# Algorithm for homogenization of non-equidistant 1-D grid

I am looking for proper algorithm for so called "delta-homogenization" of 1-D non-equidistant grid.

Definition: The sorted monotonically changing vector $$x = x_{homogeneous}$$ is $$\delta$$-homogeneous if for specified $$\delta > 1$$ , the differences ratio $$r_i(x) = (x_{i+1} - x_i)/(x_i - x_{i-1})$$ satisfy to the condition: $$1/{\delta} < r_i < \delta$$, for every $$i = 1,2, ...,N$$ where $$N = length(x)-2$$

Problem: In a case of non $$\delta$$-homogeneous vector x, how to find minimum number of additional points $$x'_j$$, to be vector $$x_{new} = sort([x , x'])$$ $$\delta$$-homogeneous?

The problem is motivated by proper design of non-equidistant grids, which are able to guarantee numerical stability of PDEs solutions.

Add Remark: When I define objective function as $$O(x,x',M) = \sum_{i=1}^M max(0,r_i(x,x')-\delta) + \sum_{i=1}^M max(0,1/\delta -r_i(x,x'))$$ , where $$O(x_{homogeneous}) = 0$$. Is it possible to re-formulate my problem effectively as mixed-integer optimization problem (Integer: number of additional points, Real: co-ordinates of $$L$$ additional points $$x'_j$$, where $$M = N + L$$ )?