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.

The computation of $f$ is very expensive, but the derivatives come almost for free (although they accumulate numerical error so I limit number of the derivatives by $m$). Because I also have derivatives doing Hermite interpolation seems most reasonable approach for me. Because of costliness of computation of the $f$ I want to make as little calls to it as possible.

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.

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?



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