Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization

From a Game Theoretic Approach to Numerical Approximation and Algorithm Design
Erscheinungsjahr 2019
ISBN: 978-1-108-48436-7
Verlag: Cambridge University Press

Buch, Englisch, Band 35, 488 Seiten, Format (B × H): 179 mm x 248 mm, Gewicht: 1078 g

Reihe: Cambridge Monographs on Applied and Computational Mathematics

Although numerical approximation and statistical inference are traditionally covered as entirely separate subjects, they are intimately connected through the common purpose of making estimations with partial information. This book explores these connections from a game and decision theoretic perspective, showing how they constitute a pathway to developing simple and general methods for solving fundamental problems in both areas. It illustrates these interplays by addressing problems related to numerical homogenization, operator adapted wavelets, fast solvers, and Gaussian processes. This perspective reveals much of their essential anatomy and greatly facilitates advances in these areas, thereby appearing to establish a general principle for guiding the process of scientific discovery. This book is designed for graduate students, researchers, and engineers in mathematics, applied mathematics, and computer science, and particularly researchers interested in drawing on and developing this interface between approximation, inference, and learning.

Autoren/Hrsg.

Owhadi, Houman

Scovel, Clint

Inhaltsverzeichnis

1. Introduction; 2. Sobolev space basics; 3. Optimal recovery splines; 4. Numerical homogenization; 5. Operator adapted wavelets; 6. Fast solvers; 7. Gaussian fields; 8. Optimal recovery games on $\mathcal{H}^{s}_{0}(\Omega)$; 9. Gamblets; 10. Hierarchical games; 11. Banach space basics; 12. Optimal recovery splines; 13. Gamblets; 14. Bounded condition numbers; 15. Exponential decay; 16. Fast Gamblet Transform; 17. Gaussian measures, cylinder measures, and fields on $\mathcal{B}$; 18. Recovery games on $\mathcal{B}$; 19. Game theoretic interpretation of Gamblets; 20. Survey of statistical numerical approximation; 21. Positive definite matrices; 22. Non-symmetric operators; 23. Time dependent operators; 24. Dense kernel matrices; 25. Fundamental concepts.

Über Autor(innen)

Owhadi, Houman
Houman Owhadi is Professor of Applied and Computational Mathematics and Control and Dynamical Systems in the Computing and Mathematical Sciences department at the California Institute of Technology. He is one of the main editors of the Handbook of Uncertainty Quantification (2016). His research interests concern the exploration of interplays between numerical approximation, statistical inference and learning from a game theoretic perspective, especially the facilitation/automation possibilities emerging from these interplays.

Scovel, Clint
Clint Scovel is a Research Associate in the Computing and Mathematical Sciences department at the California Institute of Technology, after a twenty-six-year career at Los Alamos National Laboratory, including foundational research in symplectic algorithms and machine learning. He received his Ph.D. in mathematics from the Courant Institute of Mathematics at New York University in 1983. He currently works on uncertainty quantification, Bayesian methods, incorporating computational complexity in Wald's statistical decision theory, operator adapted wavelets and fast solvers.