Inhalt

•  
   1 Introduction
•  
   Outline of the thesis
•  
  2 Linear Matrix Decompositions
•  
   2.1 Diagonalisation of a Matrix
•  
   2.2 Singular Value Decomposition
•  
   2.3 Orthogonal Projections
•  
   2.4 Gram-Schmidt Orthonormalisation
•  
   2.5 The LU Decomposition
•  
   2.5.1 Matrix Inversion by LU Decomposition
•  
   2.6 Determinant of a Matrix and Characteristic Polynomial
•  
   2.7 Inverse and Pseudo-inverse of a Matrix
•  
   2.8 The QR decomposition
•  
   2.9 The Schur decomposition
•  
   2.9.1 Ordering of the Schur decomposition
•  
   3 Partial Differential Equations
•  
   3.1 Classifications
•  
   3.2 Method of characteristics
•  
   3.3 Linearity
•  
   3.4 Well posed problems
•  
  3.5 Numerical Treatment
•  
   3.5.1 Integration of ODEs
•  
   3.5.2 Finite differencing for PDEs
•  
   Boundary conditions
•  
   Matrix notation
•  
   Boundary conditions and Matrix notation
•  
   3.5.3 Finite Element Methods for PDEs
•  
   Galerkin Methods
•  
   Implementation of Boundary Conditions
•  
   3.5.4 Spectral Methods
•  
   Boundary conditions
•  
   Derivation
•  
   4 Master Equation Description
•  
   Dimensionality versus Linearity
•  
  Stochasticity from Missing Information
•  
   Time Evolution
•  
  4.1 Constructing the Master Operator
•  
   Source and Sink Operator
•  
   Diffusion Operator
•  
   Annihilation Operator
•  
   4.1.1 Probability Conservation and Steady State
•  
  5 Dynamical Systems and Model Reduction
•  
   5.1 Problem Setup
•  
   Linear Systems
•  
   Time Discrete Systems
•  
   5.2 Model Reduction
•  
   5.2.1 Choice of error functional
•  
   5.2.2 Reduced Model
•  
   5.2.3 Linear Projection
•  
   5.2.4 Nonlinear Reductions
•  
   5.2.5 Optimal Model Reduction for Linear Dynamics
•  
  5.2.6 Optimising Model Reduction for Nonlinear Dynamics
•  
   Linearisation
•  
   Minimisation Approach
•  
   5.2.7 Proper Orthogonal Decomposition
•  
   6 Density Matrix Renormalisation Group
•  
   6.1 Infinite System Method
•  
   6.2 Finite System Iteration
•  
   6.3 Reconstruction of States
•  
  6.4 Single Particle vs. Many Particle DMRG
•  
   Single Particle Systems
•  
   Many Body Systems
•  
  7 Proposed Methods
•  
   7.1 Real Schur DMRG
•  
   7.1.1 Technical Implementation
•  
  7.2 Proper Orthogonal Decomposition DMRG
•  
   7.2.1 Technical Implementation
•  
  7.3 General Variational Method for Proper Orthogonal Decomposition
•  
   7.3.1 Technical Implementation
•  
   7.3.2 Spectral variant
•  
  8 Microscopic Models
•  
   8.1 Reaction Diffusion System
•  
  8.1.1 Numerical Results
•  
   Normalisation of the Results
•  
   Average Density Profile
•  
   Nearest Neighbour Density Correlation
•  
  8.2 Surface Deposition Model
•  
   Continuous Equation
•  
   Microscopic Model
•  
   -Model
•  
   -Model
•  
  8.2.1 Numerical Results
•  
   Normalisation of the Results
•  
   Average Surface Step Size
•  
   Correlation of Surface Steps
•  
  8.2.2 POD results
•  
   Stochastic Simulation
•  
   Simulation of the KPZ Equation
•  
  9 Proper Orthogonal Decomposition DMRG
•  
   9.1 The Linear Diffusion Equation
•  
   9.2 The Burgers Equation
•  
   9.3 Nonlinear Diffusion
•  
   9.4 Computational Load
•  
  10 General Variational Method for Proper Orthogonal Decomposition
•  
   10.1 Flow Problems
•  
   10.1.1 Navier Stokes Equations and 2D Flows
•  
   10.2 Model Problem
•  
   10.3 Numerical Integration
•  
  10.4 Numerical Results
•  
   Snapshots of the Flow
•  
  10.4.1 Comparing the Accuracy
•  
   Effect of the Reynolds Number
•  
   Effects of the Sweeps
•  
   Effects of Different Numbers of Retained States
•  
   Effect of the Spatial Resolution
•  
   Variational POD versus the Spectral Variant
•  
   Evaluation of the Relative Error
•  
   Visualisation of the POD and V-POD Modes
•  
   Visualisation of the Error Evolution
•  
  11 Conclusions
•  
   11.1 Schur DMRG
•  
   Future Prospects
•  
   11.2 POD DMRG
•  
   Future Prospects
•  
   11.3 Variational POD
•  
   Future Prospects
•  
  A Finite Numerical Precision
•  
  Fixed-Point Representation
•  
   Floating-Point Representation
•  
   Stability
•  
  B Optimal Reduction of Linear Systems
•  
   B.1 Long Time Optimised Projection
•  
   B.2 Short Time Optimised Projection
•  
   C Ordering of the Schur Decomposition
•  
  D Mathematical Addenda
•  
  D.1 Basic notation
•  
   D.1.1 Fields
•  
   D.1.2 Vectors
•  
   D.2 Vector Space Axioms
•  
   D.2.1 Scalar product, Norm
•  
   D.2.2 Basis
•  
   D.2.3 Subspaces
•  
   D.2.4 Linear Transformations, Matrixes
•  
   Range, Kernel and Rank
•  
   D.2.5 Tensors