fonts; even the cover design is undistinguished. This excellent book merits the same attention to quality found in such Cambridge University Press series as the Cambridge Texts in Applied Mathematics and the Cambridge Monographs on Applied and Computational Mathematics. I am also surprised that it is not available as an ebook.

Nevertheless, Principles of Multiscale Modeling is uniquely suitable for advanced graduate students and researchers who want to survey the field’s full range of physical ideas, mathematical techniques, and computational algorithms.