The burgeoning field of data analysis is expanding at an incredible pace due to the proliferation of data collection in almost every area of science. The enormous data sets now routinely encountered in the sciences provide an incentive to develop mathematical techniques and computational
algorithms that help synthesize, interpret and give meaning to the data in the context of its scientific setting. A specific aim of this book is to integrate standard scientific computing methods with data analysis. By doing so, it brings together, in a self-consistent fashion, the key ideas from:
DT statistics,
DT time-frequency analysis, and
DT low-dimensional reductions
The blend of these ideas provides meaningful insight into the data sets one is faced with in every scientific subject today, including those generated from complex dynamical systems. This is a particularly exciting field and much of the final part of the book is driven by intuitive examples from it,
showing how the three areas can be used in combination to give critical insight into the fundamental workings of various problems.
Data-Driven Modeling and Scientific Computation is a survey of practical numerical solution techniques for ordinary and partial differential equations as well as algorithms for data manipulation and analysis. Emphasis is on the implementation of numerical schemes to practical problems in the
engineering, biological and physical sciences.
An accessible introductory-to-advanced text, this book fully integrates MATLAB and its versatile and high-level programming functionality, while bringing together computational and data skills for both undergraduate and graduate students in scientific computing.

Prolegomenon xiii

How to Use This Book xv

About MATLAB xviii

PART I Basic Computations and Visualization

1 MATLAB Introduction 3

      1.1 Vectors and Matrices 3

      1.2 Logic, Loops and Iterations 9

      1.3 Iteration: The Newton-Raphson Method 13

      1.4 Function Calls, Input/Output Interactions and Debugging 18

      1.5 Plotting and Importing/Exporting Data 23

2 Linear Systems 31

      2.1 Direct Solution Methods for A x=b 31

      2.2 Iterative Solution Methods for A x=b 35

      2.3 Gradient(Steepest) Descent for A x=b 39

      2.4 Eigenvalues, Eigenvectors and Solvability 44

      2.5 Eigenvalues and Eigenvectors for Face Recognition 49

      2.6 Nonlinear Systems 56

3 Curve Fitting 61

      3.1 Least-Square Fitting Methods 61

      3.2 Polynomial Fits and Splines 65

      3.3 Data Fitting with MATLAB 69

4 Numerical Differentiation and Integration 77

      4.1 Numerical Differentiation 77

      4.2 Numerical Integration 83

      4.3 Implementation of Differentiation and Integration 87

5 Basic Optimization 93

      5.1 Unconstrained Optimization (Derivative-Free Methods) 93

      5.2 Unconstrained Optimization (Derivative Methods) 99

      5.3 Linear Programming 105

      5.4 Simplex Method 110

      5.5 Genetic Algorithms 113

6 Visualization 119

      6.1 Customizing Plots and Basic 2D Plotting 119

      6.2 More 2D and 3D Plotting 125

      6.3 Movies and Animations 131

PART II Differential and Partial Differential Equations

7 Initial and Boundary Value Problems of Differential Equations 137

      7.1 Initial Value Problems: Euler, Runge-Kutta and Adams Methods 137

      7.2 Error Analysis for Time-Stepping Routines 144

      7.3 Advanced Time-Stepping Algorithms 149

      7.4 Boundary Value Problems: The Shooting Method 153

      7.5 Implementation of Shooting and Convergence Studies 160

      7.6 Boundary Value Problems: Direct Solve and Relaxation 164

      7.7 Implementing MATLAB for Boundary Value Problems 167

      7.8 Linear Operators and Computing Spectra 172

8 Finite Difference Methods 180

      8.1 Finite Difference Discretization 180

      8.2 Advanced Iterative Solution Methods for A x=b 186

      8.3 Fast Poisson Solvers: The Fourier Transform 186

      8.4 Comparison of Solution Techniques for A x=b: Rules of Thumb 190

      8.5 Overcoming Computational Difficulties 195

9 Time and Space Stepping Schemes: Method of Lines 200

      9.1 Basic Time-Stepping Schemes 200

      9.2 Time-Stepping Schemes: Explicit and Implicit Methods 205

      9.3 Stability Analysis 209

      9.4 Comparison of Time-Stepping Schemes 213

      9.5 Operator Splitting Techniques 216

      9.6 Optimizing Computational Performance: Rules of Thumb 219

10 Spectral Methods 225

      10.1 Fast Fourier Transforms and Cosine/Sine Transform 225

      10.2 Chebychev Polynomials and Transform 229

      10.3 Spectral Method Implementation 233

      10.4 Pseudo-Spectral Techniques with Filtering 235

      10.5 Boundary Conditions and the Chebychev Transform 240

      10.6 Implementing the Chebychev Transform 244

      10.7 Computing Spectra: The Floquet-Fourier-Hill Method 249

11 Finite Element Methods 256

      11.1 Finite Element Basis 256

      11.2 Discretizing with Finite Elements and Boundaries 261

      11.3 MATLAB for Partial Differential Equations 266

      11.4 MATLAB Partial Differential Equations Toolbox 271

PART III Computational Methods for Data Analysis

