The situation
Imagine that you are faced with a box with a numeric display and a knob next to it that can turn in specific increments (that you can specify) and in both directions (increasing values or decreasing values).
Each time you turn the knob in any direction you see the value associated with this knob position and the output (the value on the numeric display).
The questions
Given the above situation, imagining that the box encodes a $\mathcal{C}^{\infty}(\mathbb{R})$ that you have no a priori information about (you acquire knowledge about the function by sampling it), if you are given a finite number of points (for example to plot the graph of this function) over a given interval $I = [a, b] \text{ where } (a<b)$ :
- What is the "best" (see note) way to sample this function ("box") on $I$ if you can only turn the knob in one direction (imagine that it is because a phenomenon like backlash is at work) starting at the start of the interval, $a$ ?
- Are there algorithms to deal with this problem ? If not, why ? Uniform sampling exists but seem not "optimized".
- Imagine that you can afford to make multiple passes to refine your estimates of the function, what is the best strategy to put your sampling points on each passes (that are still unidirectional: you can only go to $a$ when you reached $b$) ? Are the strategies dependent on the number of passes ?
- Is there any extension to higher dimensions to any of the previous solutions ?
Note: I do understand that the characteristics of best change given the metric used to qualify it (spectrum sampling, variation, etc). Answers on any of them would be enlightening so I remain blurry on the term.
Bonus question :
- What is the domain of science that deals with this kind of questions ? It seems like the sampling area of signal processing but I can not find any relevant answers... Being able to name a problem is already starting to solving it. Pointers appreciated !