I think most important points have been raised here by the people that know much more about this than me, but let me just add a link to this little piece of code <https://github.com/nilshg/LearningModels/blob/master/Test_Interpolations.jl>I've written that compares linear, spline and piecewise cubic hermite interpolations on regular and (where available) irregular grids using the packages ApproXD <https://github.com/floswald/ApproXD.jl>, Dierckx <\https://github.com/kbarbary/Dierckx.jl>, Grid <https://github.com/timholy/Grid.jl>, and NumericalMath <https://github.com/hwborchers/NumericalMath.jl>.
The example used here is -1/x, so a CRRA with risk averion 2 if you want. Should be easy enough to adapt to whatever function you're trying to interpolate in either one or two dimensions (and at least for Grid and ApproXD it should also be easy to extend to >2 dimensions).
