Given data points $(x\_i,y\_i)\in \mathbb{R}^m\times \mathbb{R}^n$ with $n>m$ satisfying $y\_i=f (x\_i)$ with a sufficiently smooth injective unknown function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ and a value $y \in Im (f)$, I want to find an estimate for $x\in \mathbb{R}^m$ s.t. $f (x)=y$. Does a problem of this sort have a name. Is there any literature available? I am thankful for any suggestions.



inverse interpolation
After having calculated an "explicit" interpolating function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$, satisfying $y\_i=f (x\_i)$, you can calculate the local inverse via evaluation of the [implicit taylor series](http://pdfs.semanticscholar.org/7362/e8c3ce5fc3c10403c5e93aa29e4a79333b0a.pdf) or some other series expansion of implicit functions.


If you are however satisfied with very pragmatic solution, you can chose a data point $(x\_i,y\_i)$ that minimizes $\|y-y\_i\|$ and then via the method of steepest ascend or descent identify the "nearest" $x$, for which $f(x)=y$ w.r.t. the interpolating function $f$; that should also disambiguate the calculation. 


If $f$ is smooth, it can be assumed to be (piecewise) polynomial and thus the gradients that are used in the method of steepest ascend or descent can be easily calculated.


