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

In this context, some possible keywords are "nonlinear proper orthogonal decomposition (POD)" and "(discrete) empirical interpolation methods ((D)EIM)" (the latter is the topic of Saifon Chaturantabut's thesis).

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These are appropriate references, but I'd encourage caution in only considering proper orthogonal decomposition and discrete empirical interpolation methods. More generally, searches for "reduced order model*" and "model reduction" should yield a host of methods. –  Geoff Oxberry Jan 31 '13 at 19:47
Indeed, these are only the methods I am familiar with, and not meant to be comprehensive. I'll edit the answer to make that clearer. –  Christian Clason Jan 31 '13 at 21:18
@GeoffOxberry, I'm curious: what is the reason for being cautious about POD or DEIM? –  Christian Clason Jan 31 '13 at 21:26
I did my PhD thesis on reduced order modeling in combustion. In my experience, the approximation error in the reduced model solution, at constant reduced model size (in terms of the number of state variables in the reduced model -- here, the size of the basis), depends strongly on how a method's calculated basis matches the underlying dynamics (i.e., the physics) of the model being reduced. In combustion, POD is used, but reduced models from quasi-steady state assumptions or singular perturbation analysis have been favored because they are seen as better capturing the combustion physics. –  Geoff Oxberry Jan 31 '13 at 23:36
Ah, so you meant "don't overlook other methods", not "don't use POD". I agree there: POD works well for diffusive problems, but has difficulties with convection-dominated problems, where other (specialized) methods perform better. –  Christian Clason Feb 1 '13 at 8:02
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