Using random numbers to obtain Kohn-Sham potential for a given density

Most of the density-to-potential inversion methods developed over years follow a general algorithm $ v_{xc}^{i+1}(\textbf{r}) = v_{xc}^{i}(\textbf{r}) + \Delta v_{xc}(\textbf{r})$, where $\Delta v_{xc}(\textbf{r}) \frac{\delta S[\rho]}{\delta \rho(\textbf{r})} \Big |_{ \rho_i(\textbf{r})} - \rho(\textbf{r})}\Big|_{ \rho_0(\textbf{r})}$ and $S[\rho]$ is an appropriately chosen density functional. In this work we show that can be used with random numbers to obtain exchange-correlation potential for given density. This obviates need evaluate functional in each iterative step. The method demonstrated by calculating atoms, clusters Hookium.