Numerical Analysis for Applied Science: An interview with author Myron B. Allen on the latest edition

Myron B. Allen, PhD, is Professor of Mathematics in the Department of Mathematics at the University of Wyoming. He served as provost and vice president for academic affairs from 2005-2013. His scholarly work focuses on the numerical analysis of fluid flows in porous media. The author of three books, he received his PhD in Mathematical Engineering from Princeton University. Dr. Allen is also a Consulting Editor for Wiley’s Pure and Applied Mathematics book series.

Eli L. Isaacson, PhD, is Professor Emeritus of Mathematics in the Department of Mathematics at the University of Wyoming. His scholarly work includes contributions to analytic and numerical methods for solving systems of hyperbolic conservation laws, including front-tracking methods.

Having published the succesful Numerical Analysis for Applied Science in 1998, the two mathematicians have returned with an updated 2nd edition of the book exploring new topics and expanding on those from the first. These new topics include preconditioning, kriging methods designed for stochastic data, interpolation in two and three dimensions, steady-state problems, and finite difference methods for variable-coefficient elliptic equations. This new edition also presents expanded coverage on both the finite-element method and multigrid methods.

Published in March of this year, Numerical Analysis for Applied Science, Second Edition is said to provide an excellent foundation for graduate and advanced undergraduate courses in numerical methods and numerical analysis. It is also an accessible introduction to the subject for students pursuing independent study in applied mathematics, engineering, and the physical and life sciences and a valuable reference for professionals in these areas.

To learn more about the upcoming publication, Fran McMahon spoke with Myron Allen about the book and the reasoning behind this latest edition. Read on to hear what he said.

thumbnail image: Numerical Analysis for Applied Science: An interview with aythor Myron B. Allen on the latest edition

1. You have recently published the second edition of Numerical Analysis for Applied Science, which is said to be an updated and expanded version of the previous. What led to this follow up to this area of study?

Since the publication of the first edition, the field of numerical analysis has evolved in many directions. In particular, the increasing dominance of iterative methods in numerical linear algebra has made it important to expand the sections of the book devoted to multigrid methods, preconditioners, the condition numbers of matrices arising in finite-element approximations, and similar areas.

2. Numerical analysis is a branch of mathematics that deals with the development and use of numerical methods for solving problems. For someone who may not be well versed in this subject area, how would you explain this area of mathematics to them?

Many mathematical problems that arise in engineering and applied physics are too complicated to solve exactly. The complications arise from phenomena found in nature: spatial variability, multidimensionality, nonlinearity, and the coupling of different processes such as transport and chemical reactions. Numerical analysis is the study of methods for converting impossible-to-solve exact problems to approximate problems that one can solve using high-speed computers.

3. What were your main objectives during the writing process? And which topic, that you discuss in your book, would you say was the most interesting studying?

In writing the first and second editions, I wanted to make a significant body of material accessible to engineers, geophysicists, and other scientists who may not have graduate-level mathematical backgrounds. At the same time, I wanted to make the book useful to mathematics students and others who want to learn how the methods really work from a theoretical point of view.

I find the chapter on finite-difference methods for partial differential equations particularly interesting, because it weaves together ideas from several earlier chapters.

4. If there is one piece of information or advice that you would want your audience to take away from your book, what would that be?

Numerical analysis provides a tremendously powerful tool for solving practical mathematical problems that are hard or impossible to solve otherwise. At the same time, numerical analysis is a thriving area of rigorous mathematics.

5. Who should read the book and why?

I tried to aim the book at early-career graduate students in Mathematics, Engineering, Geophysics, and Physics. For this audience, the structure of the narrative provides an introduction to the motivation and construction of methods, practical considerations in implementing them computation, and, for those who are interested, the theoretical details of why the methods work. I hope the book is also useful as a reference for practitioners who have already learned some of this material.

6. Why, do you think, this new edition may be of interest now?

With the emergence of data science, numerical analysis is at least as important now as it was when the first edition appeared. The second edition offers new material that makes the book more responsive to computational concerns that become more salient as computers have become more powerful.

7. Alongside your own text, which is a part of the Pure and Applied Mathematics: A Wiley Series of Texts, Monographs, and Tracts, what other books would you recommend to students looking to learn more about numerical analysis?

Claes Johnson’s book, Numerical Solutions of Partial Differential Equations by the Finite Element Method, is still one of the nicest treatments of finite-element methods that is aimed at similar audiences.

8. What is it about the area of numerical analysis that fascinates you?

Although I love the mathematical analysis of numerical methods, what fascinates me most about them is their utility. It’s remarkable how many practical problems one can solve, to high degrees of accuracy, by using numerical approximations.

9. What other work are you currently working on or has recently been published?

I’m working on a book that, I hope, will provide an in-depth introduction to the classical mathematics of porous-medium flows for mathematicians and engineers who model them numerically.