Beschreibung
Autorenporträt
InhaltsangabeSubgrid Phenomena and Numerical Schemes.- 1 Introduction.- 2 The Continuous Problem.- 3 From the Discrete Problem to the Augmented Problem.- 4 An Example of Error Estimates.- 5 Computational Aspects.- 5.1 First Strategy.- 5.2 Alternative Computational Strategies.- 6 Conclusions.- References.- Stability of Saddle-Points in Finite Dimensions.- 1 Introduction.- 2 Notation, and Basic Results in Linear Algebra.- 3 Existence and Uniqueness of Solutions: the Solvability Problem.- 4 The Case of Big Matrices. The Inf-Sup Condition.- 5 The Case of Big Matrices. The Problem of Stability.- 6 Additional Considerations.- References.- Mean Curvature Flow.- 1 Introduction.- 2 Some Geometric Analysis.- 2.1 Tangential Gradients and Curvature.- 2.2 Moving Surfaces.- 2.3 The Concept of Anisotropy.- 3 Parametric Mean Curvature Flow.- 3.1 Curve Shortening Flow.- 3.2 Anisotropic Curve Shortening Flow.- 3.3 Mean Curvature Flow of Hypersurfaces.- 3.4 Finite Elements on Surfaces.- 4 Mean Curvature Flow of Level Sets I.- 4.1 Viscosity Solutions.- 4.2 Regularization.- 5 Mean Curvature Flow of Graphs.- 5.1 The Differential Equation.- 5.2 Analytical Results.- 5.3 Spatial Discretization.- 5.4 Estimate of the Spatial Error.- 5.5 Time Discretization.- 6 Anisotropic Curvature Flow of Graphs.- 6.1 Discretization in Space and Estimate of the Error.- 6.2 Fully Discrete Scheme, Stability and Error Estimate.- 7 Mean Curvature Flow of Level Sets II.- 7.1 The Approximation of Viscosity Solutions.- 7.2 Anisotropic Mean Curvature Flow of Level Sets.- References.- An Introduction to Algorithms for Nonlinear Optimization.- 1 Optimality Conditions and Why They Are Important.- 1.1 Optimization Problems.- 1.2 Notation.- 1.3 Lipschitz Continuity and Taylor's Theorem.- 1.4 Optimality Conditions.- 1.5 Optimality Conditions for Unconstrained Minimization.- 1.6 Optimality Conditions for Constrained Minimization.- 1.6.1 Optimality Conditions for Equality-Constrained Minimization.- 1.6.2 Optimality Conditions for Inequality-Constrained Minimization.- 2 Linesearch Methods for Unconstrained Optimization.- 2.1 Linesearch Methods.- 2.2 Practical Linesearch Methods.- 2.3 Convergence of Generic Linesearch Methods.- 2.4 Method of Steepest Descent.- 2.5 More General Descent Methods.- 2.5.1 Newton and Newton-Like Methods.- 2.5.2 Modified-Newton Methods.- 2.5.3 Quasi-Newton Methods.- 2.5.4 Conjugate-Gradient and Truncated-Newton Methods.- 3 Trust-Region Methods for Unconstrained Optimization.- 3.1 Linesearch Versus Trust-Region Methods.- 3.2 Trust-Region Models.- 3.3 Basic Trust-Region Method.- 3.4 Basic Convergence of Trust-Region Methods.- 3.5 Solving the Trust-Region Subproblem.- 3.5.1 Solving the ?2-Norm Trust-Region Subproblem.- 3.6 Solving the Large-Scale Problem.- 4 Interior-Point Methods for Inequality Constrained Optimization.- 4.1 Merit Functions for Constrained Minimization.- 4.2 The Logarithmic Barrier Function for Inequality Constraints.- 4.3 A Basic Barrier-Function Algorithm.- 4.4 Potential Difficulties.- 4.4.1 Potential Difficulty I: Ill-Conditioning of the Barrier Hessian.- 4.4.2 Potential Difficulty II: Poor Starting Points.- 4.5 A Different Perspective: Perturbed Optimality Conditions.- 4.5.1 Potential Difficulty II. Revisited.- 4.5.2 Primal-Dual Barrier Methods.- 4.5.3 Potential Difficulty I. Revisited.- 4.6 A Practical Primal-Dual Method.- 5 SQP Methods for Equality Constrained Optimization.- 5.1 Newton's Method for First-Order Optimality.- 5.2 The Sequential Quadratic Programming Iteration.- 5.3 Linesearch SQP Methods.- 5.4 Trust-Region SQP Methods.- 5.4.1 The S?pQP Method.- 5.4.2 Composite-Step Methods.- 5.4.3 Filter Methods.- 6 Conclusion.- A Seminal Books and Papers.- B Optimization Resources on the World-Wide-Web.- B.1 Answering Questions on the Web.- B.2 Solving Optimization Problems on the Web.- B.2.1 The NEOS Server.- B.2.2 Other Online Solvers.- B.2.3 Useful Sites for Modelling Problems Prior to Online Solution.- B.2.4 Free Optimization Software.- B.3 Optim
