PDE Constrained Optimization

Taking a similar approach as the Wu and Yang method, the PDE-Constrained Optimization (PDECO) method, is refined by defining a density error so that the Lagrangian is optimized under several constraints.

More information on this methodology can be found at:

Jensen, Daniel S., and Adam Wasserman. “Numerical methods for the inverse problem of density functional theory.” International Journal of Quantum Chemistry 118.1 (2018): e25425.

and

Kanungo, Bikash, Paul M. Zimmerman, and Vikram Gavini. “Exact exchange-correlation potentials from ground-state electron densities.” Nature communications 10.1 (2019): 1-9.

The lagrangian is defined as:

\[\begin{split}&L[v_{KS}, \{\psi_i\}, \{\epsilon_i\}, \{p_i\}, \{\mu_i\}]\\ =& \int(n(\mathbf{r})-n_{in}(\mathbf{r}))^w d\mathbf{r} \\ & + \sum_{i=1}^{N/2}\int p_i(\mathbf{r})(-\frac{1}{2}\nabla^2+v_{KS}(\mathbf{r}) - \epsilon_i)\psi_i(\mathbf{r})\mathbf{dr}\\ &+\sum_{i=1}^{N/2}\mu_i(\int|\psi_i(\mathbf{r})|^2\mathbf{dr}-1),\end{split}\]

Where the set of p’s and mu’s are langange multipliers for the constraints. And a similar procedure to the Wu Yang method is followed to generate the gradient and hessians to optimize the lagrangian.