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# Factorization of Boundary Value Problems Using the Invariant Embedding Method

- 1st Edition - October 12, 2016
- Authors: Jacques Henry, A. M. Ramos
- Language: English
- Hardback ISBN:9 7 8 - 1 - 7 8 5 4 8 - 1 4 3 - 7
- eBook ISBN:9 7 8 - 0 - 0 8 - 1 0 1 0 9 0 - 7

Factorization Method for Boundary Value Problems by Invariant Embedding presents a new theory for linear elliptic boundary value problems. The authors provide a transform… Read more

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Request a sales quote*Factorization Method for Boundary Value Problems by Invariant Embedding *presents a new theory for linear elliptic boundary value problems. The authors provide a transformation of the problem in two initial value problems that are uncoupled, enabling you to solve these successively. This method appears similar to the Gauss block factorization of the matrix, obtained in finite dimension after discretization of the problem. This proposed method is comparable to the computation of optimal feedbacks for linear quadratic control problems.

- Develops the invariant embedding technique for boundary value problems
- Makes a link between control theory, boundary value problems and the Gauss factorization
- Presents a new theory for successively solving linear elliptic boundary value problems
- Includes a transformation in two initial value problems that are uncoupled

Researchers and academics working in the subjects of partial differential equations, numerical analysis, control theory

- Dedication
- Preface
- 1: Presentation of the Formal Computation of Factorization
- Abstract
- 1.1 Definition of the model problem and its functional framework
- 1.2 Direct invariant embedding
- 1.3 Backward invariant embedding
- 1.4 Internal invariant embedding

- 2: Justification of the Factorization Computation
- Abstract
- 2.1 Functional framework
- 2.2 Semi-discretization
- 2.3 Passing to the limit

- 3: Complements to the Model Problem
- Abstract
- 3.1 An alternative method for obtaining the factorization
- 3.2 Other boundary conditions
- 3.3 Explicitly taking into account the boundary conditions and the right-hand side
- 3.4 Periodic boundary conditions in x
- 3.5 An alternative but unstable formulation
- 3.6 Link with the Steklov–Poincaré operator
- 3.7 Application of the Schwarz kernel theorem: link with Green’s functions and Hadamard’s formula

- 4: Interpretation of the Factorization through a Control Problem
- Abstract
- 4.1 Formulation of problem (P
_{0}) in terms of optimal control - 4.2 Summary of results on the decoupling of optimal control problems
- 4.3 Summary of results of A. Bensoussan on Kalman optimal filtering
- 4.4 Parabolic regularization for the factorization of elliptic boundary value problems

- 5: Factorization of the Discretized Problem
- Abstract
- 5.1 Introduction and problem statement
- 5.2 Application of the factorization method to problem (P
_{h}) - 5.3 A second method of discretization
- 5.4 A third possibility: centered scheme
- 5.5 Row permutation
- 5.6 Case of a discretization of the section by finite elements

- 6: Other Problems
- Abstract
- 6.1 General second-order linear elliptic problems
- 6.2 Systems of coupled boundary value problems
- 6.3 Linear elasticity system
- 6.4 Problems of order higher than 2
- 6.5 Stokes problems
- 6.6 Parabolic problems

- 7: Other Shapes of Domain
- Abstract
- 7.1 Domain generalization: transformation preserving orthogonal coordinates
- 7.2 Quasi-cylindrical domains with normal velocity fields
- 7.3 Sweeping the domain by surfaces of arbitrary shape

- 8: Factorization by the QR Method
- Abstract
- 8.1 Normal equation for problem (P
_{0}) in section 1.1 - 8.2 Factorization of the normal equation by invariant embedding
- 8.3 The QR method

- 9: Representation Formulas for Solutions of Riccati Equations
- Abstract
- 9.1 Representation formulas
- 9.2 Diagonalization of the two-point boundary value problem
- 9.3 Homographic representation of P(x)
- 9.4 Factorization of problem (P
_{0}) with a Dirichlet condition at x = 0

- Appendix: Gaussian LU Factorization as a Method of Invariant Embedding
- A.1 Invariant embedding for a linear system
- A.2 Block tridiagonal systems

- Bibliography
- Index

- No. of pages: 256
- Language: English
- Edition: 1
- Published: October 12, 2016
- Imprint: ISTE Press - Elsevier
- Hardback ISBN: 9781785481437
- eBook ISBN: 9780081010907

JH

### Jacques Henry

Jacques Henry is Director of Research, emeritus at INRIA Bordeaux Sud-ouest, France. He graduated from École Polytechnique, Paris (1970). He has worked within INRIA (National Institute for Computer Sciences and Automatic Control, France) since 1974. His work covers control of systems governed by partial differential equations, modeling, parameter estimation and continuation-bifurcation methods applied to biological systems mainly in cardiac electrophysiology and biological sequences comparison. His current interests are on numerical analysis, inverse problems and singular perturbations for partial differential equations. He is developing research on the method of factorization of linear elliptic boundary value problems in terms of product of Cauchy problems. He was leading the INRIA project team Anubis on structured population dynamics. He has a special interest on the evolution of activity of popula- tions of neurons.

Affiliations and expertise

Research Director, INRIA, Bordeaux, FranceAR

### A. M. Ramos

His research is focused on modeling, optimization and simulation in Science and Technology, mainly using Partial Differential Equations. His research lines are the following: Epidemic modeling, spatial-stochastic individual based models, SIR models, hybrid models, risk analysis, validation with real data, control measures, economic and climate change impact analysis. He received his PhD. in Applied Mathematics from UCM< in July 1996. He is Director of the UCM Research Group Mathematical Models in Science and Engineering: Development, Analysis, Numerical Simulation and Control (MOMAT) since 2005.

Affiliations and expertise

Associate Professor, Department of Applied Mathematics, Complutense University of Madrid, SpainRead

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