Numerical Analysis and Optimization by Gregoire Allaire

Table of Contents
Introduction
1 Introduction to mathematical modelling and numerical simulation
1.1 General introduction
1.2 An example of modelling
1.3 Some classical models
1.4 Numerical calculation by finite differences
1.5 Remarks on mathematical models
2 Finite difference method
2.1 Introduction
2.2 Finite differences for the heat equation
2.3 Other models
3 Variational formulation of elliptic problems
3.1 Generalities
3.2 Variational approach
3.3 Lax–Milgram theory
4 Sobolev spaces
4.1 Introduction and warning
4.2 Square integrable functions and weak differentiation
4.3 Definition and principal properties
4.4 Some useful extra results
4.5 Link with distributions
5 Mathematical study of elliptic problems
5.1 Introduction
5.2 Study of the Laplacian
5.3 Solution of other models
6 Finite element method
6.1 Variational approximation
6.2 Finite elements in N = 1 dimension
6.3 Finite elements in N ≥ 2 dimensions
7 Eigenvalue problems
7.1 Motivation and examples
7.2 Spectral theory
7.3 Eigenvalues of an elliptic problem
7.4 Numerical methods
8 Evolution problems
8.1 Motivation and examples
8.2 Existence and uniqueness in the parabolic case
8.3 Existence and uniqueness in the hyperbolic case
8.4 Qualitative properties in the parabolic case
8.5 Qualitative properties in the hyperbolic case
8.6 Numerical methods in the parabolic case
8.7 Numerical methods in the hyperbolic case
9 Introduction to optimization
9.1 Motivation and examples
9.2 Existence of a minimum in infinite dimensions
10 Optimality conditions and algorithms
10.1 Generalities
10.2 Optimality conditions
10.3 Saddle point, Kuhn–Tucker theorem, duality
10.4 Applications
10.5 Numerical algorithms
11 Methods of operational research (Written in collaboration with Stephane Gaubert)
11.1 Introduction
11.2 Linear programming
11.3 Integer polyhedra
11.4 Dynamic programming
11.5 Greedy algorithms
11.6 Separation and relaxation
12 Appendix Review of hilbert spaces
13 Appendix Matrix Numerical Analysis
13.1 Solution of linear systems
13.2 Calculation of eigenvalues and eigenvectors
Index
Index notations