Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization
From a Game Theoretic Approach to Numerical Approximation and Algorithm Design

Presents interplays between numerical approximation and statistical inference as a pathway to simple solutions to fundamental problems.

Houman Owhadi (Author), Clint Scovel (Author)

9781108484367, Cambridge University Press

Hardback, published 24 October 2019

488 pages, 83 colour illus.
25.2 x 17.8 x 2.7 cm, 1.07 kg

'This unique book provides a novel game-theoretic approach to Probabilistic Scientific Computing by exploring the interplay between numerical approximation and statistical inference, and exploits such links to develop new fast methods for solving partial differential equations. Gamblets are magic basis functions resulting from a clever adversarial zero sum game between two players and can be used in modeling multiscale problems with no scale separation in numerical homogenization. The book provides original exposition to many topics of the modern era of scientific computing, including sparse representation of Gaussian fields, probabilistic interpretation of numerical errors, linear complexity algorithms, and rigorous settings in the Sobolev and Banach spaces of these topics. It is appropriate for graduate-level courses and as a valuable reference for any scientist who is interested in rigorous understanding and use of modern numerical algorithms in problems where data and mathematical models co-exist.' George Karniadakis, Brown University

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.

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.

Subject Areas: Game theory [PBUD], Probability & statistics [PBT], Numerical analysis [PBKS]