Numerical Analysis and Optimization

Name: Numerical Analysis and Optimization | An Introduction to Mathematical Modelling and Numerical Simulation
Brand: Oxford University Press, UK
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An Introduction to Mathematical Modelling and Numerical Simulation

Gregoire Allaire Alan Craig(Autor*in)

Oxford University Press, UK

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Erschienen am 24. Mai 2007

472 Seiten

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Inhalt

Intro
Contents
1 Introduction
1 Introduction to mathematical modelling and numerical simulation
1.1 General introduction
1.2 An example of modelling
1.3 Some classical models
1.3.1 The heat flow equation
1.3.2 The wave equation
1.3.3 The Laplacian
1.3.4 Schrödinger's equation
1.3.5 The Lamé system
1.3.6 The Stokes system
1.3.7 The plate equations
1.4 Numerical calculation by finite differences
1.4.1 Principles of the method
1.4.2 Numerical results for the heat flow equation
1.4.3 Numerical results for the advection equation
1.5 Remarks on mathematical models
1.5.1 The idea of a well-posed problem
1.5.2 Classification of PDEs
2 Finite difference method
2.1 Introduction
2.2 Finite differences for the heat equation
2.2.1 Various examples of schemes
2.2.2 Consistency and accuracy
2.2.3 Stability and Fourier analysis
2.2.4 Convergence of the schemes
2.2.5 Multilevel schemes
2.2.6 The multidimensional case
2.3 Other models
2.3.1 Advection equation
2.3.2 Wave equation
3 Variational formulation of elliptic problems
3.1 Generalities
3.1.1 Introduction
3.1.2 Classical formulation
3.1.3 The case of a space of one dimension
3.2 Variational approach
3.2.1 Green's formulas
3.2.2 Variational formulation
3.3 Lax-Milgram theory
3.3.1 Abstract framework
3.3.2 Application to the Laplacian
4 Sobolev spaces
4.1 Introduction and warning
4.2 Square integrable functions and weak differentiation
4.2.1 Some results from integration
4.2.2 Weak differentiation
4.3 Definition and principal properties
4.3.1 The space H[sup(1)](O)
4.3.2 The space H[sup(1)][sub(0)](O)
4.3.3 Traces and Green's formulas
4.3.4 A compactness result
4.3.5 The spaces H[sup(m)](O)
4.4 Some useful extra results
4.4.1 Proof of the density theorem 4.3.5
4.4.2 The space H(div)
4.4.3 The spaces W[sup(m,p)](O)
4.4.4 Duality
4.5 Link with distributions
5 Mathematical study of elliptic problems
5.1 Introduction
5.2 Study of the Laplacian
5.2.1 Dirichlet boundary conditions
5.2.2 Neumann boundary conditions
5.2.3 Variable coefficients
5.2.4 Qualitative properties
5.3 Solution of other models
5.3.1 System of linear elasticity
5.3.2 Stokes equations
6 Finite element method
6.1 Variational approximation
6.1.1 Introduction
6.1.2 General internal approximation
6.1.3 Galerkin method
6.1.4 Finite element method (general principles)
6.2 Finite elements in N = 1 dimension
6.2.1 P[sub(1)] finite elements
6.2.2 Convergence and error estimation
6.2.3 P[sub(2)] finite elements
6.2.4 Qualitative properties
6.2.5 Hermite finite elements
6.3 Finite elements in N = 2 dimensions
6.3.1 Triangular finite elements
6.3.2 Convergence and error estimation
6.3.3 Rectangular finite elements
6.3.4 Finite elements for the Stokes problem
6.3.5 Visualization of the numerical results
7 Eigenvalue problems
7.1 Motivation and examples
7.1.1 Introduction
7.1.2 Solution of nonstationary problems
7.2 Spectral theory
7.2.1 Generalities
7.2.2 Spectral decomposition of a compact operator
