Month: October 2015

Companion Authors Speaking About Their Work

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Paul Constantine

The authors of articles in The Princeton Companion to Applied Mathematics are very active in giving talks about their work and about the subject in general.

I have collected a set of links to videos (or, in some cases, audio captures with slides) of authors speaking on or around the topics of their Companion articles. These should give readers added insight into the topics and their authors.

At the time of posting all links were valid, but links have a habit of changing or disappearing. Please let me know of any new links that can be added to this list or existing ones that need changing.

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Gil Strang

Updates

June 9, 2016: Added new Villani TED talk.

Famous Mathematicians and The Princeton Companion

The Princeton Companion to Applied Mathematics has a 23-page Part I article “History of Applied Mathematics”, but apart from that it does not contain any articles with a historical or biographical emphasis. In designing the book we felt that the articles in Part II, “Equations, Laws and Functions of Applied Mathematics”, would provide a link into the history of applied mathematics through the various equations, laws, and functions included, most of which are eponymous.

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John von Neumann

The index was produced by a professional indexer, who made a judgement on which of the many names in the book had significant enough mentions to index. A phrase “Newton’s method” would not generate an index entry for “Newton”, but a phrase describing something that Newton did might.

The index revealed some interesting features. First, there are many entries for famous mathematicians and scientists: 76 in total, ranging from to Niels Henrik Abel to Thomas Young. This means that even though there are no biographical articles, authors have included plenty of historical and biographical snippets. Second, many of the mathematicians might equally well have been mentioned in a book on pure mathematics (Halmos, Poincaré, Smale, Weil), which indicates the blurred boundary between pure and applied mathematics.

A third feature of the index is that the number of locators for the mathematicians and scientists that it contains varies greatly, from 1 to 20. We can use this to produce a highly non-scientific ranking. Here is a Wordle, in which the font size is proportional to the number of times that each name occurs.

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The table of occurrences, which begins

von Neumann, John 20
Poincaré, Henri 12
Bernoulli family 9
Courant, Richard 9
Prandtl, Ludwig 9
Gauss, Carl Friedrich 8
Kac, Mark 8
Maxwell, James Clerk 8
Merton, Robert 8
Runge, Carl 8
Shannon, Claude 8

can be found in this PDF file.

John von Neumann (1903–-1957) emerges as The Companion’s “most mentioned” applied mathematician. Indeed von Neumann was a hugely influential mathematician who contributed to many fields, as his index entry shows:

von Neumann, John: applied mathematics and, 56–59, 73; computational science and, 336–37, 350; economics and, 71, 644, 650, 869; error analysis and, 77; foams and, 740; Monte Carlo method and, 57; random number generation and, 762; shock waves and, 720; spectral theory and, 239–40, 426

von Neumann’s work has strong connections with my own research interests. With Herman Goldstine he published an important rounding error analysis of Gaussian elimination for inverting a symmetric positive definite matrix. He also introduced the alternating projections method that I have used to solve the nearest correlation matrix problem. And he derived important result on unitarily invariant matrix norms and singular value inequalities

More about von Neumann can be found in the biographies

How to Use The Princeton Companion to Applied Mathematics

The Princeton Companion to Applied Mathematics, discussed in these previous posts, has a wide target audience, which includes mathematicians at undergraduate level or above; students, researchers, and professionals in other subjects who use mathematics; and mathematically interested lay readers.

Here are some examples of how different people can use the book.

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  • Undergraduate students can use it to get an overview of topics they are studying and to find out what areas of applied mathematics they might like to pursue at graduate level. Many of the articles have minimal pre-requisites (indeed some contain few, if any, equations). My article Color Spaces and Digital Imaging, for example, requires just knowledge of integration and basic linear algebra.
  • A teacher might find useful the articles The History of Applied Mathematics and the four-part article Teaching Applied Mathematics, as well as the various short articles on interesting problems and applications (e.g., Cloaking, Bubbles, The Flight of a Golf Ball, Robotics, Medical Imaging, Text Mining, and Voting Systems).
  • Researchers can use the book to find out about topics outside their area that they encounter in seminars but never have the time to study in the research literature.
  • Engineers can use the book to find out about some of the latest mathematical developments relevant to their interests. The articles Aircraft Noise, Inerters, and Signal Processing, and the index entries “aerodynamics”, “energy-efficient buildings”, and “finite-element methods”, are good starting points.
  • Students at all levels can learn about how to read and write mathematics, including the use of relevant computer tools, from several articles in Part VII, “Final Perspectives”.
  • Anyone can use the book for reference. Although it is not a dictionary, encyclopedia, or handbook, The Companion‘s extensive index makes it easy to locate material, including definitions, equations, functions, laws, theorems, and so on.

