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

3.75/5 (8 hinnangut Goodreads-ist)
(California Institute of Technology), (California Institute of Technology)
  • Formaat: Hardback, 488 pages, kõrgus x laius x paksus: 252x178x27 mm, kaal: 1070 g, Worked examples or Exercises; 83 Line drawings, color
  • Sari: Cambridge Monographs on Applied and Computational Mathematics
  • Ilmumisaeg: 24-Oct-2019
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1108484360
  • ISBN-13: 9781108484367
Teised raamatud teemal:
Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization: From a Game Theoretic Approach to Numerical Approximation and Algorithm DesignSuurem pilt
  • Formaat: Hardback, 488 pages, kõrgus x laius x paksus: 252x178x27 mm, kaal: 1070 g, Worked examples or Exercises; 83 Line drawings, color
  • Sari: Cambridge Monographs on Applied and Computational Mathematics
  • Ilmumisaeg: 24-Oct-2019
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1108484360
  • ISBN-13: 9781108484367
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781108484367.html
Märksõnad:
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.

This book, meant for graduate students and researchers, explores the connections between numerical approximation and statistical inference from a game and decision theoretic perspective, and illustrates these interplays by addressing problems related to numerical homogenization, operator adapted wavelets, and fast solvers.

Arvustused

'This is a terrific book. A hot new topic, first rate mathematics, real applications. It's an important contribution by marvelous scholars.' Persi Diaconis, Stanford University 'This book does a masterful job of bringing together the two seemingly unrelated fields of numerical approximation and statistical inference to produce a general framework for developing solvers that are both provably accurate and scale to extremely large problem sizes. It seamlessly integrates concepts from numerical approximation, statistical inference, information-based complexity, and game theory to reveal a rich mathematical structure that forms a comprehensive foundation for solver development. Of tremendous value to the practitioner is a thorough analysis of solver accuracy and computational requirements. In addition to providing a comprehensive guide to solver development and analysis this book presents a unique perspective that provides numerous valuable insights into the solution of science and engineering problems.' Don Hush, University of New Mexico '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

