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The Tenth LMS-EPSRC Numerical Analysis Summer School was held at the University of Durham, UK, from the 7th to the 19th of July 2002. This was the second of these schools to be held in Durham, having previously been hosted by the University of Lancaster and the University of Leicester. The purpose of the summer school was to present high quality instructional courses on topics at the forefront of numerical analysis research to postgraduate students. The speakers were Franco Brezzi, Gerd Dziuk, Nick Gould, Ernst Hairer, Tom Hou and Volker Mehrmann. This volume presents written contributions from all six speakers which are more comprehensive versions of the high quality lecture notes which were distributed to participants during the meeting. At the time of writing it is now more than two years since we first contacted the guest speakers and during that period they have given significant portions of their time to making the summer school, and this volume, a success. We would like to thank all six of them for the care which they took in the preparation and delivery of their material.
Inhaltsverzeichnis
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.
Zusammenfassung
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.
Zusatztext
From the reviews:"The book contains 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 … . most of the notes contain detailed proofs of the key results. The notes start from a level suitable for first year graduate students … and proceed to current research topics. The reader should therefore be able to gain quickly an insight into the important results and techniques … ." (Zentralblatt für Didaktik der Mathematik, January, 2004)“There are seven contributions from the six speakers arranged according to increasing degree of specification. … Each contribution presents a list of references, and advice for further investigation is included. Also, detailed proofs of the main results are given in most of the notes. Because of its high quality level, its dealing with topics of current research in the area, the book is to be recommended foe mathematicians, applied mathematicians and engineers as well as for graduates in mathematical sciences.” (Paula Bruggen, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)
Bericht
From the reviews:
"The book contains 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 ... . most of the notes contain detailed proofs of the key results. The notes start from a level suitable for first year graduate students ... and proceed to current research topics. The reader should therefore be able to gain quickly an insight into the important results and techniques ... ." (Zentralblatt für Didaktik der Mathematik, January, 2004)
"There are seven contributions from the six speakers arranged according to increasing degree of specification. ... Each contribution presents a list of references, and advice for further investigation is included. Also, detailed proofs of the main results are given in most of the notes. Because of its high quality level, its dealing with topics of current research in the area, the book is to be recommended foe mathematicians, applied mathematicians and engineers as well as for graduates in mathematical sciences." (Paula Bruggen, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)
