Sparse Polynomial Approximation of High-Dimensional Functions

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. Sparse Polynomial Approximation of High-Dimensional Functions presents the first comprehensive and unified treatment of polynomial approximation techniques that can mitigate the curse of dimensionality in high-dimensional approximation, including least squares and compressed sensing. It develops main concepts in a mathematically rigorous manner, with full proofs given wherever possible, and it contains many numerical examples, each accompanied by downloadable code. The authors provide an extensive bibliography of over 350 relevant references, with an additional annotated bibliography available on the book’s companion website (www.sparse-hd-book.com). This text is aimed at graduate students, postdoctoral fellows, and researchers in mathematics, computer science, and engineering who are interested in high-dimensional polynomial approximation techniques.

Ben Adcock is Professor of Mathematics at Simon Fraser University. He has published one book, 15 conference proceedings, two book chapters, and more than 50 peer-reviewed journal articles. His work has also been featured on the front page of SIAM News. His research interests include numerical analysis, mathematics of data science, approximation theory, and computational harmonic analysis. Simone Brugiapaglia is Assistant Professor of Mathematics and Statistics at Concordia University. He has published five conference proceedings, two book chapters, and 11 peer-reviewed journal articles. His research interests include mathematics of data science, computational mathematics, and numerical analysis. Clayton G. Webster is Distinguished Scientist in the Oden Institute for Computational Engineering and Sciences at the University of Texas and Distinguished Research Fellow at Lirio AI Research. He serves as Editor-in-Chief of Numerical Methods for PDEs and on several national and international conference organizing committees, as well as numerous editorial boards. His research interests include approximation theory, numerical analysis, approximation of partial differential equations, probability theory, mathematics of data science, and computational harmonic analysis.