Numerical Methods for Viscosity Solutions and Applications.
Title:
Numerical Methods for Viscosity Solutions and Applications.
Author:
Falcone, Maurizio.
ISBN:
9789812799807
Personal Author:
Physical Description:
1 online resource (249 pages)
Series:
Series on Advances in Mathematics for Applied Sciences ; v.59
Series on Advances in Mathematics for Applied Sciences
Contents:
CONTENTS -- Foreword -- Geometrical Optics and Viscosity Solutions -- 1 High frequency waves and WKB method -- 2 The Kravtsov-Ludwig method -- 3 Multivalued solutions to the eikonal equation and shadow zones -- References -- Computation of Vorticity Evolution for a Cylindrical Type-II Superconductor Subject to Parallel and Transverse Applied Magnetic Fields -- 1 Introduction -- 2 The Models -- 3 Discretization of the Models -- 4 Numerical Computations -- References -- A Characterization of the Value Function for a Class of Degenerate Control Problems -- 1 Introduction -- 2 Assumptions and definitions -- 3 A characterization of the value function -- 4 Approximation of the value function -- References -- Some Microstructures in Three Dimensions -- 1 Introduction -- 2 A three dimensional example -- 3 Minimizing sequences -- 4 Numerical experiments -- References -- Convergence of Numerical Schemes for the Approximation of Level Set Solutions to Mean Curvature Flow -- 1 Introduction -- 2 Background -- 3 The Crandall-Lions scheme -- 4 Finite element method -- References -- Optimal Discretization Steps in Semi-Lagrangian Approximation of First-Order PDEs -- 1 Introduction -- 2 Construction of the schemes and basic convergence theory -- 3 Fully discrete second and third order schemes -- 4 Relationship between time and space step -- 5 Numerical tests -- Conclusions -- References -- Convergence Past Singularities to the Forced Mean Curvature Flow for a Modified Reaction-Diffusion Approach -- 1 Introduction -- 2 Approximate Traveling Wave -- 3 Viscosity Solutions -- 4 Supersolutions -- 5 Comparison Lemma -- 6 Convergence and interfaces error estimates -- References -- The Viscosity-Duality Solutions Approach to Geometric Optics for the Helmholtz Equation -- 1 Weak solutions to the differential problem.
2 A class of numerical approximations -- 3 Numerical results -- 4 Conclusion -- References -- Adaptive Grid Generation for Evolutive Hamilton-Jacobi-Bellman Equations -- 1 Introduction -- 2 Discretization in time and space -- 3 Error estimation -- 4 Implementation details -- 5 Numerical examples -- References -- Solution and Application of Anisotropic Curvature Driven Evolution of Curves (and Surfaces) -- 1 Introduction -- 2 Direct approach using porous-medium like equations -- 3 Direct approach by intrinsic heat equations -- 4 Solution using level set equation -- 5 Phase field approximation of interface motion -- References -- An Adaptive Scheme on Unstructured Grids for the Shape-From-Shading Problem -- 1 Introduction -- 2 A fixed grid fully discrete scheme -- 3 A local error indicator -- 4 The adaptive grid algorithm -- 5 Implementation of the algorithm -- 6 Numerical experiments -- References -- On A Posteriori Error Estimation For Constant Obstacle Problems -- 1 Introduction -- 2 Results and their discussion -- 3 Proofs -- References.
Abstract:
The volume contains twelve papers dealing with the approximation of first and second order problems which arise in many fields of application including optimal control, image processing, geometrical optics and front propagation. Some contributions deal with new algorithms and technical issues related to their implementation. Other contributions are more theoretical, dealing with the convergence of approximation schemes. Many test problems have been examined to evaluate the performances of the algorithms. The volume can attract readers involved in the numerical approximation of differential models in the above-mentioned fields of applications, engineers, graduate students as well as researchers in numerical analysis. Contents: Geometrical Optics and Viscosity Solutions (A-P Blanc et al.); Computation of Vorticity Evolution for a Cylindrical Type-II Superconductor Subject to Parallel and Transverse Applied Magnetic Fields (A Briggs et al.); A Characterization of the Value Function for a Class of Degenerate Control Problems (F Camilli); Some Microstructures in Three Dimensions (M Chipot & V Lécuyer); Convergence of Numerical Schemes for the Approximation of Level Set Solutions to Mean Curvature Flow (K Deckelnick & G Dziuk); Optimal Discretization Steps in Semi-Lagrangian Approximation of First Order PDEs (M Falcone et al.); Convergence Past Singularities to the Forced Mean Curvature Flow for a Modified Reaction-Diffusion Approach (F Fierro); The Viscosity/Duality Solutions Approach to Geometric Optics for the Helmholtz Equation (L Gosse & F James); Adaptive Grid Generation for Evolutive Hamilton-Jacobi-Bellman Equations (L Grüne); Solution and Application of Anisotropic Curvature Driven Evolution of Curves (and Surfaces) (K Mikula); An Adaptive Scheme on Unstructured Grids for the Shape-From-Shading Problem (M Sagona & A Seghini); On a Posteriori
Error Estimation for Constant Obstacle Problems (A Veeser). Readership: Graduate students, researchers, academics and lecturers in numerical & computational mathematics, analysis & differential equations and mathematical modeling.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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