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Spring 2021

Introduction to Mathematical Methods for Modeling and Data Analysis

 Lecturer: Eli Galanti  eli.galanti@weizmann.ac.il   Monday 09:15-11:00 (Zoom 2020) TA: Keren Duer keren.duer@weizmann.ac.il   Wednesday 12:15-13:00 (Zoom)

Now happening!

Submit by Wednesday, July 28, 5pm

Overview and goals

Mathematical models are present in all of the scientific disciplines, providing a quantitative framework for understanding and prediction of natural phenomena. The output from such models, as well as observations, often requires complex mathematical analysis. The course provides an introduction to mathematical modeling and data analysis through in-depth discussion of a series of real examples, with an emphasis on 'hands on' exercises. Upon successful completion of this course students should be able to: (1) Understand the principles of mathematical modeling and data analysis, (2) Solve analytically and numerically a wide range of problems.

 Outline 1)  Introduction to modeling: ordinary differential equations   2)  Linear equations – Eigen values and vectors   3)  Introduction to data analysis   4)   Mining of Big Data Topics: Clustering; Classification   5)  Advanced modeling: partial differential equations   6)  Combining models and data: optimization of model initial conditions and parameters Tutorials o   Week 1 - matlab example o   Week 2 - notes o   Week 3 - notes, matlab_class, matlab_generate_sparse o   Week 4 - notes, matlab_class o   Week 5 - notes, matlab_class, data o   Week 6 - notes, matlab_ex1, matlab_ex2, data o   Week 7 - notes, matlab_class o   Week 8 - notes, matlab_class o   Week 9 - notes, examples o   Week 10 - notes o   Week 11 - notes, example1, example2 o   Week 12 - notes o   Week 13 – matlab1, matlab2, solve o   Week 14 - all Homework o   submit by 14/04/2021. o   submit by 28/04/2021. o   submit by 14/05/2021. o   submit by 02/06/2021. Image o   submit by 16/06/2021. o   submit by 30/06/2021. o   submit by 14/07/2021. Reading Syllabus 1)     Introduction to modeling: ordinary differential equations   1st order equations (Class notes) o   Motivating examples: Population dynamics, global earth temperature o   Graphical solution method and stability analysis (Strogatz 21-26) o   Numerical solutions - Euler and improved Euler (Strogatz 32-33, matlab ode45)   2nd order equations (Class notes) o   Motivating examples: Love affairs (Strogatz 138-140), pendulum (Strogatz 168-173) o   Phase space and stability analysis (Strogatz 145-154) o   Rabbits versus sheep (Strogratz 155-159) (given in tutorials)   2)     Linear equations and eigen decomposition   Over and under determined problems o   Motivating example: Medical tomography o   Row interpretations (Matlab example) o   Over-determined - least squares solution o   Under-determined - SVD/pseudo inverse o   Matlab backslash operator vs. using matrix inverse (Matlab example) o   Sparse matrices (given in tutorials)   System of linear ODEs - eigenvalues and eigenvectors o   Motivating examples: Google PageRank, Romeo & Juliet o   Review of eigenvalues and eigenvectors (Strogatz 129-140) o   Back to Romeo & Juliet o   Non-linear equations - linearizing around fixed points (Strogatz 150-151) o   Analysis of damped pendulum   3)     Data analysis   Single time series analysis o   Motivating examples – Tropical Pacific sea surface temperature o   Mean, variance, STD, auto-correlations o   Composite analysis o   Spectral analysis (FFT) (Matlab example)   Multiple time series analysis o   Motivating examples: stock prices, ENSO o   The covariance matrix o   Principal Component Analysis (PCA) using the covariance matrix o   Fraction of variance explained o   Examples in 1D and 2D   Multivariate time series analysis o   Motivating examples: stock prices from two different markets, SST+winds o   Multivariate PCA o   Singular Value Decomposition (SVD) o   MPCA vs. SVD                                         o   Image compression   Regression and interpolation o   Correlation and regression o   Quality of fit with linear regression o   ﻿Connection between a time series and a time varying field o   ﻿Interpolation via nonlinear polynomial fit (notes)   4)     Mining of Big Data   Unsupervised learning: Clustering (notes) o   Motivating examples o   Distances: Euclidian (L2), Manhattan (L1), maximum (L∞), Jaccard. o   Hierarchical clustering, dendrogram, elbow plot (Matlab) o   K-means (Matlab)   Supervised learning: Classification (notes) o   Motivating examples o   Introduction to machine learning o   Perceptrons (Matlab) o   Neural networks (Matlab)   5)     Advanced modeling: partial differential equations   Diffusion-advection equations o   Motivating examples: HIV, Computer virus, WalMart o   Derivation of the diffusion-advection equation o   Setting the boundary condition o   Numerical solution - Finite differences (Matlab example) o   Steady state solutions using eigenvalues o   Implicit solver – Euler backward   Wave equations o   Motivating examples o   Derivation of the Wave equation o   Setting the initial and boundary conditions o   Numerical solution   6)     Combining models and data: optimization of model initial conditions and parameters o   Motivating example – Juno gravity measurements   Optimization of unconstrained linear problems o   Regression – linear model fitting o   A general linear case with no constraints o   Steepest descent (Gill 4.3.1, 4.3.2.2) o   Conjugate gradient (Gill 4.8.3) o   Iterative optimization of an unconstrained linear model o   Optimizing the strike of a pendulum   Optimization of constrained nonlinear problems o   The effect of nonlinearity on optimization – nonlinear pendulum o   Motivating example - diffusion-advection with a source o   Matlab `fmincon` (webpage) o   Solving a bounded and constrained problem (Matlab code) Administrative Prerequisites: This is a basic introductory breadth course and should be accessible to all Weizmann graduate students, given that they took university level introductory courses in linear algebra and calculus.  Requirements: o   Attendance of at least 80% of the lectures. o   Submission of all homework assignments by deadline. o   Submission of final assignment by deadline.  Grading:  Home assignments 70 points, final assignment 30 points.