Over seventy years ago, Richard Bellman coined the term “the curse of dimensionality” to describe phenomena and computational challenges that arise in high dimensions. These challenges, in tandem with the ubiquity of high-dimensional functions in real-world applications, have led to a lengthy, focused research effort on high-dimensional approximation—that is, the development of methods for approximating functions of many variables accurately and efficiently from data.

This book provides an in-depth treatment of one of the latest installments in this long and ongoing story: sparse polynomial approximation methods. These methods have emerged as useful tools for various high-dimensional approximation tasks arising in a range of applications in computational science and engineering. It begins with a comprehensive overview of best s-term polynomial approximation theory for holomorphic, high-dimensional functions, as well as a detailed survey of applications to parametric differential equations. It then describes methods for computing sparse polynomial approximations, focusing on least squares and compressed sensing techniques.

MATLAB code

The book contains many numerical examples, each accompanied by downloadable code. This is maintained by the authors in the GitHub repository

https://github.com/simone-brugiapaglia/sparse-hd-book

Annotated bibliography

The book includes an extensive bibliography of over 350 relevant references, with an additional annotated bibliography. You can find the annotated bibliography on Overleaf:

https://www.overleaf.com/read/qhjgtyjtwcwg