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E-raamat: Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications: Cetraro, Italy 2011, Editors: Paola Loreti, Nicoletta Anna Tchou

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  • Sari: C.I.M.E. Foundation Subseries 2074
  • Ilmumisaeg: 24-May-2013
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783642364334
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Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications: Cetraro, Italy 2011, Editors: Paola Loreti, Nicoletta Anna TchouSuurem pilt
  • Formaat: PDF+DRM
  • Sari: C.I.M.E. Foundation Subseries 2074
  • Ilmumisaeg: 24-May-2013
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783642364334
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These Lecture Notes contain the material relative to the courses given at the CIME summer school held in Cetraro, Italy from August 29 to September 3, 2011. The topic was "Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications". The courses dealt mostly with the following subjects: first order and second order Hamilton-Jacobi-Bellman equations, properties of viscosity solutions, asymptotic behaviors, mean field games, approximation and numerical methods, idempotent analysis. The content of the courses ranged from an introduction to viscosity solutions to quite advanced topics, at the cutting edge of research in the field. We believe that they opened perspectives on new and delicate issues. These lecture notes contain four contributions by Yves Achdou (Finite Difference Methods for Mean Field Games), Guy Barles (An Introduction to the Theory of Viscosity Solutions for First-order Hamilton-Jacobi Equations and Applications), Hitoshi Ishii (A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations) and Grigory Litvinov (Idempotent/Tropical Analysis, the Hamilton-Jacobi and Bellman Equations).
Finite Difference Methods for Mean Field Games
1(48)
Yves Achdou
1 Introduction
1(3)
2 Finite Difference Schemes
4(15)
2.1 Description of the Schemes
4(5)
2.2 Existence and A priori Bounds
9(5)
2.3 A Fundamental Identity
14(2)
2.4 Uniqueness
16(1)
2.5 A priori Estimates for (21)-(22) with Local Operators Φ
16(3)
3 Examples of Convergence Results
19(5)
4 Algorithms for Solving the Discrete Linear Systems
24(7)
4.1 Newton Methods for Solving (21)-(22)
24(2)
4.2 Iterative Strategies for Solving (54) Based on Eliminating U
26(5)
5 Some Simulations
31(4)
6 The Planning Problem
35(14)
6.1 Description of the Planning Problem
35(1)
6.2 The Finite Difference Scheme and an Optimal Control Formulation
36(8)
6.3 Uniqueness
44(1)
6.4 A Penalty Method
45(1)
References
45(4)
An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton-Jacobi Equations and Applications
49(62)
Guy Barles
1 Introduction
49(2)
2 Preliminaries: A Running Example
51(2)
3 The Notion of Continuous Viscosity Solutions: Definition(s) and First Properties
53(7)
3.1 Why a "Good" Notion of Weak Solution is Needed?
53(1)
3.2 Continuous Viscosity Solutions
54(2)
3.3 Back to the Running Example (I): The Value Function U is a Viscosity Solution of (7)
56(2)
3.4 An Equivalent Definition and Its Consequences
58(2)
4 The First Stability Result for Viscosity Solutions
60(4)
5 Uniqueness: The Basic Arguments and Additional Recipes
64(10)
5.1 A First Basic Result
64(6)
5.2 Several Variations
70(2)
5.3 Finite Speed of Propagation
72(2)
6 Discontinuous Viscosity Solutions, Discontinuous Nonlinearities and the "Half-Relaxed Limits" Method
74(8)
6.1 Discontinuous Viscosity Solutions
74(2)
6.2 Back to the Running Example (II): The Dirichlet Boundary Condition for the Value-Function
76(1)
6.3 The Half-Relaxed Limit Method
77(4)
6.4 Strong Comparison Results
81(1)
7 Existence of Viscosity Solutions: Perron's Method
82(4)
8 Regularity Results
86(3)
9 Convex Hamiltonians, Barron-Jensen Solutions
89(3)
10 Large Time Behavior of Solutions of Hamilton-Jacobi Equations
