$$ \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\EE}{\mathbb{E}} \newcommand{\HH}{\mathbb{H}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\length}{\operatorname{length}} \newcommand{\uppersum}[1]{{\textstyle\sum^+_{#1}}} \newcommand{\lowersum}[1]{{\textstyle\sum^-_{#1}}} \newcommand{\upperint}[1]{{\textstyle\smallint^+_{#1}}} \newcommand{\lowerint}[1]{{\textstyle\smallint^-_{#1}}} \newcommand{\rsum}[1]{{\textstyle\sum_{#1}}} \newcommand{\partitions}[1]{\mathcal{P}_{#1}} \newcommand{\erf}{\operatorname{erf}} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\khat}{\hat{k}} \newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} \newcommand{\smat}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)} $$

Assignment 8 Extra: Fall Break Optional

Problems

Heatshields on Rockets

You are working with several research teams at NASA who are trying to understand and model the heat flow along the nose cone of a rocket during re-entry. Below is a picture from a real simulation of this just for context (though you do not need this picture to solve the problem)

Waves of hot and cold on a nose-cone for a rocket traveling at hypersonic speeds.

You and your team use the heat equation as a model of this: if \(x\) is the distance along the nose cone (starting with \(x=0\) at the tip) and \(t\) is the time, then the temperature \(H(x,t)\) of the metal at point \(x\) and time \(t\) satisfies the equation

\[\frac{\partial H}{\partial t} = \lambda^2 \frac{\partial ^2 H}{\partial x^2}\]

Where \(\lambda\) is a positive constant that depends on the metal you choose to make the nose of the rocket out of (larger values of \(\lambda\) means that it conducts heat faster, smaller values mean it conducts heat slower).

Your team of atmospheric scientists tell you that during re-entry waves of plasma will be crashing over the nose cone, heating it up in a complicated pattern. Once this portion of the flight is over, the temperature distribution along the nose cone is expected to look like

Temperature distribution (as a function of \(x\) along the nose cone) at time \(t=0\).

Your team of theorists find an exact solution to the heat equation describing these plasma waves: it looks like this

\[H(x,t)=e^{-\lambda^2 t}\sin(\lambda x)\]

(You don’t have to turn this in, but it’s a good idea to verify this is a solution to the heat equation, by taking the derivatives on both sides and seeing the results are equal!)

Your team of mechanical engineers says that the materials they have built the nose cone out of have \(\lambda = 4\), but their measurement is only accurate to within \(\pm 0.1\). The rocket team plans to measure the temperature at the location \(x=\pi/2\) as a function of time. However, because the rocket is vibrating during re-entry, they can only measure position to within \(\pm 0.05\).

• Use error analysis to figure out how far off the actual temperature reading might be from the predicted theoretical value by \(H(x,t)\) at \(t=0\).

\[dH = H_x dx + H_\lambda d \lambda\]

Optimization

You are once again working on the engineering team developing new rocket engines for the Artemis mission, where on a previous homework you submitted recommendations to your team members on how to modify the nozzle width and fuel temperature to increase efficiency. Now, you are tasked with taking this efficient design and trying to maximize the thrust during the initial burn to leave earth’s atmosphere, which is where most fuel gets used up.

Having already set the nozzle diameter and fuel temperature, you look to study the effect of two other variables: the overall radius of the rocket and the slope of the nose-cone (both of which greatly affect air resistance) on the net thrust. Your colleagues run a computational simulation of the atmospheric physics, and determine that the net force varies as a function of the radius \(r\) and slope \(s\) as

\[F(r,s)=8+4rs-4r^4-4s^4\]

Find the critical points of \(F\): these are the configurations that you and your engineering team should investigate more closely.

After figuring this out, what radius and slope should you recommend to the manufacturing team? (Remember radius should be a positive number, as its a length).

High Dimensional Optimization

In class we have mainly focused on understanding maxima minima and saddles for functions of two variables. THe main reason for this is just that the computations remain manageable - there are only four second derivatives to deal with, and they all fit nicely into a \(2\times 2\) array. However problems in the real world often have to deal with many many more variables, and this problem is your opportunity to think a bit through how the theory might work here.

Consider you have a function \(f(x_1,\ldots, x_n)\) of \(n\) variables. Such a function has \(n\) first derivatives, so to find its critical points we need to solve a *system of \(n\) equations: this might take some work! But assume that we have such a critical point \(c\), and now we want to know if this is a max, a min or a saddle.

Question: If its a random critical point, what is the probability that its a maximum? Whats the probability is a minimum? A saddle?

Hint: Think about slices! A hill has all of its slices downward shaped parabolas, and a bowl has all of its slices upward facing parabolas, and a saddle is anything else. How many options are there, and how many of them are maxima?*

The answer might surprise you: its rarer than you may think to find a maximum or a minimum in high dimensions! This has implications for optimization in large real world problems like machine learning: you may need to think a little harder to make sure you actually reach a minimum than you’d originally hoped.