MTH 420-
520: MODELS AND METHODS OF APPLIED MATHEMATICS - Spring 2015
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Schedule and assignments |
Extra exercises (do not turn in, use to prepare for Exams)
- 1.4.2, 1.4.7, 1.4.9
- 1.5.1, 1.5.4-6, 1.5.24-25
- 2.1.3-5, 2.1.8-9, 2.1.12
- 2.2.1-2, 2.2.3-2.2.5
- 3.1.1, 3.1.3, 3.1.5-6
- 3.2.1-2; 3.2.5-6
- 3.6.1, 3.6.3
- 4.1.1, 4.1.2
- 3/28/16: Introduction and overview. Spring and mass model in equilibrium.
Read 1.3, 1.4 from book (review 1.1-1.2 if necessary.).
- 3/30/16: Symmetric positive definite matrices = postive eigenvalues.
Balance of forces and balance of external/internal work/minimization of energy functional.
Examples with n=2 and C=I.
- 4/1/16: Lab1: introduction, (positive definite) quadratic forms and (unconstrained) minimization
Report due 4/8/16 in class.
- 4/4/16: Symmetric positive definiteness = postive eigenvalues = positive pivots.
- 4/6/16: Least squares. Solving an overdetermined system using the normal equations
AT Ax=AT b. Fitting a linear model. The solution is a criticial point of a function phi(x)=||x-Ab||2.
- 4/8/16: Lab2 (Kidd 033): least squares (linear and nonlinear) Report due 4/16/16 in class
Also Extra materials to improve in MATLAB
- 4/11/16: Discrete models beyond mechanics. Stationary continuum
models: mechanics, heat conduction, diffusion,
flow/filtration. Hooke's law, Fourier's law, Fick's, law, Darcy's
law. Conservation of force, energy, mass.
Read 3.1 from book.
- 4/13/16: Discrete models of flow on networks.
Direction of maximum strech = direction of dominant eigenvector.
Read 2.1 from book. Handout from the book by Lay to prepare for Lab 3.
- 4/15/16: Lab 3: network models and Google PageRank.
Report due 4/22 in class
- 4/18/16: Minimization under constraints. Projections, LSQ, and Lagrange multipliers.
Read 2.1-2.2. Extra reading: 2.3 and 2.4.
- 4/20/16: Best approximation in inner product spaces. Continuum example: space C0[-1,1]. Handout MIN.
- 4/22/16: Lab 4: Best approximation in inner product spaces. Report due 5/2 in class.
- 4/25/16: Continuum models revisited. Solve a Boundary Value
Problem. [Read 3.1-3.2]. Continue example of best approximation in
P2 with inner product defined with integrals.
- 4/27/16: Continuum models: Neumann boundary conditions. Energy
functional and its minimization: Euler-Lagrange equations. [Read 3.6]
- 4/29/16: MIDTERM
- 5/2/16: finish E-L. Handout on Euler-Lagrange equations
- 5/4/16: Fourier series! [Read 4.1]
Extra Credit Opportunity DUE Friday, May 5: Solve problems in Parts A, B, and C
in Eigenvalues and Dynamics worksheet.
- 5/6/16: Lab 5:
Fourier series and applications. Report due 5/13 in class.
For the restless, Lab 5 EXTRA CREDIT.
- 5/9/16: Why Fourier basis ? (Find eigenvalues and eigenfunctions of the differential operator, with boundary conditions).
EXTRA OFFICE HOURS MONDAY, May 9, 5:00-6:00pm
EXTRA OFFICE HOURS TUESDAY, May 10, 12:00-1:00, and 2:00-3:00
- 5/11/16: (Guest lecture:) Complex form of Fourier series. [Read 4.2]
- 5/13/16: LAB 6: Discrete Fourier analysis and Music. Report due 5/23 in class
- 5/16/16: Guest lecture on Kalman filter. [Read 2.5]
OFFICE HOURS CANCELLED Monday and Wednesday
- 5/18/16: No class today.
"Take-home" MIDTERM 2 will be posted Wednesday May 18 at noon and will
be due Friday 5/20/16 in class.
- 5/20/16: LAB 7. Demonstration of machine learning and exercises, in three versions.
Instructions for alphas and mus: choose one of the levels and follow instructions in the code.
Omegas should find their own sets of "faces" (images) instead of those used in the code.
Report due 5/27 in class
- 5/23/16: SVD as a result of maximization under constraints. Sensitivity of solving linear systems [Re-read 1.6]
- 5/25/16: Introduction to inverse problems.
Gravity surveying problem. Fredholm Integral equation of
the first kind, and its numerical solution. Rough input becomes smooth
output in forward problem. Smooth input becomes rough output in an
inverse problem. Sensitivity of solving a linear system measured with
a condition number of a matrix.
- 5/27/16: LAB 8 SVD and Inverse problems lab. Report due 6/3 in class
- 6/1/16: GPS and geodesy models. Newton's method:
"When she was good, she was really really good. When she was bad, she was horrid"
- 6/3/16: CLASS REVIEW (meet in STAG 160). THE END.
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