7.3 Eigenvalues of an elliptic problem
7.3.1 Variational problem
7.3.2 Eigenvalues of the Laplacian
7.3.3 Other models
7.4 Numerical methods
7.4.1 Discretization by finite elements
7.4.2 Convergence and error estimates
8 Evolution problems
8.1 Motivation and examples
8.1.1 Introduction
8.1.2 Modelling and examples of parabolic equations
8.1.3 Modelling and examples of hyperbolic equations
8.2 Existence and uniqueness in the parabolic case
8.2.1 Variational formulation
8.2.2 A general result
8.2.3 Applications
8.3 Existence and uniqueness in the hyperbolic case
8.3.1 Variational formulation
8.3.2 A general result
8.3.3 Applications
8.4 Qualitative properties in the parabolic case
8.4.1 Asymptotic behaviour
8.4.2 The maximum principle
8.4.3 Propagation at infinite velocity
8.4.4 Regularity and regularizing effect
8.4.5 Heat equation in the entire space
8.5 Qualitative properties in the hyperbolic case
8.5.1 Reversibility in time
8.5.2 Asymptotic behaviour and equipartition of energy
8.5.3 Finite velocity of propagation
8.6 Numerical methods in the parabolic case
8.6.1 Semidiscretization in space
8.6.2 Total discretization in space-time
8.7 Numerical methods in the hyperbolic case
8.7.1 Semidiscretization in space
8.7.2 Total discretization in space-time
9 Introduction to optimization
9.1 Motivation and examples
9.1.1 Introduction
9.1.2 Examples
9.1.3 Definitions and notation
9.1.4 Optimization in finite dimensions
9.2 Existence of a minimum in infinite dimensions
9.2.1 Examples of nonexistence
9.2.2 Convex analysis
9.2.3 Existence results
10 Optimality conditions and algorithms
10.1 Generalities
10.1.1 Introduction
10.1.2 Differentiability
10.2 Optimality conditions
10.2.1 Euler inequalities and convex constraints
10.2.2 Lagrange multipliers
10.3 Saddle point, Kuhn-Tucker theorem, duality
10.3.1 Saddle point
10.3.2 The Kuhn-Tucker theorem
10.3.3 Duality
10.4 Applications
10.4.1 Dual or complementary energy
10.4.2 Optimal command
10.4.3 Optimization of distributed systems
10.5 Numerical algorithms
10.5.1 Introduction
10.5.2 Gradient algorithms (case without constraints)
10.5.3 Gradient algorithms (case with constraints)
10.5.4 Newton's method
11 Methods of operational research (Written in collaboration with Stéphane Gaubert)
11.1 Introduction
11.2 Linear programming
11.2.1 Definitions and properties
11.2.2 The simplex algorithm
11.2.3 Interior point algorithms
11.2.4 Duality
11.3 Integer polyhedra
11.3.1 Extreme points of compact convex sets
11.3.2 Totally unimodular matrices
11.3.3 Flow problems
11.4 Dynamic programming
11.4.1 Bellman's optimality principle
11.4.2 Finite horizon problem
11.4.3 Minimum cost path, or optimal stopping, problem
11.5 Greedy algorithms
11.5.1 General points about greedy methods
11.5.2 Kruskal's algorithm for the minimum spanning tree problem
11.6 Separation and relaxation
11.6.1 Separation and evaluation (branch and bound)
11.6.2 Relaxation of combinatorial problems
12 Appendix: Review of hilbert spaces
13 Appendix: Matrix Numerical Analysis
13.1 Solution of linear systems
13.1.1 Review of matrix norms
13.1.2 Conditioning and stability
13.1.3 Direct methods
13.1.4 Iterative methods
13.1.5 The conjugate gradient method
13.2 Calculation of eigenvalues and eigenvectors
13.2.1 The power method
13.2.2 The Givens-Householder method
13.2.3 The Lanczos method
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Index notations

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