I am not sure if this is the right stack to ask this question, but since there is a definite fractal dimension to it, I thought I'd give it a go.
The problem I am facing is one of calculating an expensive function $f(x,y)$ in a finite 2D domain $D$. I perform the evaluation of $f$ at each point in $(x,y)$ independently from all other $(x,y)$ points.
The dumb way to parallelise the process of "scanning" $f$ over $D$ was to define a fixed rectangular grid of points covering $D$. By creating a trivial correspondence between integers and the position in the grid, I can assign each parallel task a range of integers to work with. I.e., each integer maps to only one point in $D$.
The solution above has a huge drawback: I cannot have a quick preview of what $f(x,y)$ looks like. The problem is akin to the one solved by progressive image encoding: I want to be able to refine the picture of $f(x,y)$ as I add more points.
The way I am framing the problem as is: how to map integers to a series of $(x,y)$ points which never repeat while becoming denser on an average way as the integer value increases.
This made me go straight to space-filling curves or trees: I could triangulate $D$ and in each triangle iterate to the desired scale. In this scheme, the number of points evaluated at iteration $n$ is $4\times(4-1)^{n-1}$. For instance, at iteration 3 there are 36 new points (which is different from the points in this example), and I would easily find myself "filling in" a certain corner of the triangle before others, resulting in a rather "non-progressive" way of filling in the triangle.
So, even if this would be a way to iteratively improve the density, in each iteration the overall sampling density would not be balanced (uniform) across $D$.
Any ideas on how to make the sampling in each iteration more uniform are very welcome.
Would the Peano or Hilbert curve do the trick?