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It's funny how different backgrounds bring you to having different ideas of what is easy and hard. As a mathematician, to me the easy problem is finding the min

It's funny how different backgrounds bring you to having different ideas of what is easy and hard. As a mathematician, to me the easy problem is finding the minimum of the functional (where "easy" means "we can at least try", certainly not "trivial"; some minimum problems are actually relatively easy, some others are very difficult and open), while proving existence of PDEs is exactly why calculus of variations (and many other theories) were invented in the first place.

ur-whale 8y ago
I didn't mean to imply that PDE's were an easy problem. They clearly aren't, especially when it comes to proving some sort of formal property about them (which is something a mathematician would worry about, but which is rarely an engineer's first concern: they'd only worry about that type of thing when numerical integration starts producing "crazy" results). However, there is a very intuitive way to compute an approximate solution to a system of PDE's, namely increase time in very tiny steps and solve the resulting system of equations each time, rinse and repeat. Now, when you deal with arbitrary functions, there are so many ways to "represent" them: as various kind of series, as algebraic composition of elementary functions, as solutions to implicit equations, as solutions to PDE's, as polynomial approximations, as neural networks, etc ... These representations are usually dense in "regular" functional spaces and you do get a lot of mileage out of them. Whichever way you chose to represent functions in your functional space, the problem of finding an extrema over that representation is very much not obvious in the calculus of variation setting (or even that the extremum will be itself be representable, for that matter), whereas the conversion to a system of PDE gives you a clear path to victory - as long as the representation can be differentiated.
- giomasce 8y ago
  That's exactly why I say that different viewpoints give different concept of easiness. To me there are a lot of way to find minima (or, at least, critical points) of a functional: the direct method in CV, mountain pass theorems, etc. Most of them just need to know very general information about your space and functional and are actually even simple to visualize.

ur-whale 8y ago

I didn't mean to imply that PDE's were an easy problem. They clearly aren't, especially when it comes to proving some sort of formal property about them (which is something a mathematician would worry about, but which is rarely an engineer's first concern: they'd only worry about that type of thing when numerical integration starts producing "crazy" results). However, there is a very intuitive way to compute an approximate solution to a system of PDE's, namely increase time in very tiny steps and solve the resulting system of equations each time, rinse and repeat. Now, when you deal with arbitrary functions, there are so many ways to "represent" them: as various kind of series, as algebraic composition of elementary functions, as solutions to implicit equations, as solutions to PDE's, as polynomial approximations, as neural networks, etc ... These representations are usually dense in "regular" functional spaces and you do get a lot of mileage out of them. Whichever way you chose to represent functions in your functional space, the problem of finding an extrema over that representation is very much not obvious in the calculus of variation setting (or even that the extremum will be itself be representable, for that matter), whereas the conversion to a system of PDE gives you a clear path to victory - as long as the representation can be differentiated.

giomasce 8y ago
That's exactly why I say that different viewpoints give different concept of easiness. To me there are a lot of way to find minima (or, at least, critical points) of a functional: the direct method in CV, mountain pass theorems, etc. Most of them just need to know very general information about your space and functional and are actually even simple to visualize.