# 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.

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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 $\{f(\tilde y_1),f(\tilde y_2),\dots\}$.