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Data-Driven Modeling & Scientific Computation.pdf

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    Cover
    Contents
    Prolegomenon
    How to Use This Book
    About MATLAB
    PART I: Basic Computations and Visualization
    1 MATLAB Introduction
    1.1 Vectors and Matrices
    1.2 Logic, Loops and Iterations
    1.3 Iteration: The Newton–Raphson Method
    1.4 Function Calls, Input/Output Interactions and Debugging
    1.5 Plotting and Importing/Exporting Data
    2 Linear Systems
    2.1 Direct Solution Methods for Ax = b
    2.2 Iterative Solution Methods for Ax = b
    2.3 Gradient (Steepest) Descent for Ax = b
    2.4 Eigenvalues, Eigenvectors and Solvability
    2.5 Eigenvalues and Eigenvectors for Face Recognition
    2.6 Nonlinear Systems
    3 Curve Fitting
    3.1 Least-Square Fitting Methods
    3.2 Polynomial Fits and Splines
    3.3 Data Fitting with MATLAB
    4 Numerical Differentiation and Integration
    4.1 Numerical Differentiation
    4.2 Numerical Integration
    4.3 Implementation of Differentiation and Integration
    5 Basic Optimization
    5.1 Unconstrained Optimization (Derivative-Free Methods)
    5.2 Unconstrained Optimization (Derivative Methods)
    5.3 Linear Programming
    5.4 Simplex Method
    5.5 Genetic Algorithms
    6 Visualization
    6.1 Customizing Plots and Basic 2D Plotting
    6.2 More 2D and 3D Plotting
    6.3 Movies and Animations
    PART II: Differential and Partial Differential Equations
    7 Initial and Boundary Value Problems of Differential Equations
    7.1 Initial Value Problems: Euler, Runge–Kutta and Adams Methods
    7.2 Error Analysis for Time-Stepping Routines
    7.3 Advanced Time-Stepping Algorithms
    7.4 Boundary Value Problems: The Shooting Method
    7.5 Implementation of Shooting and Convergence Studies
    7.6 Boundary Value Problems: Direct Solve and Relaxation
    7.7 Implementing MATLAB for Boundary Value Problems
    7.8 Linear Operators and Computing Spectra
    8 Finite Difference Methods
    8.1 Finite Difference Discretization
    8.2 Advanced Iterative Solution Methods for Ax = b
    8.3 Fast Poisson Solvers: The Fourier Transform
    8.4 Comparison of Solution Techniques for Ax = b: Rules of Thumb
    8.5 Overcoming Computational Difficulties
    9 Time and Space Stepping Schemes: Method of Lines
    9.1 Basic Time-Stepping Schemes
    9.2 Time-Stepping Schemes: Explicit and Implicit Methods
    9.3 Stability Analysis
    9.4 Comparison of Time-Stepping Schemes
    9.5 Operator Splitting Techniques
    9.6 Optimizing Computational Performance: Rules of Thumb
    10 Spectral Methods
    10.1 Fast Fourier Transforms and Cosine/Sine Transform
    10.2 Chebychev Polynomials and Transform
    10.3 Spectral Method Implementation
    10.4 Pseudo-Spectral Techniques with Filtering
    10.5 Boundary Conditions and the Chebychev Transform
    10.6 Implementing the Chebychev Transform
    10.7 Computing Spectra: The Floquet–Fourier–Hill Method
    11 Finite Element Methods
    11.1 Finite Element Basis
    11.2 Discretizing with Finite Elements and Boundaries
    11.3 MATLAB for Partial Differential Equations
    11.4 MATLAB Partial Differential Equations Toolbox
    PART III: Computational Methods for Data Analysis
    12 Statistical Methods and Their Applications
    12.1 Basic Probability Concepts
    12.2 Random Variables and Statistical Concepts
    12.3 Hypothesis Testing and Statistical Significance
    13 Time–Frequency Analysis: Fourier Transforms and Wavelets
    13.1 Basics of Fourier Series and the Fourier Transform
    13.2 FFT Application: Radar Detection and Filtering
    13.3 FFT Application: Radar Detection and Averaging
    13.4 Time–Frequency Analysis: Windowed Fourier Transforms
    13.5 Time–Frequency Analysis and Wavelets
    13.6 Multi-Resolution Analysis and the Wavelet Basis
    13.7 Spectrograms and the Gábor Transform in MATLAB
    13.8 MATLAB Filter Design and Wavelet Toolboxes
    14 Image Processing and Analysis
    14.1 Basic Concepts and Analysis of Images
    14.2 Linear Filtering for Image Denoising
    14.3 Diffusion and Image Processing
    15 Linear Algebra and Singular Value Decomposition
    15.1 Basics of the Singular Value Decomposition (SVD)
    15.2 The SVD in Broader Context
    15.3 Introduction to Principal Component Analysis (PCA)
    15.4 Principal Components, Diagonalization and SVD
    15.5 Principal Components and Proper Orthogonal Modes
    15.6 Robust PCA
    16 Independent Component Analysis
    16.1 The Concept of Independent Components
    16.2 Image Separation Problem
    16.3 Image Separation and MATLAB
    17 Image Recognition: Basics of Machine Learning
    17.1 Recognizing Dogs and Cats
    17.2 The SVD and Linear Discrimination Analysis
    17.3 Implementing Cat/Dog Recognition in MATLAB
    18 Basics of Compressed Sensing
    18.1 Beyond Least-Square Fitting: The L[sup(1)] Norm
    18.2 Signal Reconstruction and Circumventing Nyquist
    18.3 Data (Image) Reconstruction from Sparse Sampling
    19 Dimensionality Reduction for Partial Differential Equations
    19.1 Modal Expansion Techniques for PDEs
    19.2 PDE Dynamics in the Right (Best) Basis
