# How to pick a basis for the result of a non-linear function given a basis for its argument

I am trying to represent the result of a non-linear function in a small basis, given another small basis that does a good job a representing the argument of the function.

More specifically, there is a non-linear map, $$f: \left< r \middle| \rho \middle| r \right> \longrightarrow \left< r \middle| V \middle| r \right>$$ or alternatively $f[\rho(r)] = V(r)$, where $\rho$ and $V$ are Hermitian and $V$ is diagonal in real-space. I already have a great (small) basis $u_i$ that represents $\rho$ perfectly: $$\sum_i \left< r \middle| \rho \middle| r \right> \left<r\middle|u_i\right> \left<u_i\middle|r\right> = \left< r \middle| \rho \middle| r \right>$$

With access to $f$, is there anything I can do to the basis $u_i$ to make it represent $V$ more completely?

$f$ generally has terms that look like $\rho(r)^{1/3}$, plus some linear terms, but can be more general. I don't need to evaluate $f$ in this basis; I just want to be able to transform $V$ into it after it is computed.

Scientific back-story: $\rho$ is an electron density and $V$ is the local potential. Real-space is really big, so I'm trying to get out of it as quickly as possible.

This sounds pretty close to what people doing reduced order modeling for optimization of nonlinear differential equations are working on. They have a reduced basis for expressing (and computing) an approximation $\tilde y$ of the solution $y$, but evaluation of the nonlinearities $f(\tilde y)$ in the equation still have to be performed in the full space. To circumvent this, they construct a reduced model $\widetilde{f(\tilde y)}$ from a basis $\{f(\tilde y_1),f(\tilde y_2),\dots\}$.