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Schedule and assignments |
- 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].
- 4/2/14: Positive definite linear systems and minimization of a
quadratic function in Rn. [Please read 1.1-1.3 for review].
- 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
- 4/7/14: 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)
- 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.
- Let K=A^TCA, and A is the "difference" matrix. Write out the general form of K.
- Interpret the modeling assumptions behind the determinate case as in in LAB1.F.
- 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:
- 1.4.2, 1.4.7, 1.4.9 (do not turn in)
- 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.
- 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:
- 1.5.1, 1.5.4-6, 1.5.24-25 (do not turn in)
- Read 1.6 (review of matrix theory)
- 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.
- 4/18/14: Meet in computer lab Kidd 033.
LAB3 report due 4/25/14
LAB3: Google search, Markov chains, and PageRank.
- 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):
- 2.1.3, 2.1.5, 2.1.8-9
- 3.1.1, 3.1.3, 3.1.5-6
- 3.2.1-2; 3.2.5-6
- 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
- 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]
- 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):
- 3.6.1, 3.6.3
- 4.1.1, 4.1.2
- 4/30/14: Review
- 5/2/14: Exam (in class)
- 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.
- 5/7/14: Continue Fourier series, and Fourier transform.
Midterm (additional) problems (and corrections) due in class Friday, May
9. Individual work only.
- 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 !
- 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):
- 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.
- 5/14/14: SVD and low rank approximations.
- 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.
- 5/19/14: Introduction to inverse problems. Why are they
ill-posed. Smoothing and roughing behavior. Sensitivity measured by
condition number of a matrix.
- 5/21/14: Examples of inverse problems: gravity survey and barcode reading.
Worksheet on SVD.
- 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.
- 5/28/14: Midterm in class.
- 5/30/14: Meet in computer lab Kidd 033.
LAB8: finish LAB7.
- 6/2/14: MIDTERM discussion. Examples of regularization.
- 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].
- 6/6/14: In class REVIEW.
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