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Schedule and assignments
  1. 3/31/14: Introduction. First day handout and questionnaire.
    Discrete equilibrium model: mass and springs [Read 1.4] and its continuous version [Read 3.1].
  2. 4/2/14: Positive definite linear systems and minimization of a quadratic function in Rn. [Please read 1.1-1.3 for review].
  3. 4/4/14: Meet in computer lab Kidd 033.
    LAB1 report due 4/11/14
    LAB1: introduction, syntax; positive definite systems and minimizing quadratic functions
  4. 4/7/14: Spd matrices three ways: (i) definition, (ii) characterization with eigenvalues, (iii) characterization with pivots. LDL^T decomposition. "Elimination"="Completing the square". Cholesky decomposition: why and how. [Continue 1.3-1.4].
    Extra exercises, part I: (do not turn in)
    1. Assume K is spd. Write out details of the general case why the minimizer of phi(x)=1/2 x^TKx-x^tf must satisfy Kx=f.
    2. Let K=A^TCA, and A is the "difference" matrix. Write out the general form of K.
    3. Interpret the modeling assumptions behind the determinate case as in in LAB1.F.

  5. 4/9/14: Least squares for overdetermined systems: direct, and through ATA. Nonlinear least squares examples: rotation; yaw, pitch, and roll.
    Extra exercises, part II:
    1. 1.4.2, 1.4.7, 1.4.9 (do not turn in)

  6. 4/11/14: Meet in computer lab Kidd 033.
    LAB2 report due 4/18/14
    LAB2: least squares (linear and nonlinear); orthogonal transformations, yaw, pitch and roll.
  7. 4/14/14: Mechanics models (balls and springs) versus Flow models (fluids, diffusion, heat flow, electrical networks). Same model ATCAx=f and ATCAx=f, but different A.
    Continuum model counterpart of balls and springs. A=d/dx and A^T = -d/dx. Stress/strain relationships. [Read 3.1]
    Dynamics and eigenvalues [Read 1.5]
    Extra exercises, part III:
    1. 1.5.1, 1.5.4-6, 1.5.24-25 (do not turn in)
    2. Read 1.6 (review of matrix theory)

  8. 4/16/14: Dynamics of vibrations in discrete and continuous case. [Read 1.5, 2.1, 3.1]
    Develop eigenfunctions and eigenvalues for the discrete and continuous case and how to use them.
    Start network models; incidence vs adjacency matrix.
    Read handout on Markov chain models in preparation for LB3.
  9. 4/18/14: Meet in computer lab Kidd 033.
    LAB3 report due 4/25/14
    LAB3: Google search, Markov chains, and PageRank.
  10. 4/21/14: The meaning of ^T in A^T when A=d/dx in the continuum case. Function space L^2 as an inner product space. Solving Ku=f for some examples of boundary conditions.
    Extra exercises, part IV (do not turn in):
    1. 2.1.3, 2.1.5, 2.1.8-9
    2. 3.1.1, 3.1.3, 3.1.5-6
    3. 3.2.1-2; 3.2.5-6

  11. 4/23/14: Introduction to variational calculus [3.6]: minimization of functionals using first variation (discrete and continuous case).
    Example: finding projections in lower dimensional subspaces
  12. 4/25/14: Meet in computer lab Kidd 033. WATER DAY
    LAB4 report due 5/2/14
    LAB4: Inner Product Spaces, Best Approximations, Projections, and Fourier coefficients [4.1]
  13. 4/28/14: Details of how Fourier coefficients relate to least squares.
    Solving evolution problems x'+Ax=0 with spd Ax using eigenvalue decomposition. Next, the same for u_t-u_xx=0 with Fourier series. Examples of Fourier series (MATLAB tool)

    Extra exercises, part V (do not turn in):
    1. 3.6.1, 3.6.3
    2. 4.1.1, 4.1.2

  14. 4/30/14: Review
  15. 5/2/14: Exam (in class)
  16. 5/5/14: Fourier series [4.1, skip after p 276]. How to compute Fourier coefficients. Fourier series of a major fifth triad in C major scale. Fourier series as a method of data compression.
  17. 5/7/14: Continue Fourier series, and Fourier transform.

    Midterm (additional) problems (and corrections) due in class Friday, May 9. Individual work only.
  18. 5/9/14: Meet in computer lab Kidd 033.
    LAB5 report due 5/16/14
    LAB5: Continuous and discrete Fourier analysis and applications. Gibbs phenomenon and convergence of Fourier series. Bessel's inequality. Discrete FT and FFT. Sound !
  19. 5/12/14: Optimization under constraints: the general setting with a Lagrangian, and Lagrange multipliers. [Read 2.2]
    Extra exercises, part VI (do not turn in):
    1. 2.2.1-2, 2.2.3-2.2.5

    Max/Min/imize a quadratic function with a constraint.
    Max/Min/imize a quadratic function over unit ball.
    Overview of how SVD ([Read 1.6 (7)]) connects to inverse problems and those with convolutions to FFT.
  20. 5/14/14: SVD and low rank approximations.
  21. 5/16/14: Meet in computer lab Kidd 033.
    LAB6 report due 5/23/14
    LAB6: Lagrange multipliers. Low rank approximations and image compression. Random realizations via PCA=Karhunen-Loeve expansions.
  22. 5/19/14: Introduction to inverse problems. Why are they ill-posed. Smoothing and roughing behavior. Sensitivity measured by condition number of a matrix.
  23. 5/21/14: Examples of inverse problems: gravity survey and barcode reading. Worksheet on SVD.
  24. 5/23/14: Meet in computer lab Kidd 033.
    LAB7 report due 6/4/14
    LAB7: iill-conditioned problems. Inverse problems. Regularization (Tikhonov).
    EXTRA office hours Friday 5/23/14, 4:00-5:30.
    All HW and LAB corrections due May 30. All solutions are due Wed. June 4.
  25. 5/28/14: Midterm in class.
  26. 5/30/14: Meet in computer lab Kidd 033.
    LAB8: finish LAB7.
  27. 6/2/14: MIDTERM discussion. Examples of regularization.
  28. 6/4/14: The Math behind GPS (Global Positioning System). Least squares problems and Newton's method. Newton's method is the iterative linearization. Outlook to Kalman filter. [Read 2.5 and/or Borre/Strang book].
  29. 6/6/14: In class REVIEW.