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
- 11 Banach Space Basics
- 12 Optimal Recovery Splines
- 13 Gamblets
- 14 Bounded Condition Numbers
- 15 Exponential Decay
- 16 Fast Gamblet Transform
- Part IV Game Theoretic Approach on Banach Spaces
- Part V Applications, Developments, and Open Problems
- Part VI Appendix
- Bibliography
- Algorithms
- Glossary
- Nomenclature
- Index
- Identities
16 - Fast Gamblet Transform
from Part III - The Banach Space Setting
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
- 11 Banach Space Basics
- 12 Optimal Recovery Splines
- 13 Gamblets
- 14 Bounded Condition Numbers
- 15 Exponential Decay
- 16 Fast Gamblet Transform
- Part IV Game Theoretic Approach on Banach Spaces
- Part V Applications, Developments, and Open Problems
- Part VI Appendix
- Bibliography
- Algorithms
- Glossary
- Nomenclature
- Index
- Identities
Summary
The computation ofgamblets is accelerated by localizing their computation in a hierarchical manner (using a hierarchy of distances), and the approximation errors caused by these localization steps are bounded based on three properties: nesting, the well-conditioned nature of the linear systems solved in the Gamblet Transform, and theexponential decay of the gamblets. These efficiently computed, accurate, andlocalized gamblets are shown to producea Fast Gamblet Transform of near-linear complexity. Application to the three primary classes ofmeasurement functions in Sobolev spaces are developed.
Keywords
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- Operator-Adapted Wavelets, Fast Solvers, and Numerical HomogenizationFrom a Game Theoretic Approach to Numerical Approximation and Algorithm Design, pp. 297 - 344Publisher: Cambridge University PressPrint publication year: 2019