Multiscale Problems and Methods in Numerical Simulations: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 9-15, 2001

Title......Page 1
Dedication......Page 5
Preface......Page 7
Contents......Page 11
1 Introduction......Page 14
2 A Simple Example......Page 17
3 The Haar System and Thresholding......Page 20
4 Linear Uniform Approximation......Page 22
5 Nonlinear Adaptive Approximation......Page 28
6 Data Compression......Page 31
7 Statistical Estimation......Page 34
8 Adaptive Numerical Simulation......Page 37
9 The Curse of Dimensionality......Page 39
References......Page 41
1 Introduction......Page 43
2.1 Sparse Representations of Functions, an Example......Page 44
2.2 (Quasi-) Sparse Representation of Operators......Page 48
2.3 Preconditioning......Page 49
3.1 The General Format......Page 51
3.3 Main Features......Page 52
4.1 What Could Additional Conditions Look Like?......Page 57
4.2 Fourier- and Basis-free Criteria......Page 58
5.1 Multiresolution......Page 63
5.2 Stability of Multiscale Transformations......Page 64
5.4 Refinement Relations......Page 65
5.5 Structure of Multiscale Transformations......Page 67
5.6 Parametrization of Stable Completions......Page 68
6.1 Problem Setting......Page 69
6.3 Global Operators – Boundary Integral Equations......Page 71
6.4 Saddle Point Problems......Page 73
7 An Equivalent  $l_2$-Problem......Page 78
7.1 Connection with Preconditioning......Page 79
8.1 Introductory Comments......Page 80
8.3 The Basic Paradigm......Page 82
8.4 (III) Convergent Iteration for the $\\infty$-dimensional Problem......Page 83
8.5 (IV) Adaptive Application of Operators......Page 86
8.6 The Adaptive Algorithm......Page 87
8.8 Compressible Matrices......Page 88
8.9 Fast Approximate Matrix/Vector Multiplication......Page 89
8.11 Main Result – Convergence/Complexity......Page 91
8.12 Some Ingredients of the Proof of Theorem 8......Page 92
8.13 Approximation Properties and Regularity......Page 97
9.1 Nonlinear Problems......Page 100
10.1 Function Spaces......Page 102
10.2 Local Polynomial Approximation......Page 103
10.3 Condition Numbers......Page 104
References......Page 105
1.1 Sobolev Spaces......Page 109
1.2 A Model Problem......Page 110
1.3 Finite Element Approximation of the Model Problem......Page 112
1.4 The Stiffness Matrix and its Condition Number......Page 113
1.5 A Two-Level Multigrid Method......Page 114
2 Multigrid I......Page 118
2.2 The Multilevel Framework......Page 119
2.3 The Abstract V-cycle Algorithm, I......Page 120
2.4 The Two-level Error Recurrence......Page 121
2.5 The Braess-Hackbusch Theorem......Page 122
3.1 Introduction and Preliminaries......Page 124
3.2 The Multiplicative Error Representation......Page 128
3.3 Some Technical Lemmas......Page 129
3.4 Uniform Estimates......Page 131
4.1 Non-nested Spaces and Varying Forms......Page 133
4.2 General Multigrid Algorithms......Page 134
4.3 Multigrid V-cycle as a Reducer......Page 137
4.4 Multigrid W-cycle as a Reducer......Page 139
4.5 Multigrid V-cycle as a Preconditioner......Page 143
5.1 Introduction......Page 145
5.2 A Norm Equivalence Theorem......Page 147
5.3 Development of Preconditioners......Page 150
5.5 A Simple Approximation Operator   $\\tilde{Q}_k$......Page 151
5.6 Applications......Page 157
References......Page 162
List of Participants......Page 164
LIST OF C.I.M.E. SEMINARS......Page 168
2004 COURSES LIST......Page 173