MTH 420- 520: MODELS AND METHODS OF APPLIED MATHEMATICS - Spring 2015
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General information
Assignments
Schedule and assignments
  1. 3/30/15: Introduction and overview. Spring and mass model in equilibrium. Positive definite matrices.
  2. 4/1/15: Balance of forces and balance of external/internal work/minimization of energy functional. Examples with n=2 and C=I.
    Read 1.3, 1.4 from book (review 1.1-1.2 if necessary. ).
  3. 4/3/15: Lab1 (Kidd 033): introduction, (postivie definite) quadratic forms and (unconstrained) minimization Report due 4/10/15 in class.
  4. 4/6/15: Symmetric positive definiteness = postive eigenvalues = positive pivots. Direction of maximum strech = direction of dominant eigenvector.
  5. 4/8/15: 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.
  6. 4/10/15: Lab2 (Kidd 033): least squares (linear and nonlinear) Report due 4/17/15 in class
  7. 4/13/15: 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.
  8. 4/15/15: Discrete models of flow on networks.
    Read 2.1 from book. Handout to prepare for Lab 3.
  9. 4/17/15: Lab 3: network models and Google PageRank Report due 4/24 in class
  10. 4/20/15: Minimization under constraints. Projections, LSQ, and Lagrange multipliers.
    Read 2.1-2.2. Extra reading: 2.3 and 2.4.
  11. 4/22/15: Best approximation in inner product spaces. Continuum example: space L2(0,1). [Read ahead 3.1-3.2]
  12. 4/24/15: Lab 4: Best approximation in inner product spaces and Fourier coefficients. Report due 5/1 in class. Extra: solve (B) also with v*=[1,1,0]T.
  13. 4/27/15: Continuum models revisited. [Read 3.1-3.2]. Find Fourier series for a function in L^2(0,1). [Read 4.1] Why Fourier basis ? (Find eigenvalues and eigenfunctions of the differential operator, with boundary conditions).
  14. 4/30/15: Fourier series on L^2(-1,1): need sine and cosinee basis functions. [Read 4.1]. Adjoint operators in R^N and on L^2(0,1). Continuum (equilibrium) model of elastic bar/vertical string displacement/stationary heat conduction or diffusion/ in strong form and weak (variational) form, as a critical point of the energy functional.
  15. 5/1/15: Lab 5: Fourier series and applications. Report due 5/8 in class.
  16. 5/4/15:
  17. 5/6/15: Review worksheet distributed. More examples of Euler-Lagrange equations.
  18. 5/8/15: Lab 6: Discrete Fourier transform and music. Report due 5/22 in class
    (There was a typo in PROJECT on DFT, page 2, in the paragraph starting with "We proceed". You can obtain lab6.pdf as usual from class website).
    Extra office hours 5/8/15, 4:00-5:00 (Kidd 292A)
  19. 5/11/15: NO CLASS TODAY
    Extra office hours (with Mr McClelland, Kidd 288): Monday 5/11, 11:00-12:00, Tuesday 5/12, 1;00-2:00.
    Mr McClelland MLC Hours Mondays 9-10, and Tuesdays 9-11.
  20. 5/13/15: MIDTERM
  21. 5/15/15: Lab 7: more on Fourier analysis. Applications to image processing. Report due 5/22 in class
  22. 5/18/15: Introduction to inverse problems. SVD as a result of maximization under constraints. [Read 1.6]
  23. 5/20/15: 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.
  24. 5/22/15: Lab 8: Inverse problems lab. Report due 6/1 in class
  25. 5/27/15: GPS and geodesy models; use of Newton's method.
    "When she was good, she was really really good. When she was bad, she was horrid"
  26. 5/29/15: Lab 9: randomness and Monte Carlo simulations.
    Take home MIDTERM due 6/3 in class.
  27. 6/1/15: Kalman filter: recursive and weighted least squares. [2.5]
  28. 6/3/15: Worksheet and group work on Green's function: link between inverse problems, Fourier series, and equilibrium differential equation.
  29. 6/5/15: THE END, lab and class review.
    ImageEigenvectors activity: face recognition. Vote for class content !
    Extra exercises (do not turn in, use to prepare for Exams)
    1. 1.4.2, 1.4.7, 1.4.9
    2. 1.5.1, 1.5.4-6, 1.5.24-25
    3. 2.1.3, 2.1.5, 2.1.8-9
    4. 2.2.1-2, 2.2.3-2.2.5
    5. 3.1.1, 3.1.3, 3.1.5-6
    6. 3.2.1-2; 3.2.5-6
    7. 3.6.1, 3.6.3
    8. 4.1.1, 4.1.2