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 $M = N + L$ )?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.