I have a multi-variate, continuous function $f$ from $R^n$ to $R$, which I can query for its output for any input. I would like to create interpolation polynomial for it.In one-dimensional case choosing points is relatively straight forward. I have error estimation:$$f(x)-H(x)={\frac  {f^{{(K)}}(c)}{K!}}\prod _{{i}}(x-x_{i})^{{k_{i}}},$$and to choose the interpolation points, I need to solve corresponding minmax problem, as I'm looking to minimize the maximum error for a given number of interpolation points. I'm willing to go with any class of  multivariate polynomials. I'm even willing to go with something other that polynomials if helps me interpolate my function, takes into account derivatives and provides theoretical guarantees (for example, in form of error bound).   In multi-variate case I cannot find such error bounds in the literature nor I unable to derive them myself. Any advice on how should I pick the interpolation points for this function?  
How to optimally choose points for multivariable Hermite interpolation?
optimization;interpolation;error estimation;polynomials;grid