Automatically choosing the "right" interpolation type

[cstjean]

continuous_y = value(y, linear_interpolation); is nice, but shouldn't InfiniteOpt be able to choose which interpolation type is appropriate automatically? For instance, when I use constant_over_collocation, the appropriate interpolation is a step function, and likewise when I use OrthogonalCollocation(3) it should use a cubic to represent the solution (IIUC). It would help to understand what's going on when I make different choices.

[pulsipher]

Thanks for the suggestion. To clarify, the interpolation is based on all the supports (even the collocation nodes) and a cubic interpolation is not equivalent to an orthogonal collocation with 3 nodes (since it is based on Lobatto quadrature). However, automatically choosing constant interpolation with constant_over_collocation makes sense.

[cstjean]

Perhaps I am talking out of my depth again, but my comment was based on https://youtu.be/wlkRYMVUZTs?t=2291, and on Copilot's

Yes — with Lobatto collocation the solution is a (piecewise) polynomial spline, provided you are using it in the usual time‑interval or mesh‑based way.

So that it'd be cool to show the solution as a polynomial spline of the appropriate degree (eg. if the soluton for x is piecewise-linear and dy/dt = x then the solution for y should be a quadratic spline).

[pulsipher]

While it is true that orthogonal collocation integrates ODEs based on a polynomial spline, the nodes used for that spline vary based on the selected quadrature method. For instance, a 3rd order collocation uses different nodes depending on whether one chooses Lobatto, Legendre, or Radau quadrature.

Another way to think about this is that orthogonal collocation over finite elements (as implemented in InfiniteOpt) is stringing multiple Lagrange interpolations schemes together, whereas Interpolations.jl relies on B-splines which are not exactly equivalent.