Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization From a Game Theoretic Approach to Numerical Approximation and Algorithm Design
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Reading Guide
- 1 Introduction
- Part I The Sobolev Space Setting
- Part II The Game Theoretic Approach
- Part III The Banach Space Setting
- Part IV Game Theoretic Approach on Banach Spaces
- 17 Gaussian Measures, Cylinder Measures, and Fields on B
- 18 Optimal Recovery Games on B
- 19 Game Theoretic Interpretation of Gamblets
- 20 Survey of Statistical Numerical Approximation
- Part V Applications, Developments, and Open Problems
- Part VI Appendix
- Bibliography
- Algorithms
- Glossary
- Nomenclature
- Index
- Identities
18 - Optimal Recovery Games on B
from Part IV - Game Theoretic Approach on Banach Spaces
Published online by Cambridge University Press: 10 October 2019
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Reading Guide
- 1 Introduction
- Part I The Sobolev Space Setting
- Part II The Game Theoretic Approach
- Part III The Banach Space Setting
- Part IV Game Theoretic Approach on Banach Spaces
- 17 Gaussian Measures, Cylinder Measures, and Fields on B
- 18 Optimal Recovery Games on B
- 19 Game Theoretic Interpretation of Gamblets
- 20 Survey of Statistical Numerical Approximation
- Part V Applications, Developments, and Open Problems
- Part VI Appendix
- Bibliography
- Algorithms
- Glossary
- Nomenclature
- Index
- Identities
Summary
This chapter introduces optimal recovery games on Banach spaces, presents their natural lift to mixed strategies, and then characterizes their saddle points interms of Gaussian measures, cylinder measures, and fields. The canonical Gaussian field is shown to be a universal field in the sense that its conditioningwith respect to linear measurements producesoptimal strategies. When those measurements form a nested hierarchy, hierarchies of optimal approximations form a martingale obtained by conditioningthe Gaussian field on the filtration formed by those measurements.
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- Operator-Adapted Wavelets, Fast Solvers, and Numerical HomogenizationFrom a Game Theoretic Approach to Numerical Approximation and Algorithm Design, pp. 360 - 369Publisher: Cambridge University PressPrint publication year: 2019