Numerical methods for viscosity solutions and applications (Series on advances in mathematics for applied sciences, vol. 59)

• Main Authors: Falcone, Maurizio, Makridakis, Charalampos
• Format: eBook Book
• Language: English
• Published: Singapore World Scientific Publishing Co. Pte. Ltd 2001; World Scientific; World Scientific Publishing Company; WORLD SCIENTIFIC; World Scientific Publishing
• Edition: 1
• Series: Series on Advances in Mathematics for Applied Sciences
• Subjects: Aerospace engineering; Applied Mathematics; MATHEMATICS; Numerical & Computational Mathematics; Numerical Analysis and Computational Mathematics; Pure Mathematics; Viscosity solutions; Error Estimates; Adaptive Schemes; Viscosity Solutions; Geometrical Optics; Semi-Lagrangian Schemes; Front Propagation; Image Processing; Hamilton-Jacobi Equations; Finite Differences; Finite Elements
• ISBN: 9789810247171; 9810247176; 9789812799807; 981279980X
• DOI: 10.1142/4781

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 relate...

Table of Contents:

• Numerical methods for viscosity solutions and applications (Series on advances in mathematics for applied sciences, vol. 59) -- Foreword -- Contents -- Geometrical Optics and Viscosity Solutions -- Computation of Vorticity Evolution for a Cylindrical Type-II Superconductor Subject to Parallel and Transverse Applied Magnetic Fields -- A Characterization of the Value Function for a Class of Degenerate Control Problems -- Some Microstructures in Three Dimensions -- Convergence of Numerical Schemes for the Approximation of Level Set Solutions to Mean Curvature Flow -- Optimal Discretization Steps in Semi-Lagrangian Approximation of First-Order PDES -- Convergence Past Singularities to the Forced Mean Curvature Flow for a Modified Reaction-Diffusion Approach -- The Viscosity-Duality Solutions Approach to Geometric Optics for the Helmholtz Equation -- Adaptive Grid Generation for Evolutive Hamilton-Jacobi-Bellman Equations -- Solution and Application of Anisotropic Curvature Driven Evolution of Curves (and Surfaces) -- An Adaptive Scheme on Unstructured Grids for the Shape-From-Shading Problem -- On a Posteriori Error Estimation for Constant Obstacle Problems.
• Intro -- 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