Fall 2024 
Introduction to
Mathematical Methods for Modeling and Data Analysis
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Lecturer: Eli Galanti
Monday, 09:15 to 11:00, FGS, Rm B |
TA: Maayan Ziv
Wednesday, 11:15 to 12:00, FGS, Rm A |
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.
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1) Introduction
to modeling: ordinary differential equations Topics:
First order equations; Second order equations 2) Linear
equations – Eigen values and vectors Topics: Over and under
determined problems; System of linear ODEs
3) Introduction
to data analysis Topics:
Single time series analysis; Multiple time series analysis; Multivariate time series analysis 4) Mining
of Big Data Topics:
Clustering; Classification 5) Advanced
modeling: partial differential equations Topics:
Diffusion-advection equations 6) Combining
models and data: optimization of model initial conditions and parameters Topics: Optimization of
unconstrained linear problems, Optimization
of constrained nonlinear problems |
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Week 1
– Matlab example o
Week 2
– Notes o
Week 3 – Notes
Matlab example generate_sparse publish o
Week 4
– Notes
Matlab
example |
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Homework
1 - submit by 20.11.2024 o
Homework
2 – submit by 11.12.2024 |
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Reading |
o Strogarz,
S.H: Nonlinear Dynamics and Chaos. Perseus books, 1994 o
Cleve Moler: Numerical
Computing with MATLAB,
2004 o Gill,
Murray, and Wright: practical optimization (1981) |
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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 (Class
notes, Matlab examples) o Motivating
example: Medical tomography o Row
interpretations o Over-determined
- least squares solution o Under-determined
- SVD/pseudo inverse o Matlab
backslash operator vs. using matrix inverse o Sparse
matrices System of
linear ODEs - eigenvalues and eigenvectors (Class
notes) 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 (Class
notes) o Motivating
examples – Tropical Pacific sea surface temperature o Auto-correlations,
Composite analysis o Spectral
analysis (FFT) (Matlab
example) o Regression
and interpolation (Notes) Multiple
time series analysis (Class
notes) 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 (Class
notes) 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 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) o Convolution
Neural networks (CNNs) for image processing 5) Advanced
modeling: partial differential equations Diffusion-advection
equations (Notes) o Motivating
examples: COVID-19, WalMart o Derivation
of the diffusion-advection equation o Setting the
boundary condition o Numerical
solution - Finite differences (Matlab) o Steady state
solutions using eigenvalues o Implicit
solver – Euler backward 6) Combining models
and data: optimization of model initial conditions and parameters o Motivating
example – Juno gravity measurements Optimization
of unconstrained linear problems (Class notes) 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 (notes) 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) |
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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.
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