12 Statistical Methods and Their Applications 279

      12.1 Basic Probability Concepts 279

      12.2 Random Variables and Statistical Concepts 286

      12.3 Hypothesis Testing and Statistical Significance 294

13 Time-Frequency Analysis: Fourier Transforms and Wavelets 301

      13.1 Basics of Fourier Series and the Fourier Transform 301

      13.2 FFT Application: Radar Detection and Filtering 308

      13.3 FFT Application: Radar Detection and Averaging 316

      13.4 Time-Frequency Analysis: Windowed Fourier Transforms 322

      13.5 Time-Frequency Analysis and Wavelets 328

      13.6 Multi-Resolution Analysis and the Wavelet Basis 335

      13.7 Spectrograms and the Gabor Transform in MATLAB 340

      13.8 MATLAB Filter Design and Wavelet Toolboxes 346

14 Image Processing and Analysis 358

      14.1 Basic Concepts and Analysis of Images 358

      14.2 Linear Filtering for Image Denoising 364

      14.3 Diffusion and Image Processing 369

15 Linear Algebra and Singular Value Decomposition 376

      15.1 Basics of the Singular Value Decomposition (SVD) 376

      15.2 The SVD in Broader Context 381

      15.3 Introduction to Principal Component Analysis (PCA) 387

      15.4 Principal Components, Diagonalization and SVD 391

      15.5 Principal Components and Proper Orthogonal Modes 395

      15.6 Robust PCA 403

16 Independent Component Analysis 412

      16.1 The Concept of Independent Components 412

      16.2 Image Separation Problem 419

      16.3 Image Separation and MATLAB 424

17 Image Recognition: Basics of Machine Learning 431

      17.1 Recognizing Dogs and Cats 431

      17.2 The SVD and Linear Discrimination Analysis 436

      17.3 Implementing Cat/Dog Recognition in MATLAB 445

18 Basics of Compressed Sensing 449

      18.1 Beyond Least-SquareFitting: The L1 Norm 449

      18.2 Signal Reconstruction and Circumventing Nyquist 456

      18.3 Data (Image) Reconstruction from Sparse Sampling 464

19 Dimensionality Reduction for Partial Differential Equations 472

      19.1 Modal Expansion Techniques for PDEs 472

      19.2 PDE Dynamics in the Right (Best) Basis 478

      19.3 Global Normal Forms of Bifurcation Structures in PDEs 482

      19.4 The POD Method and Symmetries/In variances 492

      19.5 POD Using Robust PCA 499

20 Dynamic Mode Decomposition 506

      20.1 Theory of Dynamic Mode Decomposition (DMD) 506

      20.2 Dynamics of DMD Versus POD 510

      20.3 Applications of DMD 515

21 Data Assimilation Methods 521

      21.1 Theory of Data Assimilation 521

      21.2 Data Assimilation, Sampling and Kalman Filtering 526

      21.3 Data Assimilation for the Lorenz Equation 529

22 Equation-Free Modeling 537

      22.1 Multi-Scale Physics: An Equation-Free Approach 537

      22.2 Lifting and Restricting in Equation-Free Computing 542

      22.3 Equation-Free Space-Time Dynamics 547

23 Complex Dynamical Systems: Combining Dimensionality Reduction,Compressive Sensing and Machine Learning 551

      23.1 Combining Data Methods for Complex Systems 551

      23.2 Implementing a Dynamical Systems Library 556

      23.3 Flow Around a Cylinder: A Prototypical Example 564

PART IV Scientific Applications

24 Applications of Differential Equations and Boundary Value Problems 573

      24.1 Neuroscience and the Hodgkin-Huxley Model 573

      24.2 Celestial Mechanics and the Three-Body Problem 577

      24.3 Atmospheric Motion and the Lorenz Equations 581

      24.4 Quantum Mechanics 585

      24.5 Electromagnetic Waveguides 588

25 Applications of Partial Differential Equations 590

      25.1 The Wave Equation 590

      25.2 Mode-Locked Lasers 593

      25.3 Bose-Einstein Condensates 600

      25.4 Advection-Diffusion and Atmospheric Dynamics 604

      25.5 Introduction to Reaction-Diffusion Systems 611

      25.6 Steady State Flow Over an Airfoil 616

26 Applications of Data Analysis 620

      26.1 Analyzing Music Scores and the Gabor Transform 620

      26.2 Image Denoising through Filtering and Diffusion 622

      26.3 Oscilating Mass and Dimensionality Reduction 625

      26.4 Music Genre Identification 626

References 629

Index of MATLAB Commands 634

Index 636

J. Nathan Kutz, Professor of Applied Mathematics, University of Washington PA\Professor Kutz is the Robert Bolles and Yasuko Endo Professor of Applied Mathematics at the University of Washington. Prof. Kutz was awarded the B.S. in physics and mathematics from the University of Washington (Seattle, WA) in 1990 and the PhD in Applied Mathematics from Northwestern University (Evanston, IL) in 1994. He joined the Department of Applied Mathematics, University of Washington in 1998 and became Chair in 2007. PA\Professor Kutz is especially interested in a unified approach to applied mathematics that includes modeling, computation and analysis. His area of current interest concerns phenomena in complex systems and data analysis (dimensionality reduction, compressive sensing, machine learning), neuroscience (neuro-sensory systems, networks of neurons), and the optical sciences (laser dynamics and modelocking, solitons, pattern formation in nonlinear optics).

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