This concise text presents an introduction to the emerging area of reducing complex nonlinear differential equations or time-resolved data sets to spectral submanifolds (SSMs). SSMs are ubiquitous low-dimensional attracting invariant manifolds that can be constructed systematically, building on the spectral properties of the linear part of a ......
Most books on algorithms are narrowly focused on a single field of application. This unique book cuts across discipline boundaries, exposing readers to the most successful algorithms from a variety of fields. Algorithm derivation is a legitimate branch of the mathematical sciences driven by hardware advances and the demands of many scientific ......
Data Structures, Algorithms, and Invariants: A Practical Guide provides students with the knowledge and understanding they need to make sensible decisions regarding the choice of an abstract data type (ADT) for particular use cases. The book is a practical guide that helps readers with the process of building a software application utilizing a ......
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 ......
Nonstandard Interaction Domains and Finite Element Discretizations
The book presents the state of the art of nonlocal modeling and discretization and novel analyses of a class of nonstandard nonlocal 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 ......
The method of least squares, discovered by Gauss in 1795, is a principal tool for reducing the influence of errors when fitting a mathematical model to given observations. Applications arise in a great number of areas in sciences and engineering. The increased use of automatic data capturing frequently leads to large-scale least squares problems. ......
The goal of Algorithmic Mathematics in Machine Learning is to explore several well-known machine learning and data analysis algorithms from a mathematical and programming perspective. In this unique book, the authors: Present machine learning methods, review the underlying mathematics, and provide programming exercises intended to deepen the ......
This concise, self-contained volume introduces convex analysis and optimization algorithms, with an emphasis on bridging the two areas. It explores cutting-edge algorithms--such as the proximal gradient, Douglas-Rachford, Peaceman-Rachford, and FISTA--that have applications in machine learning, signal processing, image reconstruction, and other ......
Optimization is presented in most multivariable calculus courses as an application of the gradient, and while this treatment makes sense for a calculus course, there is much more to the theory of optimization. Optimization problems are generated constantly, and the theory of optimization has grown and developed in response to the challenges ......
