# Numerical Methods for the Inverse Problem of Density Functional Theory

## TL;DR

This paper reviews numerical methods for solving the inverse problem in density functional theory, focusing on algorithms, error analysis, and comparing different inversion techniques on model systems.

## Contribution

It introduces and compares two inversion methods based on PDE-constrained optimization and variational ideas for the inverse DFT problem.

## Key findings

- Two inversion methods are introduced and analyzed.
- Comparison of methods on finite and periodic systems.
- Insights into error behavior and algorithm performance.

## Abstract

The inverse problem of Kohn-Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to one-dimensional finite and periodic model systems.