Muu info

Presents interplays between numerical approximation and statistical inference as a pathway to simple solutions to fundamental problems.
Preface xiii
Acknowledgments xiv
Reading Guide xv
1 Introduction
1(22)
1.1 Statistical Numerical Approximation
1(3)
1.2 The Game Theoretic Perspective
4(3)
1.3 In the Setting of Sobolev Spaces
7(12)
1.4 Uncertainty Quantification and Probabilistic Numerics
19(1)
1.5 Structure of the Book
20(3)
Part I The Sobolev Space Setting
23(80)
2 Sobolev Space Basics
25(9)
2.1 The Sobolev Space
25(2)
2.2 The Operator and Its Corresponding Energy Norm
27(7)
3 Optimal Recovery Splines
34(4)
3.1 Information-Based Complexity
34(1)
3.2 Optimal Recovery
35(1)
3.3 Variational Properties of Optimal Recovery Splines
36(2)
4 Numerical Homogenization
38(25)
4.1 A Short Review of Classical Homogenization
38(9)
4.2 The Numerical Homogenization Problem
47(4)
4.3 Indicator and Dirac Delta Functions as φ
51(3)
4.4 Accuracy
54(1)
4.5 Exponential Decay
54(4)
4.6 Local Polynomials as φiα
58(1)
4.7 A Short Review of the Localization Problem
59(2)
4.8 A Short Review of Optimal Recovery Splines in Numerical Analysis
61(2)
5 Operator-Adapted Wavelets
63(27)
5.1 A Short Review
63(2)
5.2 Overview of the Construction of Operator-Adapted Wavelets
65(1)
5.3 Non-adapted Prewavelets as Φi(k)
66(7)
5.4 Operator-Adapted Prewavelets
73(1)
5.5 Multiresolution Decomposition of Hs0(Ω)
74(2)
5.6 Operator-Adapted Wavelets
76(3)
5.7 Uniformly Bounded Condition Numbers
79(2)
5.8 Multiresolution Decomposition of u ε Hs0 (Ω)
81(3)
5.9 Interpolation Matrix R(k-1,k)
84(2)
5.10 The Discrete Gamblet Decomposition
86(2)
5.11 Local Polynomials as Φ(k)
88(2)
6 Fast Solvers
90(13)
6.1 A Short Review
90(2)
6.2 The Gamblet Transform and Solve
92(2)
6.3 Sparse and Rank-Revealing Representation of the Green's Function
94(1)
6.4 Numerical Illustrations of the Gamblet Transform and Solve
95(4)
6.5 The Fast Gamblet Transform
99(4)
Part II The Game Theoretic Approach
103(46)
7 Gaussian Fields
105(14)
7.1 Gaussian Random Variable
105(1)
7.2 Gaussian Random Vector
106(2)
7.3 Gaussian Space
108(1)
7.4 Conditional Covariance and Precision Matrix
109(3)
7.5 Gaussian Process
112(1)
7.6 Gaussian Measure on a Hilbert Space
113(2)
7.7 Gaussian Field on a Hilbert Space
115(1)
7.8 Canonical Gaussian Field on (Hs0(Ω), || ·e; ||) in Dual Pairing with (H-s(Omega;), || ·e; ||*)
116(2)
7.9 Degenerate Noncentered Gaussian Fields on Hs0(Ω) in Dual Pairing with H-s(Ω)
118(1)
8 Optimal Recovery Games on Hs0(Ω)
119(12)
8.1 A Simple Finite Game
119(3)
8.2 A Simple Optimal Recovery Game on Rn
122(2)
8.3 An Optimal Recovery Game on Hs0(Ω)
124(1)
8.4 Randomized Strategies
124(2)
8.5 Optimal Mixed Strategies
126(5)
9 Gamblets
131(6)
9.1 Elementary Gambles/Bets
131(2)
9.2 Conditional Distribution of the Gaussian Field
133(1)
9.3 Screening Effect
134(3)
10 Hierarchical Games
137(12)
10.1 Introduction
137(2)
10.2 Downscaling Game
139(3)
10.3 The Sequence of Approximations Is a Martingale
142(2)
10.4 Sparse Representation of Gaussian Fields
144(1)
10.5 Probabilistic Interpretation of Numerical Errors
145(1)
10.6 Upscaling with Nested Games
146(3)
Part III The Banach Space Setting
149(196)
11 Banach Space Basics
151(3)
12 Optimal Recovery Splines
154(6)
12.1 Projection Properties
154(2)
12.2 Optimal Recovery
156(2)
12.3 Variational Properties
158(1)
12.4 Duality
158(2)
13 Gamblets
160(35)
13.1 Prewavelets
160(2)
13.2 Multiresolution Decomposition of B
162(1)
13.3 Operator-Adapted Wavelets
163(2)
13.4 Dual Wavelets
165(3)
13.5 Multiresolution Decomposition of u ε B
168(2)
13.6 Interpolation Matrices
170(2)
13.7 The Gamblet Transform and Gamblet Decomposition
172(2)
13.8 Multiresolution Representation of Q
174(1)
13.9 The Schur Complement O(k)/O(k-1)and B(k)
174(6)
13.10 Geometry of Gamblets
180(13)
13.11 Table of Gamblet Identities
193(2)
14 Bounded Condition Numbers
195(57)
14.1 Notation and Structure Constants
195(1)
14.2 Bounds on A(k)
196(1)
14.3 Bounds on B(k)
196(2)
14.4 Bounds on N(k), T N(k)
198(4)
14.5 Alternate Bounding Mechanism for B(k)
202(2)
14.6 Stability Conditions
204(2)
14.7 Minimum Angle between Gamblets
206(2)
14.8 Sobolev Spaces
208(42)
14.9 Useful Properties of the Structure Constants
250(2)
15 Exponential Decay
252(45)
15.1 Introduction
252(1)
15.2 Subspace Decomposition
253(11)
15.3 Frame Inequalities in Dual Norms
264(5)
15.4 Sobolev Spaces
269(28)
16 Fast Gamblet Transform
297(48)
16.1 Hierarchy of Distances
297(5)
16.2 Hierarchy of Localized Gamblets
302(3)
16.3 The Fast Gamblet Transform and Gamblet Decomposition
305(5)
16.4 Accuracy vs. Complexity Estimates
310(31)
16.5 Sobolev Spaces
341(4)
Part IV Game Theoretic Approach on Banach Spaces
345(42)
17 Gaussian Measures, Cylinder Measures, and Fields on B
347(13)
17.1 Gaussian Measure
347(2)
17.2 Gaussian Field
349(1)
17.3 Gaussian Field and Duality Pairing
350(1)
17.4 Weak Distributions and Cylinder Measures
351(2)
17.5 Gaussian Cylinder Measures as Weak Limits of Gaussian Measures
353(1)
17.6 Canonical Gaussian Field
353(1)
17.7 Canonical Construction
354(1)
17.8 Conditional Expectation and Covariance
355(3)
17.9 When B = Rn
358(2)
18 Optimal Recovery Games on B
360(10)
18.1 Optimal Recovery Game
360(3)
18.2 Optimal Strategies
363(7)
19 Game Theoretic Interpretation of Gamblets
370(8)
19.1 With Two Scales
370(1)
19.2 With Multiple Scales
371(2)
19.3 Conditional Covariances
373(2)
19.4 Sparse Representation of Gaussian Processes
375(1)
19.5 Table of Gaussian Process Regression Identities
376(2)
20 Survey of Statistical Numerical Approximation
378(9)
Part V Applications, Developments, and Open Problems
387(40)
21 Positive Definite Matrices
389(17)
21.1 The Setting
389(1)
21.2 The Hierarchy of Labels and Measurement Matrices
389(1)
21.3 The Gamblet Transform and Gamblet Decomposition
390(3)
21.4 Multiresolution Decomposition of A-1
393(2)
21.5 Bounded Condition Numbers
395(6)
21.6 Exponential Decay
401(3)
21.7 The Fast Gamblet Transform on RN
404(1)
21.8 On Universality
405(1)
22 Nonsymmetric Operators
406(4)
22.1 Example: Nondivergence Form Operators
407(1)
22.2 Example: Symmetrization with the Inverse Laplacian
408(2)
23 Time-Dependent Operators
410(11)
23.1 Sealar-Wave PDEs
410(9)
23.2 Parabolic PDEs
419(2)
24 Dense Kernel Matrices
421(6)
24.1 The Problem
421(1)
24.2 The Algorithm
422(2)
24.3 Why Does It Work?
424(3)
Part VI Appendix
427(17)
25 Fundamental Concepts
429(15)
25.1 Spaces and Mappings
429(2)
25.2 Banach and Hilbert Spaces
431(5)
25.3 The Euclidean Space RN
436(2)
25.4 Measure and Integration
438(2)
25.5 Random Variables
440(3)
25.6 Reproducing Kernel Hilbert Spaces
443(1)
Bibliography 444(16)
Algorithms 460(1)
Glossary 461(2)
Nomenclature 463(4)
Index 467(4)
Identities 471
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. 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.