I would like to interpolate (not fit) a data set whose points have been rounded.
Lets say I have some observations $y_i$ of a function sampled at $x_i$. The sample locations $x_i$ may be considered exact, but the observations $y_i$ were rounded to the hundredths place (except perhaps the first/last points, which may be considered exact if possible).
I would like to interpolate (say a piecewise cubic Bezier) where the interpolation passes within the rounding error of each observation.
I don't want to fit a global function (say a high order polynomial) because of all the problems with high order fits.
I don't want traditional interpolation because the $x_i$ are tightly sampled enough that the errors in $y_i$ give my later analysis problems.
The only approach I can think of so far is a general nonlinear optimization where the optimization variables are assumed error values $\epsilon_i$ for each observation, and the Bezier interpolation goes through $y_i+\epsilon_i$. Each $\epsilon_i$ would be constrained to stay within the appropriate bounds.
The objective function would be (something like) to minimize the changes in 3rd derivative at the Bezier segment joints (assuming the interpolation used was C2).
Hopefully this formulation wouldn't suffer lots of local optima, and an initial guess of $\epsilon_i=0$ would suffice.
As an Engineer, I would approach this problem numerically -- I could probably use an unconstrained optimizer with bounds constraints. If I squint, I see there is hope that the objective function might actually be linear. Good news for the lack of local optima -- but I probably wouldn't try to formulate it as a linear programming problem.
Is there a better way? Does this problem have a name? Any better ideas?