2. Dogs and Cats AMATH 582 – Computational Methods for Data Analysis CatsDogs

3. Low-Dimensional Dogs and Cats: “Eigen- Dogs/Cats”

4. Linear Discrimination Analysis R. A. Fisher 1936: Classification of Species Find a projection that maximizes the statistical distance between two random data sets

5. Histogram of the Data 95% Accurate

6. Misclassified Dogs It’s the ears!!!!

7. PDEs A backbone of AMATH (fluids) Getting the right basis - POD modes Minimal dynamics - dynamical systems of the cheap Getting the right basis - POD modes Minimal dynamics - dynamical systems of the cheap

8. How to solve a PDE: 3 easy steps * Sad fact: This is all you know how to do! PDE ODE Algebra* Undo 12 3 Eigenfunction expansion (separation of variables) - u(x,t) = Σ a n (t) ϕ n (x)

9. Two Soliton Solution of a nonlinear PDE Kutz, Data Driven Modeling Sciand entific Computing (Oxford)

10. SVD on Data Kutz, Data Driven Modeling Sciand entific Computing (Oxford)

11. Galerkin Expansion Kutz, Data Driven Modeling Sciand entific Computing (Oxford)

12. Nonlinear Schroedinger Equation Kutz, Data Driven Modeling Sciand entific Computing (Oxford)

13. Water Waves: Governing Equations V. E. Zakharov. “Stability of periodic waves of finite amplitude on the surface of a deep fluid.” Zh. Prikl. Mekh. Tekh. Fiz. (1968).

14. Low Amplitude Waves

15. High Amplitude Waves

16. Mode Structure

17. PCA (POD) Convergence P. Holmes, J. Lumley, and G. Berkooz. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge U. Press (1996) The n mode POD basis of a set of data captures more of the L 2 norm of that data than any other linear set of n modes:

20. www.mathworks/moler/eigs.pdf See also http://people.maths.ox.ac.uk/trefethen/text.html