CiNii 図書 - Frontiers in numerical analysis

TheEleventh LMS-EPSRCComputational MathematicsandScienti?cC- puting Summer School was held at the University of Durham, UK, from the 4th of July to the 9th of July 2004. This was the third of these schools to be held in Durham, having previously been hosted by the University of L- caster and the University of Leicester. The purpose of the summer school was to present high quality instructional courses on topics at the forefront of computational mathematics and scienti?c computing research to postgra- ate students. The main speakers were Emmanuel Candes, Markus Melenk, Joe Monaghan and Alex Schweitzer. This volume presents written contributions three of our speakers which are more comprehensive versions of the high quality lecture notes which were distributedtoparticipantsduringthemeeting.Wearealsoextremelypleased that Angela Kunoth was able to make an additional contribution from the ill-fated ?rst week. At the time of writing it is now more than two years since we ?rst contacted theguestspeakersandduringthatperiodtheyhavegivensigni?cantportions of their time to making the summer school, and this volume, a success. We wouldliketothankallofthemforthecarewhichtheytookinthepreparation and delivery of their material.

Wavelet-Based Multiresolution Methods for Stationary PDEs and PDE-Constrained Control Problems.- On Approximation in Meshless Methods.- Theory and Applications of Smoothed Particle Hydrodynamics.- Efficient Implementation and Parallelization of Meshfree and Particle Methods-The Parallel Multilevel Partition of Unity Method.

巻冊次: Durham 2002 : pbk ISBN 9783540443193

内容説明

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.

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.- 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 Optimization Reports on the Web.- C Sketches of Proofs.- GniCodes - Matlab Programs for Geometric Numerical Integration.- 1 Problems to be Solved.- 1.1 Hamiltonian Systems.- 1.2 Reversible Differential Equations.- 1.3 Hamiltonian and Reversible Systems on Manifolds.- 2 Symplectic and Symmetric Integrators.- 2.1 Simple Symplectic Methods.- 2.2 Simple Reversible Methods.- 2.3 Stoermer/Verlet Scheme.- 2.4 Splitting Methods.- 2.5 High Order Geometric Integrators.- 2.6 Rattle for Constrained Hamiltonian Systems.- 3 Theoretical Foundation of Geometric Integrators.- 3.1 Backward Error Analysis.- 3.2 Properties of the Modified Equation.- 3.3 Long-Time Behaviour of Geometric Integrators.- 4 Matlab Programs of 'GniCodes'.- 4.1 Standard Call of Integrators.- 4.2 Problem Description.- 4.3 Event Location.- 4.4 Program gni_irk2.- 4.5 Program gni_limn2.- 4.6 Program gni.comp.- 5 Some Typical Applications.- 5.1 Comparison of Geometric Integrators.- 5.2 Computation of Poincare Sections.- 5.3 'Rattle' as a Basic Integrator for Composition.- References.- Numerical Approximations to Multiscale Solutions in PDEs.- 1 Introduction.- 2 Review of Homogenization Theory.- 2.1 Homogenization Theory for Elliptic Problems.- 2.2 Homogenization for Hyperbolic Problems.- 2.3 Convection of Microstructure.- 3 Numerical Homogenization Based on Sampling Techniques.- 3.1 Convergence of the Particle Method.- 3.2 Vortex Methods for Incompressible Flows.- 4 Numerical Homogenization Based on Multiscale FEMs.- 4.1 Multiscale Finite Element Methods for Elliptic PDEs.- 4.2 Error Estimates (h?).- 4.4 The Over-Sampling Technique.- 4.5 Performance and Implementation Issues.- 4.6 Applications.- 5 Wavelet-Based Homogenization (WBH).- 5.1 Wavelets.- 5.2 Introduction to Wavelet-Based Homogenization (WBH).- 6 Variational Multiscale Method.- References.- Numerical Methods for Eigenvalue and Control Problems.- 1 Introduction.- 2 Classical Techniques for Eigenvalue Problems.- 2.1 The Schur Form and the QR-Algorithm.- 2.2 The Generahzed Schur Form and the QZ-Algorithm.- 2.3 The Singular Value Decomposition (SVD).- 2.4 The Arnoldi Algorithm.- 3 Basics of Linear Control Theory.- 3.1 Controllability and Stabilizability.- 3.2 System Equivalence.- 3.3 Optimal Control.- 4 Hamiltonian Matrices and Riccati Equations.- 4.1 The Hamiltonian Schur Form.- 4.2 Solution of the Optimal Control Problem via Riccati Equations.- 5 Numerical Solution of Hamiltonian Eigenvalue Problems.- 5.1 Subspace Computation.- 6 Large Scale Problems.- 7 Conclusion.- References.

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Frontiers in numerical analysis

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