Data-Driven Modeling. Scientific Computation J. NATHAN KUTZ OXPORD. Methods for Complex Systems & Big Data

Transcription

1 Data-Driven Modeling & Scientific Computation Methods for Complex Systems & Big Data J. NATHAN KUTZ Department ofapplied Mathematics University of Washington OXPORD UNIVERSITY PRESS

2 Contents Prolegomenon How to Use This Book About MATLAB xiii xv xviii PART I Basic Computations and Visualization MATLAB Introduction Vectors and Matrices Logic, Loops and Iterations Iteration: The Newton-Raphson Method Function Calls, Input/Output Interactions and Debugging Plotting and Importing/Exporting Data 23 tfl Linear Systems Direct Solution Methods for Ax = b Iterative Solution Methods for Ax = b Gradient (Steepest) Descent for Ax = b Eigenvalues, Eigenvectors and Solvability Eigenvalues and Eigenvectors for Face Recognition Nonlinear Systems 56 Bfl Curve Fitting Least-Square Fitting Methods Polynomial Fits and Splines Data Fitting with MATLAB 69 Numerical Differentiation and Integration Numerical Differentiation Numerical Integration Implementation of Differentiation and Integration 87

3 Vlii CONTENTS ^9 Basic Optimization Unconstrained Optimization (Derivative-Free Methods) Unconstrained Optimization (Derivative Methods) Linear Programming Simplex Method Genetic Algorithms 113 ^9 Visualization Customizing Plots and Basic 2D Plotting More 2D and 3D Plotting Movies and Animations 131 ^^^Jf^0ifFerentiaf and Partial Differential Equations ^pnitial and Boundary Value Problems of Differential Equations Initial Value Problems: Euler, Runge-Kutta and Adams Methods Error Analysis for Time-Stepping Routines Advanced Time-Stepping Algorithms Boundary Value Problems: The Shooting Method Implementation of Shooting and Convergence Studies Boundary Value Problems: Direct Solve and Relaxation Implementing MATLAB for Boundary Value Problems Linear Operators and Computing Spectra 172 Kfl Finite Difference Methods Finite Difference Discretization Advanced Iterative Solution Methods for Ax b = Fast Poisson Solvers: The Fourier Transform Comparison of Solution Techniques for Ax = b: Rules of Thumb Overcoming Computational Difficulties 195 UgM Time and Space Stepping Schemes: Method of Lines Basic Time-Stepping Schemes Time-Stepping Schemes: Explicit and Implicit Methods Stability Analysis 209

4 CONTENTS ix 9.4 Comparison of Time-Stepping Schemes Operator Splitting Techniques Optimizing Computational Performance: Rules of Thumb 219 MUM Spectral Methods Fast Fourier Transforms and Cosine/Sine Transform Chebychev Polynomials and Transform Spectral Method Implementation Pseudo-Spectral Techniques with Filtering Boundary Conditions and the Chebychev Transform Implementing the Chebychev Transform Computing Spectra: The Floquet-Fourier-Hill Method 249! Finite Element Methods Finite Element Basis Discretizing with Finite Elements and Boundaries MATLAB for Partial Differential Equations MATLAB Partial Differential Equations Toolbox 271 PART III Computational Methods for Data Analysis M Statistical Methods and Their Applications Basic Probability Concepts Random Variables and Statistical Concepts Hypothesis Testing and Statistical Significance 294 EQ Time-Frequency Analysis: Fourier Transforms and Wavelets Basics of Fourier Series and the Fourier Transform FFT Application: Radar Detection and Filtering FFT Application: Radar Detection and Averaging Time-Frequency Analysis: Windowed Fourier Transforms Time-Frequency Analysis and Wavelets Multi-Resolution Analysis and the Wavelet Basis Spectrograms and the Gabor Transform in MATLAB MATLAB Filter Design and Wavelet Toolboxes 346

5 X CONTENTS ULM Image Processing and Analysis Basic Concepts and Analysis of Images Linear Filtering for Image Denoising Diffusion and Image Processing 369 M Linear Algebra and Singular Value Decomposition Basics of the Singular Value Decomposition (SVD) The SVD in Broader Context Introduction to Principal Component Analysis (PCA) Principal Components, Diagonalization and SVD Principal Components and Proper Orthogonal Modes Robust PCA Independent Component Analysis The Concept of Independent Components Image Separation Problem Image Separation and MATLAB 424 M Image Recognition: Basics of Machine Learning Recognizing Dogs and Cats The SVD and Linear Discrimination Analysis Implementing Cat/Dog Recognition in MATLAB 445 Bfl Basics of Compressed Sensing Beyond Least-Square Fitting: The L1 Norm Signal Reconstruction and Circumventing Nyquist Data (Image) Reconstruction from Sparse Sampling 464 UigM Dimensionality Reduction for Partial Differential Equations Modal Expansion Techniques for PDEs PDE Dynamics in the Right (Best) Basis Global Normal Forms of Bifurcation Structures in PDEs The POD Method and Symmetries/lnvariances POD Using Robust PCA 499

6 CON1ENTS Xi Mil Dynamic Mode Decomposition Theory of Dynamic Mode Decomposition (DMD) Dynamics of DMD Versus POD Applications of DMD 515 WSM Data Assimilation Methods Theory of Data Assimilation Data Assimilation, Sampling and Kalman Filtering Data Assimilation for the Lorenz Equation 529 M Equation-Free Modeling Multi-Scale Physics: An Equation-Free Approach Lifting and Restricting in Equation-Free Computing Equation-Free Space-Time Dynamics 547 E l Complex Dynamical Systems: Combining Dimensionality Reduction, Compressive Sensing and Machine Learning Combining Data Methods for Complex Systems Implementing a Dynamical Systems Library Flow Around a Cylinder: A Prototypical Example 564 PART IV Scientific Applications E J Applications of Differential Equations and Boundary Value Problems Neuroscience and the Hodgkin-Huxley Model Celestial Mechanics and the Three-Body Problem Atmospheric Motion and the Lorenz Equations Quantum Mechanics Electromagnetic Waveguides 588 HJ Applications of Partial Differential Equations The Wave Equation Mode-Locked Lasers Bose-Einstein Condensates 600

7 Xii CONTENTS 25.4 Advection-Diffusion and Atmospheric Dynamics Introduction to Reaction-Diffusion Systems Steady State Flow Over an Airfoil 616 Hj9 Applications of Data Analysis Analyzing Music Scores and the Cabor Transform Image Denoising through Filtering and Diffusion Oscillating Mass and Dimensionality Reduction Music Genre Identification 626 References 629 Index ofmatlab Commands 634 Index 636