Sparse Polynomial Approximation of High-Dimensional Functions
Sparse Polynomial Approximation of High-Dimensional Functions

Author: Ben Adcock

Publisher: SIAM

Published: 2022-02-16

Total Pages: 310

ISBN-13: 161197688X

DOWNLOAD EBOOK

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.

High-Dimensional Optimization and Probability
High-Dimensional Optimization and Probability

Author: Ashkan Nikeghbali

Publisher: Springer Nature

Published: 2022-08-04

Total Pages: 417

ISBN-13: 3031008324

DOWNLOAD EBOOK

This volume presents extensive research devoted to a broad spectrum of mathematics with emphasis on interdisciplinary aspects of Optimization and Probability. Chapters also emphasize applications to Data Science, a timely field with a high impact in our modern society. The discussion presents modern, state-of-the-art, research results and advances in areas including non-convex optimization, decentralized distributed convex optimization, topics on surrogate-based reduced dimension global optimization in process systems engineering, the projection of a point onto a convex set, optimal sampling for learning sparse approximations in high dimensions, the split feasibility problem, higher order embeddings, codifferentials and quasidifferentials of the expectation of nonsmooth random integrands, adjoint circuit chains associated with a random walk, analysis of the trade-off between sample size and precision in truncated ordinary least squares, spatial deep learning, efficient location-based tracking for IoT devices using compressive sensing and machine learning techniques, and nonsmooth mathematical programs with vanishing constraints in Banach spaces. The book is a valuable source for graduate students as well as researchers working on Optimization, Probability and their various interconnections with a variety of other areas. Chapter 12 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

Numerical Analysis meets Machine Learning
Numerical Analysis meets Machine Learning

Author:

Publisher: Elsevier

Published: 2024-06-13

Total Pages: 590

ISBN-13: 0443239851

DOWNLOAD EBOOK

Numerical Analysis Meets Machine Learning series, highlights new advances in the field, with this new volume presenting interesting chapters. Each chapter is written by an international board of authors. - Provides the authority and expertise of leading contributors from an international board of authors - Presents the latest release in the Handbook of Numerical Analysis series - Updated release includes the latest information on the Numerical Analysis Meets Machine Learning

Compressed Sensing and its Applications
Compressed Sensing and its Applications

Author: Holger Boche

Publisher: Birkhäuser

Published: 2018-01-17

Total Pages: 402

ISBN-13: 3319698028

DOWNLOAD EBOOK

This contributed volume contains articles written by the plenary and invited speakers from the second international MATHEON Workshop 2015 that focus on applications of compressed sensing. Article authors address their techniques for solving the problems of compressed sensing, as well as connections to related areas like detecting community-like structures in graphs, curbatures on Grassmanians, and randomized tensor train singular value decompositions. Some of the novel applications covered include dimensionality reduction, information theory, random matrices, sparse approximation, and sparse recovery. This book is aimed at both graduate students and researchers in the areas of applied mathematics, computer science, and engineering, as well as other applied scientists exploring the potential applications for the novel methodology of compressed sensing. An introduction to the subject of compressed sensing is also provided for researchers interested in the field who are not as familiar with it.

Data-Driven Methods for Dynamic Systems
Data-Driven Methods for Dynamic Systems

Author: Jason Bramburger

Publisher: SIAM

Published: 2024-11-05

Total Pages: 180

ISBN-13: 1611978165

DOWNLOAD EBOOK

As experimental data sets have grown and computational power has increased, new tools have been developed that have the power to model new systems and fundamentally alter how current systems are analyzed. This book brings together modern computational tools to provide an accurate understanding of dynamic data. The techniques build on pencil-and-paper mathematical techniques that go back decades and sometimes even centuries. The result is an introduction to state-of-the-art methods that complement, rather than replace, traditional analysis of time-dependent systems. Data-Driven Methods for Dynamic Systems provides readers with methods not found in other texts as well as novel ones developed just for this book; an example-driven presentation that provides background material and descriptions of methods without getting bogged down in technicalities; and examples that demonstrate the applicability of a method and introduce the features and drawbacks of their application. The online supplementary material includes a code repository that can be used to reproduce every example and that can be repurposed to fit a variety of applications not found in the book. This book is intended as an introduction to the field of data-driven methods for graduate students. It will also be of interest to researchers who want to familiarize themselves with the discipline. It can be used in courses on dynamical systems, differential equations, and data science.

Compressive Imaging: Structure, Sampling, Learning
Compressive Imaging: Structure, Sampling, Learning

Author: Ben Adcock

Publisher: Cambridge University Press

Published: 2021-08-31

Total Pages: 300

ISBN-13: 9781108421614

DOWNLOAD EBOOK

Accurate, robust and fast image reconstruction is a critical task in many scientific, industrial and medical applications. Over the last decade, image reconstruction has been revolutionized by the rise of compressive imaging. It has fundamentally changed the way modern image reconstruction is performed. This in-depth treatment of the subject commences with a practical introduction to compressive imaging, supplemented with examples and downloadable code, intended for readers without extensive background in the subject. Next, it introduces core topics in compressive imaging - including compressed sensing, wavelets and optimization - in a concise yet rigorous way, before providing a detailed treatment of the mathematics of compressive imaging. The final part is devoted to recent trends in compressive imaging: deep learning and neural networks. With an eye to the next decade of imaging research, and using both empirical and mathematical insights, it examines the potential benefits and the pitfalls of these latest approaches.

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

Author: Spencer J. Sherwin

Publisher: Springer Nature

Published: 2020-08-11

Total Pages: 658

ISBN-13: 3030396479

DOWNLOAD EBOOK

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions.

Nonlocal Integral Equation Continuum Models
Nonlocal Integral Equation Continuum Models

Author: Marta D'Elia

Publisher: SIAM

Published: 2024-09-12

Total Pages: 187

ISBN-13: 161197805X

DOWNLOAD EBOOK

The book presents the state of the art of nonlocal modeling and discretization and provides a practical introduction to nonlocal modeling for readers who are not familiar with such models. These models have recently become a viable alternative to classical partial differential equations when the latter are unable to capture effects such as discontinuities and multiscale behavior in a system of interest. Because of their integral nature, nonlocal operators allow for the relaxation of regularity requirements on the solution and thus allow for the capture of multiscale effects, the result of which is their successful use in many scientific and engineering applications. The book also provides a thorough analysis and numerical treatment of nonstandard nonlocal models, focusing on both well-known and nonstandard interaction neighborhoods. In addition, the book delivers an extensive practical treatment of the implementation of discretization strategies via finite element methods. Numerous figures are provided as concrete examples to illustrate both the analytic and computational results. Nonlocal Integral Equation Continuum Models: Nonstandard Interaction Neighborhoods and Finite Element Discretizations is intended for mathematical and application researchers interested in alternatives to using partial differential equation models that better describe the phenomena they are interested in. The book will also be of use to computational scientists and engineers who need to make sense of how to use available software, improve existing software, or develop new software tailored to their application interests.