92(19)
10.1 Introduction
92(1)
10.2 Existence and Regularity of the Solution
93(1)
10.3 Ergodic Behavior
94(3)
10.4 Asymptotic Behavior of u (x, t) --- ct
97(1)
10.5 The Namah-Roquejoffre Framework
98(2)
10.6 The "Strictly Convex" Framework
100(6)
10.7 Concluding Remarks
106(1)
References
107(4)
A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations
111(140)
Hitoshi Ishii
1 Introduction to Viscosity Solutions
114(29)
1.1 Hamilton-Jacobi Equations
114(1)
1.2 An Optimal Control Problem
115(6)
1.3 Characterization of the Value Function
121(7)
1.4 Semicontinuous Viscosity Solutions and the Perron Method
128(11)
1.5 An Example
139(2)
1.6 Sup-convolutions
141(2)
2 Neumann Boundary Value Problems
143(5)
3 Initial-Boundary Value Problem for Hamilton-Jacobi Equations
148(17)
3.1 Initial-Boundary Value Problems
148(4)
3.2 Additive Eigenvalue Problems
152(3)
3.3 Proof of Comparison Theorem
155(10)
4 Stationary Problem: Weak KAM Aspects
165(20)
4.1 Aubry Sets and Representation of Solutions
166(8)
4.2 Proof of Theorem 4.2
174(11)
5 Optimal Control Problem Associated with (ENP)-(ID)
185(26)
5.1 Skorokhod Problem
185(6)
5.2 Value Function I
191(3)
5.3 Basic Lemmas
194(8)
5.4 Value Function II
202(6)
5.5 Distance-Like Function d
208(3)
6 Large-Time Asymptotic Solutions
211(40)
6.1 Preliminaries to Asymptotic Solutions
214(5)
6.2 Proof of Convergence
219(3)
6.3 Representation of the Asymptotic Solution u∞
222(4)
6.4 Localization of Conditions (A9)±
226(3)
A.1 Local maxima to global maxima
229(1)
A.2 A Quick Review of Convex Analysis
230(5)
A.3 Global Lipschitz Regularity
235(3)
A.4 Localized Versions of Lemma 4.2
238(4)
A.5 A Proof of Lemma 5.4
242(3)
A.6 Rademacher's Theorem
245(2)
References
247(4)
Idempotent/Tropical Analysis, the Hamilton-Jacobi and Bellman Equations
251
Grigory L. Litvinov
1 Introduction
251(2)
2 The Maslov Dequantization
253(1)
3 Semirings and Semifields: The Idempotent Correspondence Principle
254(1)
4 Idempotent Analysis
255(1)
5 The Superposition Principle and Linear Equations
256(4)
5.1 Heuristics
256(3)
5.2 The Cauchy Problem for the Hamilton-Jacobi Equations
259(1)
6 Convolution and the Fourier-Legendre Transform
260(1)
7 Idempotent Functional Analysis
261(14)
7.1 Idempotent Semimodules and Idempotent Linear Spaces
262(3)
7.2 Basic Results
265(1)
7.3 Idempotent b-semialgebras
266(1)
7.4 Linear Operator, b-semimodules and Subsemimodules
267(1)
7.5 Functional Semimodules
268(2)
7.6 Integral Representations of Linear Operators in Functional Semimodules
270(2)
7.7 Nuclear Operators and Their Integral Representations
272(1)
7.8 The b-approximation Property and b-nuclear Semimodules and Spaces
272(1)
7.9 Kernel Theorems for Functional b-Semimodules
273(1)
7.10 Integral Representations of Operators in Abstract Idempotent Semimodules
273(2)
8 The Dequantization Transform, Convex Geometry and the Newton Poly topes
275(5)
8.1 Dequantization Transform: Algebraic Properties
276(1)
8.2 Generalized Polynomials and Simple Functions
277(1)
8.3 Subdifferentials of Sublinear Functions
278(1)
8.4 Newton Sets for Simple Functions
279(1)
9 Dequantization of Set Functions and Measures on Metric Spaces
280(1)
10 Dequantization of Geometry
281(1)
11 Some Semiring Constructions and the Matrix Bellman Equation
282(5)
11.1 Complete Idempotent Semirings and Examples
282(1)
11.2 Closure Operations
282(1)
11.3 Matrices Over Semirings
283(1)
11.4 Discrete Stationary Bellman Equations
284(1)
11.5 Weighted Directed Graphs and Matrices Over Semirings
284(3)
12 Universal Algorithms
287(1)
13 Universal Algorithms of Linear Algebra Over Semirings
288(5)
14 The Correspondence Principle for Computations
293(1)
15 The Correspondence Principle for Hardware Design
293(2)
16 The Correspondence Principle for Software Design
295(1)
17 Interval Analysis in Idempotent Mathematics
296
References
297
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