Data-Driven Modeling

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: * statistics, * time-frequency analysis, and * 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.I BASIC COMPUTATIONS AND VISUALIZATION ; 1. MATLAB Introduction ; 2. Linear Systems ; 3. Curve Fitting ; 4. Numerical Differentiation and Integration ; 5. Basic Optimization ; 6. Visualization ; II DIFFERENTIAL AND PARTIAL DIFFERENTIAL EQUATIONS ; 7. Initial and Boundary Value Problems of Differential Equations144 ; 8. Finite Difference Methods ; 9. Time and Space Stepping Schemes: Method of Lines ; 10. Spectral Methods ; 11. Finite Element Methods ; III COMPUTATIONAL METHODS FOR DATA ANALYSIS ; 12. Statistical Methods and Their Applications ; 13. Time-Frequency Analysis: Fourier Transforms and Wavelets ; 14. Image Processing and Analysis ; 15. Linear Algebra and Singular Value Decomposition ; 16. Independent Component Analysis ; 17. Image Recognition ; 18. Basics of Compressed Sensing ; 19. Dimensionality Reduction for Partial Differential Equations ; 20. Dynamic Mode Decomposition ; 21. Data Assimilation Methods ; 22. Equation Free Modeling ; IV SCIENTIFIC APPLICATIONS ; 23. Applications of Differential Equations and Boundary Value Problems ; 24. Quantum Mechanics ; 25. Applications of Partial Differential Equations ; 26. Applications of Data Analysis