This is a domain where experimental calibration is the correct approach. Simply because there are far too many variables to determine mathematically.
- Pasta starts cooking at 80C, as starch starts absorbing water then. So the moment you add pasta matters.
- The absorption speed largely depends on mechanical aid of stirring - as water boils, motion of the bubbles stirs the pasta. This process is far too complex to model mathematically.
- Pasta of different cross-sections, composition, dryness will cook at different rate. This is again not something you determine through models, but experimentally.
- there are other engineering factors to consider: splashing water (and pasta) when boiling too intensely, ratio of heat transfer into water and modelling temperature in the pot (hotter by bottom, cooler by surface); wasting energy through heat escaping through the sides, pasta sticking to the bottom of the pot or forming clumps if not stirred, is the water salted? (boiling point rise up to 4 Celsius degrees), is the pot covered (higher temperature near the top, which may speed up cooking but will build pressure and splash worse).
- How do you define 'cooked' - the most accepted method involves taste test, that is experimental. For your model you'd need to develop a hardness function of pasta, determined through dedicated devices.
- How uniformly will the pasta cook? If you cook thick noodles in very hot water shortly, they will be overcooked on the surface and hard inside.
In short, that all depends. Generally boiling stronger will speed the process up, but may not necessarily provide preferable results than gentle simmering.
You've actually added some answerable questions.
- how do we define cooked for this problem?
The 'correct' baseline is organoleptic, 'tastes just right'. If you want this to be more 'scientific' you can go with tests spread across many people, and do statistics. This is not necessary though, because you're not discovering a new field; pasta science is long known and established, so you can stand on the shoulders of giants: apply standard cooking conditions to pasta with known recipe, and it will result in 'cooked pasta'. So you have the baseline 'cooked pasta' template. Now for further tests you will need a more simple test, based on properties less cumbersome to measure - for example, mass. If the mass of drained pasta rose by a factor of 2.25, you have 'al dente'. With a factor of 2.4, you have it thoroughly cooked. It's an easy, reliable test usable in practice.
- does heat exchange coefficient change with fire size?
Heat exchange coefficient is completely insignificant here. Pasta has a very low specific heat coefficient comparing to water, and with it being usually thin-walled, reaches the temperature of surrounding water within seconds after immersion; especially that it soaks the hot water rapidly. So, no, except for very low heat where you dump a lot of pasta into little water significantly reducing the water temperature (and resulting in lousy clumped cooked pasta) you're not changing regarding heat exchange coefficient.
Cooked pasta differs from raw, soggy pasta in content of water absorbed. (after all, if you cool down cooked pasta, it doesn't revert to uncooked...) And that is what changes with cooking intensity. Wet pasta secretes starch, which forms a weak barrier obstructing further absorption. Stirring washes it off, and allows water to penetrate deeper; motion moves water through pores and promotes penetrating cellular membranes. In short, water that boils hard stirs the pasta and lets it cook a little faster. Temperature and heat transfer have little influence here: if you stir a slow-simmering pasta with a ladle, you'll get it cooked just as fast as if you cook it on high fire.