    19.3 Global Normal Forms of Bifurcation Structures in PDEs
    19.4 The POD Method and Symmetries/Invariances
    19.5 POD Using Robust PCA
    20 Dynamic Mode Decomposition
    20.1 Theory of Dynamic Mode Decomposition (DMD)
    20.2 Dynamics of DMD Versus POD
    20.3 Applications of DMD
    21 Data Assimilation Methods
    21.1 Theory of Data Assimilation
    21.2 Data Assimilation, Sampling and Kalman Filtering
    21.3 Data Assimilation for the Lorenz Equation
    22 Equation-Free Modeling
    22.1 Multi-Scale Physics: An Equation-Free Approach
    22.2 Lifting and Restricting in Equation-Free Computing
    22.3 Equation-Free Space–Time Dynamics
    23 Complex Dynamical Systems: Combining Dimensionality Reduction, Compressive Sensing and Machine Learning
    23.1 Combining Data Methods for Complex Systems
    23.2 Implementing a Dynamical Systems Library
    23.3 Flow Around a Cylinder: A Prototypical Example
    PART IV: Scientific Applications
    24 Applications of Differential Equations and Boundary Value Problems
    24.1 Neuroscience and the Hodgkin–Huxley Model
    24.2 Celestial Mechanics and the Three-Body Problem
    24.3 Atmospheric Motion and the Lorenz Equations
    24.4 Quantum Mechanics
    24.5 Electromagnetic Waveguides
    25 Applications of Partial Differential Equations
    25.1 The Wave Equation
    25.2 Mode-Locked Lasers
    25.3 Bose–Einstein Condensates
    25.4 Advection–Diffusion and Atmospheric Dynamics
    25.5 Introduction to Reaction–Diffusion Systems
    25.6 Steady State Flow Over an Airfoil
    26 Applications of Data Analysis
    26.1 Analyzing Music Scores and the Gábor Transform
    26.2 Image Denoising through Filtering and Diffusion
    26.3 Oscillating Mass and Dimensionality Reduction
    26.4 Music Genre Identification
    References
    Index of MATLAB Commands
    A
    B
    C
    D
    E
    F
    G
    H
    I
    L
    M
    N
    O
    P
    Q
    R
    S
    T
    U
    V
    W
    X
    Y
    Index
    A
    B
    C
    D
    E
    F
    G
    H
    I
    J
    K
    L
    M
    N
    O
    P
    Q
    R
    S
    T
    U
    V
    W
    Data-Driven Modeling & Scientific Computation
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    Data-Driven Modeling & Scientific Computation Methods for Complex Systems & Big Data J. NATHAN KUTZ Department of Applied Mathematics University of Washington 3
    3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c J. Nathan Kutz 2013 The moral rights of the author have been asserted First Edition published in 2013 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2013937977 ISBN 978–0–19–966033–9 (hbk.) ISBN 978–0–19–966034–6 (pbk.) Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.
    Dedication For Kristy, Lauren and Katie
    Acknowledgments The idea of the first part of this book began as a series of conversations with Dave Muraki. It then grew into the primary set of notes for a scientific computing course whose ambition was to provide a truly versatile and useful course for students in the engineering, biological and phys- ical sciences. And over the last couple of years, the book expanded to included methods for data analysis, thus bolstering the intellectual scope of the book significantly. Unbeknownst to them, much of the data analysis portion of the book was heavily inspired by the fantastic works of Emmanuel Candés, Yannis Kevrekidis and Clancy Rowley and various conversations I had with each of them. I’ve also benefitted greatly from early discussions with James Rossmanith, and with implementation ideas with Peter Blossey and Sorin Mitran; and more recently on dimension- ality reduction methods with Steven Brunton, Edwin Ding, Joshua Proctor, Peter Schmid, Eli Shlizerman, Jonathan Tu and Matthew Williams. Leslie Butson, Sarah Hewitt and Jennifer O’Neil have been very helpful in editing the book so that it is more readable, useful and error-free. A special thanks should also be given to all the many wonderful students who have provided so much critical commentary and vital feedback for improving the delivery, style and correctness of the book. Of course, all errors in this book are the fault of my daughters’ hamsters Fluffy and Quickles.
    Contents Prolegomenon How to Use This Book About MATLAB PART I Basic Computations and Visualization 1 MATLAB Introduction 1.1 Vectors and Matrices 1.2 Logic, Loops and Iterations 1.3 Iteration: The Newton–Raphson Method 1.4 Function Calls, Input/Output Interactions and Debugging 1.5 Plotting and Importing/Exporting Data 2 Linear Systems 2.1 Direct Solution Methods for Ax = b 2.2 Iterative Solution Methods for Ax = b 2.3 Gradient (Steepest) Descent for Ax = b 2.4 Eigenvalues, Eigenvectors and Solvability 2.5 Eigenvalues and Eigenvectors for Face Recognition 2.6 Nonlinear Systems 3 Curve Fitting 3.1 Least-Square Fitting Methods 3.2 Polynomial Fits and Splines 3.3 Data Fitting with MATLAB 4 Numerical Differentiation and Integration 4.1 Numerical Differentiation 4.2 Numerical Integration 4.3 Implementation of Differentiation and Integration xiii xv xviii 3 3 9 13 18 23 31 31 35 39 44 49 56 61 61 65 69 77 77 83 87
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