Grid interpolation

[PabRod]

Problem

The communication between multiple numerical models for a boundary problem operating within the same geometry requires to efficiently sharing information. Some of these pieces of information are relativelly trivial, such as state, time, and parameters, but others are way less straightforward, such as integration_times and, most importantly, integration_grid.

Suggested approach

0. Understand the problem.
   • Brainstorm about the problem.
   • Google a bit. It is hard to believe this is a novel problem.
   • Ask internally. Good candidates are Lourens and Johan.
1. Implement a solution. Here is my suggestion:
   • Minimal: enforce the use of the same grid ab initio for all integrators.
   • Intermediate: use direct interpolation, i.e.:, assume that if $I(x)$ returns the interpolated value of $x$, then $F(I(x)) \approx I(F(x))$
   • Excellent: use automatic differentiation for a more system-informed interpolation, i.e.:, assume that $F(I(x)) \approx F(x) + F'(x) \epsilon + O(\epsilon^2)$, where $\epsilon$ depends on the interpolation method $I$.