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  • The book, produced with \LaTeX, is a great example of how to typeset mathematics, with examples of all kinds of equations, figures, and tables. For those learning \LaTeX or new to mathematical typesetting it should be a source of ideas and inspiration. The \LaTeX source code is not provided, but feel free to contact me with questions about how things were done and I will write a post that answers the most common questions.
  • The final collection of articles, by mathematicians from China, France, the UK, and the USA, gives advice on how to make the case for mathematics to politicians, and will be of interest to anyone who wishes to promote the importance of mathematics.

The Rise of Mixed Precision Arithmetic

For the last 30 years, most floating point calculations in scientific computing have been carried out in 64-bit IEEE double precision arithmetic, which provides the elementary operations of addition, subtraction, multiplication, and division at a relative accuracy of about 10^{-16}. We are now seeing growing use of mixed precision, in which different floating point precisions are combined in order to deliver a result of the required accuracy at minimal cost.

Single precision arithmetic (32 bits) is an attractive alternative to double precision because it halves the costs of storing and transferring data, and on Intel chips the SSE extensions allow single precision arithmetic to run twice as fast as double.

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” /> The Mandelbrot set computed in double and quadruple precision. Image taken from https://www.thasler.com/blog/blog/glsl-part5.

Quadruple precision arithmetic, which was included in the 2008 revision of the IEEE standard, is supported by some compilers, and it can be implemented in terms of double precision arithmetic via double-double arithmetic. Arbitrary precision floating point arithmetic is available through, for example, the GNU MPFR library, the mpmath library for Python, the core data type BigFloat in the new language Julia, VPA arithmetic in the MATLAB Symbolic Math Toolbox, or the Advanpix Multiprecision Computing Toolbox for MATLAB.

Half precision arithmetic, in which a number occupies 16 bits, is supported by the IEEE standard for storage but not for computation. It has been argued that for deep learning half precision, with its relative accuracy of about 10^{-4}, is good enough for training and running neural networks. Here are some of the ways in which extra precision is currently being used.

  • Iterative refinement, in the traditional form that first became popular in the 1960s, improves the quality of an approximate solution to a linear system via updates obtained from residuals computed in extra precision.
  • When an algorithm suffers instability it may be possible to overcome it by using extra precision in just a few, key places. This has been done recently in eigensolvers and for matrix orthonormalization.
  • Any iterative algorithm that accepts an arbitrary starting point can be run once at a given precision and the solution used to “warm start” a second run of the same algorithm at higher precision. This idea has been used recently in linear programming.
  • Numerical integration of differential equations over long time periods may need higher precision in order to allow the phenomena of interest to be observed. A recent example is in the study of Kerr (rotating) black holes, where the underlying hyperbolic partial differential equation is solved using quadruple precision arithmetic running on GPUs.
  • When one is developing error bounds or testing algorithms, one needs in principle the exact solution. In practice, a solution computed at high precision and rounded to the working precision is usually adequate, and this is an approach I frequently use in my work in numerical linear algebra.

As the relative costs and ease of computing at different precisions evolve, due to changing architectures and software, as well as the disruptive influence of accelerators such as GPUs, we will see an increasing development and use of mixed precision algorithms. In some ways this is analogous to the increasing interoperability of programming languages (illustrated by C++, Julia, and Python, for example): one uses the main tool (precision) one would like to work with and brings in other tools (precisions) as necessary in order to complete the task.

Update: linear programming link updated December 18, 2018.

What is Applied Mathematics For?

Those of us working in applied mathematics are well aware that our field has many important uses in the real world. But if we are put on the spot during a conversation and asked to give some examples it can be difficult to conjure up a convincing list.

One response is to point people to The Princeton Companion to Applied Mathematics. Its 186 articles contains a large number of examples of how applied mathematics is put to work in fields such as sport, engineering, economics, physics, biology, computer science, and finance.

Another way to convince people of the value of applied mathematics is to get them to watch the 1-minute SIAM video below. It was constructed from interviews conducted at a variety of SIAM conferences and comprises snippets of 25 mathematicians saying what they use mathematics for.

Well done to Karthika Swamy Cohen and Michelle Montgomery at SIAM, Adam Bauser and his team at Bauser Media Group, and Sonja Stark at PilotGirl Productions, for producing this great advertisement for applied mathematics!