Fall 2022 
Introduction to
Mathematical Methods for Modeling and Data Analysis
|
Lecturer: Eli Galanti
Monday 09:15-11:00 (FGS, Rm A) |
TA: Keren Duer
Wednesday 11:15-12:00 (FGS, Rm C) |
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.
|
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; Regression and interpolation 4) Mining
of Big Data Topics:
Clustering; Classification 5) Advanced
modeling: partial differential equations Topics:
Diffusion-advection equations; Wave equations 6) Combining
models and data: optimization of model initial conditions and parameters Topics: Optimization of
unconstrained linear problems, Optimization
of constrained nonlinear problems |
|
||
|
o
Week 1
- matlab example o
Week 2 - notes o
Week 3 - notes,
example,
generate_sparse o
Week 5 - notes,
example,
data o
Week 6 - notes,
example1,
example2,
data o
Week 8 - notes,
example1,
example2,
data2.1,
data2.2 o
Week 10 - notes,
example,
data o
Week 12 - notes o
Week 13 -
notes,
example,
fun o
Week 14 - zip |
|
||
|
o Homework
1 - submit by 23.11.2022. P53, P185, matlab |
|
||
|
|
o
Homework
2 - submit by 07.12.2022. o
Homework
3 - submit by 21.12.2022. o
Homework
4 - submit by 04.01.2023. image o
Homework
5 - submit by 18.01.2023. o
Homework
6 - submit by 01.02.2023. o
Homework
7 -
submit by 19.02.2023. Solution |
|
|
|
|
|
|
|
|
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) |
|
|
|
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) 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 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) Recording 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 (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 Regression and interpolation (Class
notes) 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) pw=mynewnotes o Motivating
examples o Introduction
to machine learning o Perceptrons
(Matlab) o Neural
networks (Matlab) 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 Wave
equations (Notes) (Recording passcode: r3nD14) 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 (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) |
||
|
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.
|
|
||
