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E-raamat: Factorization of Boundary Value Problems Using the Invariant Embedding Method

(Research Director, INRIA, Bordeaux, France), (Associate Professor, Department of Applied Mathematics, Complutense University of Madrid, Spain)
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 09-Nov-2016
  • Kirjastus: ISTE Press Ltd - Elsevier Inc
  • Keel: eng
  • ISBN-13: 9780081010907
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Factorization of Boundary Value Problems Using the Invariant Embedding MethodSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 09-Nov-2016
  • Kirjastus: ISTE Press Ltd - Elsevier Inc
  • Keel: eng
  • ISBN-13: 9780081010907
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The book 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

Muu info

This innovative book presents a new theory for successively solving linear elliptic boundary value problems, including a transformation in two initial value problems that are uncoupled
Preface ix
Chapter 1 Presentation of the Formal Computation of Factorization
1(22)
1.1 Definition of the model problem and its functional framework
1(3)
1.2 Direct invariant embedding
4(8)
1.3 Backward invariant embedding
12(6)
1.4 Internal invariant embedding
18(5)
Chapter 2 Justification of the Factorization Computation
23(18)
2.1 Functional framework
23(3)
2.2 Semi-discretization
26(5)
2.3 Passing to the limit
31(10)
Chapter 3 Complements to the Model Problem
41(28)
3.1 An alternative method for obtaining the factorization
41(2)
3.2 Other boundary conditions
43(9)
3.2.1 Boundary conditions on the lateral boundary Σ
43(5)
3.2.2 Boundary conditions on the faces Γ0 and Γa
48(1)
3.2.3 Robin-to-Neumann operator
49(2)
3.2.4 Neumann problem
51(1)
3.3 Explicitly taking into account the boundary conditions and the right-hand side
52(6)
3.4 Periodic boundary conditions in x
58(1)
3.5 An alternative but unstable formulation
59(2)
3.6 Link with the Steklov--Poincare operator
61(2)
3.7 Application of the Schwarz kernel theorem; link with Green's functions and Hadamard's formula
63(6)
Chapter 4 Interpretation of the Factorization through a Control Problem
69(30)
4.1 Formulation of problem (ρq) in terms of optimal control
69(4)
4.2 Summary of results on the decoupling of optimal control problems
73(4)
4.3 Summary of results of A. Bensoussan on Kalman optimal filtering
77(1)
4.4 Parabolic regularization for the factorization of elliptic boundary value problems
78(21)
4.4.1 Convergence of the operator Pε
84(10)
4.4.2 Parabolic regularization for the Neumann-to-Dirichlet operator
94(5)
Chapter 5 Factorization of the Discretized Problem
99(28)
5.1 Introduction and problem statement
99(3)
5.2 Application of factorization method to problem (ρh)
102(6)
5.3 A second method of discretization
108(3)
5.4 A third possibility: centered scheme
111(3)
5.5 Row permutation
114(4)
5.6 Case of a discretization of the section by finite elements
118(9)
Chapter 6 Other Problems
127(42)
6.1 General second-order linear elliptic problems
127(6)
6.1.1 Problem statement
127(1)
6.1.2 Factorization by invariant embedding
128(5)
6.2 Systems of coupled boundary value problems
133(8)
6.2.1 Global approach
134(1)
6.2.2 Sequential approach
135(6)
6.3 Linear elasticity system
141(8)
6.3.1 Problem statement and transformation
141(4)
6.3.2 Derivation of the decoupled system
145(2)
6.3.3 Associated control problem
147(2)
6.4 Problems of order higher than 2
149(6)
6.4.1 A factorization of the bilaplacian
149(3)
6.4.2 Another (unstable) factorization of the bilaplacian
152(3)
6.5 Stokes problems
155(8)
6.6 Parabolic problems
163(6)
Chapter 7 Other Shapes of Domain
169(30)
7.1 Domain generalization: transformation preserving orthogonal coordinates
169(7)
7.1.1 Hypotheses on the domain
170(2)
7.1.2 Formal derivation
172(4)
7.2 Quasi-cylindrical domains with normal velocity fields
176(5)
7.3 Sweeping the domain by surfaces of arbitrary shape
181(18)
Chapter 8 Factorization by the QR Method
199(14)
8.1 Normal equation for problem (ρn) in section 1.1
199(2)
8.2 Factorization of the normal equation by invariant embedding
201(6)
8.3 The QR method
207(6)
Chapter 9 Representation Formulas for Solutions of Riccati Equations
213(8)
9.1 Representation formulas
213(2)
9.2 Diagonalization of the two-point boundary value problem
215(2)
9.3 Homographic representation of P(x)
217(3)
9.4 Factorization of problem (ρq) with a Dirichlet condition at x = 0
220(1)
Appendix 221(12)
Bibliography 233(4)
Index 237
Jacques Henry is Director of Research, emeritus at INRIA Bordeaux Sud-ouest, France. He graduated from Ecole 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. 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.
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