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

Description

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. Cover Front Matter Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization: From a Game Theoretic Approach to Numerical Approximation and Algorithm Design Copyright Dedication Contents Preface Acknowledgments Reading Guide 1 Introduction Part I: The Sobolev Space Setting 2 Sobolev Space Basics 3 Optimal Recovery Splines 4 Numerical Homogenization 5 Operator-Adapted Wavelets 6 Fast Solvers Part II: The Game Theoretic Approach 7 Gaussian Fields 8 Optimal Recovery Games on Hs0 () 9 Gamblets 10 Hierarchical Games Part III: The Banach Space Setting 11 Banach Space Basics 12 Optimal Recovery Splines 13 Gamblets 14 Bounded Condition Numbers 15 Exponential Decay 16 Exponential Decay 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 21 Positive Definite Matrices 22 Nonsymmetric Operators 23 Time-Dependent Operators 24 Dense Kernel Matrices Part VI: Appendix 25 Fundamental Concepts Bibliography Algorithms Glossary Nomenclature Index Identities