Ben Adcock, Simone Brugiapaglia, Clayton G. Webster
SIAM 2022, 292 PAGES
PRICE (PAPERBACK) £72.00 ISBN 978-1-61197-687-8
There’s a saying that: ‘to a person with a hammer, every problem looks like a nail’. Of course, there are some cases where a widely applicable tool is of great value, whether the tool be a hammer or a mathematical technique. Conversely, at least in my experience, this saying most often comes with negative connotations, because using the wrong tool rarely leads to good outcomes. Whilst their impressive performance cannot be denied, from a personal perspective, I’m just a little bit wary the large models that are at the forefront of artificial intelligence may be hammer-like, encouraging us to see too many problems merely as nails. Against that background, the publication of this text is very welcome.
In the context of this book, we are interested in approximating functions with hundreds or thousands of dimensions, or even an infinite number of dimensions. However, high-dimensionality is just one of the four challenges that characterise the situations that are addressed. The others are: generating data is expensive; data is corrupted by errors; and the function may be function-space valued.
The intended audience includes graduate students, postdoctoral fellows and researchers, across mathematics, computer science and engineering. This breadth of application areas means the authors could have focused on applications, spending the majority of the text on practical implementation issues. Whilst these are not ignored, there is a deliberate intent to be mathematically rigorous, with either full proofs or detailed sketches being provided. This approach means this is a text that rewards detailed study.
The book begins with an excellent introductory chapter. This provides a solid grounding for the rest of the text, for example, covering unstructured grids, anisotropic functions and different types of approximating polynomial (including Chebyshev and Legendre). The chapter also introduces four main criteria for an approximation method, these being: accuracy; sample efficiency; stability; and computational efficiency.
After the introduction comes the first of the two main parts of the book. This first part concerns ‘fundamental polynomial approximation theory for smooth functions’. It includes key notation and terminology, concentrating on pure function approximation and applications to parametric differential equations. The descriptions in this part of the book are detailed and instructive, with theory being supported by numerical experimentations, code from which is linked from a companion website (and hosted in GitHub). Whilst being pedagogical in nature, the text remains accessible and capable of capturing a reader’s interest.
The second main part of the book concerns ‘techniques for computing polynomial approximations from sample values’. Understandably, these begin with least-squares approximation. They also cover approaches associated with compressed sensing. This part of the text continues the excellent balance between theory and numerical examples. The remainder of the book comprises a brief chapter on additional topics, including ways of enhancing performance and open problems, and a couple of appendixes.
Throughout, there is evidence of the detailed care that has gone into preparation of this manuscript. For example, the introductory matter includes sections on notation and abbreviations, which aid accessibility for the rest of the volume. There is also an extensive bibliography of over 360 references, an annotated version of which is available from the companion website (and hosted on Overleaf).
Despite the book’s obvious strengths, I was initially tempted to ask for a little bit more. The authors’ deliberate focus on methods that admit a full theoretical analysis, coupled with an intentional bias towards recent results, means anyone wanting a grounding of the entire field of polynomial approximation will need supplementary material. However, on reflection, I was being a bit greedy: I was looking for a metaphorical hammer, forgetting that not all problems are nails! Polynomial approximation is such a broad topic that either it must be treated superficially, or multiple sources must be used. In the latter context, this text covers an important part of this important domain, and it does so very well indeed.
Rob Ashmore CMath CSci FIMA, Dstl
The views and opinions expressed herein are those of the author and do not necessarily reflect those of the Defence Science and Technology Laboratory.
Book review published directly onto IMA website
