Warm-up
- Name, area, and a favorite textbook (take notes!)
- If you are elsewhere, we’ll find someone for you to do this with!
Logistics
- Waitlist notes
- Time zone issues
- Julia and other languages
- Readings
Optimization background
- Variational notation
- Reminder that Lagrange multipliers exist!
- Direct solves, fixed point iterations
- Models of functions and models of solver dynamics
Least squares
- Normal equations and calculus
- Alternate inner products
- Cholesky, QR, and SVD
- Aside re min norm connection
Activity (submit a notebook or PDF for points)
- Show that if $\phi : \mathcal{V} \rightarrow \mathbb{R}$ is a
quadratic form, then $a(v,w) = (\phi(v+w)-\phi(v)-\phi(w))/2$ is a
bilinear form for which $\phi(v) = a(v,v)$.
- Show that adding rows to a matrix (e.g. more data in a least squares
problem) can only increase the smallest singular value.
- Write a code to minimize
where $x_j$ are points in a uniform mesh on $[-\pi, \pi]$,
with $p(x) = c_0 + c_1 x^2 + c_2 x^4 + c_3 x^6$.
- Argue that the coefficients converge to the coefficients that minimize
$\int_{-\pi}^\pi |p(x)-\cos(x)|^2 \, dx$?
- Design (and answer) your own question that would be appropriate for
this type of exercise!