3.30.09

Today's the first day of the spring quarter, which means I had to spend most of the day fussing with bullshit errands like getting my parking permit and paying rent. Other than that, I spent some time taking a break from SPGD and working on my interior point idea.

Basically, the general plan is to find the command vector c that minimizes the 2-norm of the total wavefront. This requires the linear model of the DM using the poke matrix, G, and can be generalized using modes if desired. Without saturation the optimal DM surface would be the projection of the static wavefront onto the range space of G. But, limits on the actuator commands add inequality constraints. This means that the true optimal command vector must lie in the union of the feasible set AND the orthogonal compliment to the null space of G. Note that if modes are used, the inequality constraints involve the modal poke matrix, since you can't place explicit constraints on each modal command.

This kind of optimization problem can be solved by replacing the inequality constraints with logorithmic barrier functions that are tacked onto the objective. These beasts are designed to go to infinity as the limits are approached. By eliminating the inequality constraints, we're left with (in this case with no equality constraints) an unconstrained problem. The new objective function to be minimized now consists of a weighted sum of the wavefront norm, and the barrier function terms. This unconstrained problem can then be solved using Newton's method iteratively for more and more accuracy.

Wow, that's possibly the worst explanation of the barrier method ever committed to words. See the notes on Nonlinear Optimization for the real deal.