A set of detailed lecture notes on six topics at the forefront of current research in numerical analysis and applied mathematics. Each set of notes presents a self-contained guide to a current research area. Detailed proofs of key results are provided. The notes start from a level suitable for first year graduate students in applied mathematics, mathematical analysis or numerical analysis, and proceed to current research topics. Current (unsolved) problems are also described and directions for future research are given. This book is also suitable for professional mathematicians.

Includes supplementary material: sn.pub/extras

Inhalt
Subgrid 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.- 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.- 3 Parametric Mean Curvature Flow.- 4 Mean Curvature Flow of Level Sets I.- 5 Mean Curvature Flow of Graphs.- 6 Anisotropic Curvature Flow of Graphs.- 7 Mean Curvature Flow of Level Sets II.- 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.- 2 Linesearch Methods for Unconstrained Optimization.- 3 Trust-Region Methods for Unconstrained Optimization.- 4 Interior-Point Methods for Inequality Constrained Optimization.- 5 SQP Methods for Equality Constrained Optimization.- 6 Conclusion.- GniCodes - Matlab Programs for Geometric Numerical Integration.- 1 Problems to be Solved.- 2 Symplectic and Symmetric Integrators.- 3 Theoretical Foundation of Geometric Integrators.- 4 Matlab Programs of 'GniCodes'.- 5 Some Typical Applications.- References.- Numerical Approximations to Multiscale Solutions in PDEs.- 1 Introduction.- 2 Review of Homogenization Theory.- 3 Numerical Homogenization Based on Sampling Techniques.- 4 Numerical Homogenization Based on Multiscale FEMs.- 5 Wavelet-Based Homogenization (WBH).- 6 Variational MultiscaleMethod.- References.- Numerical Methods for Eigenvalue and Control Problems.- 1 Introduction.- 2 Classical Techniques for Eigenvalue Problems.- 3 Basics of Linear Control Theory.- 4 Hamiltonian Matrices and Riccati Equations.- 5 Numerical Solution of Hamiltonian Eigenvalue Problems.- 6 Large Scale Problems.- 7 Conclusion.- References.