Numerical analysis and optimization : an introduction to mathematical modelling and numerical simulation /
This text, based on the author's teaching at Ecole Polytechnique, introduces the reader to the world of mathematical modelling and numerical simulation. Covering the finite difference method; variational formulation of elliptic problems; Sobolev spaces; elliptical problems; the finite element m...
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| Format: | Book |
| Language: | English French |
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Oxford :
Oxford University Press,
2007.
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| Series: | Numerical mathematics and scientific computation
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| Subjects: |
Table of Contents:
- An example of modelling
- Some classical models
- The heat flow equation
- The wave equation
- The Laplacian
- Schrodinger's equation
- The Lame system
- The Stokes system
- The plate equations
- Numerical calculation by finite differences
- Principles of the method
- Numerical results for the heat flow equation
- Numerical results for the advection equation
- Remarks on mathematical models
- The idea of a well-posed problem
- Classification of PDEs
- Finite difference method
- Finite differences for the heat equation
- Various examples of schemes
- Consistency and accuracy
- Stability and Fourier analysis
- Convergence of the schemes
- Multilevel schemes
- The multidimensional case
- Other models
- Advection equation
- Wave equation
- Variational formulation of elliptic problems
- Classical formulation
- The case of a space of one dimension
- Variational approach
- Green's formulas
- Variational formulation
- Lax-Milgram theory
- Abstract framework
- Application to the Laplacian
- Sobolev spaces
- Introduction and warning
- Square integrable functions and weak differentiation
- Some results from integration
- Weak differentiation
- Definition and principal properties
- The space H[superscript 1][ohm]
- The space [Characters not reproducible] [ohm]
- Traces and Green's formulas
- A compactness result
- The spaces H[superscript m] [ohm]
- Some useful extra results
- Proof of the density theorem 4.3.5
- The space H(div)
- The spaces W[superscript m,p] [ohm]
- Duality
- Link with distributions
- Mathematical study of elliptic problems
- Study of the Laplacian
- Dirichlet boundary conditions
- Neumann boundary conditions
- Variable coefficients
- Qualitative properties
- Solution of other models
- System of linear elasticity
- Stokes equations
- Finite element method
- Variational approximation
- Genera] internal approximation
- Galerkin method
- Finite element method (general principles)
- Finite elements in N = 1 dimension
- P[subscript 1] finite elements
- Convergence and error estimation
- P[subscript 2] finite elements
- Qualitative properties
- Hermite finite elements
- Finite elements in N [greater than or equal] 2 dimensions
- Triangular finite elements
- Convergence and error estimation
- Rectangular finite elements
- Finite elements for the Stokes problem
- Visualization of the numerical results
- Eigenvalue problems
- Motivation and examples
- Solution of nonstationary problems
- Spectral theory
- Spectral decomposition of a compact operator
- Eigenvalues of an elliptic problem
- Variational problem
- Eigenvalues of the Laplacian
- Other models
- Numerical methods
- Discretization by finite elements
- Convergence and error estimates
- Evolution problems
- Motivation and examples
- Modelling and examples of parabolic equations
- Modelling and examples of hyperbolic equations
- Existence and uniqueness in the parabolic case
- Variational formulation
- A general result
- Applications
- Existence and uniqueness in the hyperbolic case
- Variational formulation
- A general result
- Applications
- Qualitative properties in the parabolic case
- Asymptotic behaviour
- The maximum principle
- Propagation at infinite velocity
- Regularity and regularizing effect
- Heat equation in the entire space
- Qualitative properties in the hyperbolic case
- Reversibility in time
- Asymptotic behaviour and equipartition of energy
- Finite velocity of propagation
- Numerical methods in the parabolic case
- Semidiscretization in space
- Total discretization in space-time
- Numerical methods in the hyperbolic case
- Semidiscretization in space
- Total discretization in space-time
- Motivation and examples
- Definitions and notation
- Optimization in finite dimensions
- Existence at a minimum in infinite dimensions
- Examples of nonexistence
- Convex analysis
- Existence results
- Optimality conditions and algorithms
- Differentiability
- Optimality conditions
- Euler inequalities and convex constraints
- Lagrange multipliers
- Saddle point, Kuhn-Tucker theorem, duality
- Saddle point
- The Kuhn-Tucker theorem
- Duality
- Applications
- Dual or complementary energy
- Optimal command
- Optimization of distributed systems
- Numerical algorithms
- Gradient algorithms (case without constraints)
- Gradient algorithms (case with constraints)
- Newton's method
- Methods of operational research / Stephane Gaubert
- Linear programming
- Definitions and properties
- The simplex algorithm
- Interior point algorithms
- Duality
- Integer polyhedra
- Extreme points of compact convex sets
- Totally unimodular matrices
- Flow problems
- Dynamic programming
- Bellman's optimality principle
- Finite horizon problem
- Minimum cost path, or optimal stopping, problem
- Greedy algorithms
- General points about greedy methods
- Kruskal's algorithm for the minimum spanning tree problem
- Separation and relaxation
- Separation and evaluation (branch and bound)
- Relaxation of combinatorial problems
- Appendix Review of hilbert spaces
- Appendix Matrix Numerical Analysis
- Solution of linear systems
- Review of matrix norms
- Conditioning and stability
- Direct methods
- Iterative methods
- The conjugate gradient method
- Calculation of eigenvalues and eigenvectors
- The power method
- The Givens-Householder method
- The Lanczos method.