Summer of Math Exposition

Clear explanation. The choice of how many points to use is the hard part. I originally thought of using cubic polynomials, with first and second derivatives constrained at adjoining sections so the error can be estimated (finite element method). However, you skipped the need to join sections by using only one function over the domain.

The introduction is good. It establishes exactly what the audience wants to know: What question are we answering, and why do we care? One suggestion is that the first 4 seconds of silence drag down the initial energy. In the future, you could start the voiceover during the animation, or introduce the animation later.

After the introduction, you start discussing optimal control problems and optimizers. The way you talk about them seems to assume the viewer has heard of these before. Maybe the target audience is engineers who are already familiar with these terms, but I’m not. Instead of “What are the parts of an optimal control problem?”, something like “The brachistochrone problem is an example of an optimal control problem, which is …” or “We’ll put the brachistochrone problem into the context of optimal control problems, which are a class of …” wouldn’t sound like the viewer missed something. Similarly “fed to the optimizer itself” — what is the optimizer?

… After watching more of the video, now I understand that it’s for a fairly specialized audience. So it would be great to state upfront who that is and what background will be assumed.

A couple times during the discussion of differentiation and integration corresponding to matrix multiplication, the voiceover stops while the animations play out. In making my own videos, I often plan segments like this, but then when I watch them back I realize that some voiceover is necessary to accompany the viewer and narrate what’s going on. Otherwise it’s too easy for the viewer to not follow the transformations.

It may be useful to give a broad outline of the method at the beginning of the video before going into details.

Overall, the animations look great and are helpful for understanding what’s going on. The animations showing the optimizer iterating are really cool! I said “woah!” out load during the first one. Maybe start the video with that! I think there is definite potential for more videos demonstrating this type of convergence visually.

I had never heard of the LGL method for choosing better interpolation points. This sounds really interesting, and I would love to see a video about this topic specifically! (Just look up the pronunciation of “Legendre”.)

The video went through a lot of topics, that one might encounter through high school and university, and that was quite cool. Cool setup - I love brachistochrones.

Pretty nice visuals and design

Very interesting taking a classic calculus problem and reframing it using engineering techniques. Well-produced and engaging. However, it needed more explanation of what the Lagrange polynomials are and what they do, even without the full derivation.

The audio quality is quite bad compared to other videos: awkward pauses and cuts, a missed take or two that wasn’t edited out, quite a bit of background noise.

Additionally, I would say that the lagrange polynomial explanation was unclear. Perhaps motivate why someone would use the lagrange polys, move the explanation of the basis functions earlier. What do you mean by “basis” here? What do you mean by “most accurate” here?

There was very little motivation to the method or the problem, this seems like powerful technique, but is it actually used? If so where and how? I’m also still a little confused how formulate the problem using Lagrange.

Overall good video it make me want to learn more, thanks for posting.

This gives great insight into how, in optimisation problems, formulation can sometimes be more valuable than the optimiser itself. I think a few more animations could be provided (for example for the sanity checks of the equations), and maybe a few more words could be said about the lagrange interpolation polynomials as I felt you went over it fairly quickly. But the problem was well motivated, and the narration very pleasant and humorous!

I like the animation. I think this is a difficult topic and it might be hard for some viewers to understand.

Finite difference and basic area rules are also just matrix multiplication. You did not explain why the Verein you legged on is better.

Very intuitive video.

• Learned something new
• Numerical methods is sick as hell very cool
• Explained what our math does and why we use it
• Showed the solution running and gave another example of it running

Great example. It exemplifies the mean features of setting an optimal control problem.

I will start by saying I’m not an engineer, but it’s an interesting video nonetheless.

I will say that your video inspired me to look up and learn some stuff about Lagrange polynomials! I was aware that Lagrange polynomials don’t always converge to the sampled function as you sample more and more points, but didn’t know that there was an optimal choice of sample points that does work (at least if the function is sufficiently smooth).

It looks like it was Chebychev, Gauss and Lobatto that figured out this optimal choice of points, not Legendre, Gauss and Lobatto, as claimed, though?

I would’ve liked a bit more explanation about why we want to use these pseudospectral operators rather than the discrete operators from earlier. And maybe a brief explanation about what IPOPT is and how it works? But otherwise, good work!

I really liked the video! Good explanation, everything is very clear, pacing is good, animations are nice. I appreciated the part at the end where you fix the angle to 90 degrees and the solver automatically comes up with a different solution to satisfy the constraint. Makes me wonder if you could take this further and use the algorithm to generate a full-blown roller coaster or something, optimized for maximum speed.

The voice over seems a bit quiet though, and I can hear background noises occasionally. Could use some background music as well.

Really nice approach to the Brachistochrone problem! Good explanation, the video also is of a very nice quality.

Too many math and intro at the beginning. The animation in the middle is good, helpful for understanding. Too many equations, hard to follow step by step.

The initial placement of formulae on screen may be somewhat intimidating to students “at the Calculus level”, though Engineers may be used to seeing it. Would “Calculus students” know what Jacobians & Hessians are prior to Vector Calculus (Calc III)? May wish to explain more or explain when those come into play. Good “aha!” moment when the iterated curve “popped” into a cycloid. Some students may not know what it is, so you did a good job anticipating this with the introduction of the dotted red cycloid. Changing a constraint at the end was a good extension of the problem to show the power of the method.

It is indeed a very nice video, and the same approache would have helped when i learn it for the first time, but i think it could still have been made a bit smooth, its not bad in anyway, just in comparioson to the other posts that i came across