I'm using a Sobol sequence to numerically integrate a function $f$ over $[0,1)^d$. Additionally, I need to evaluate $f$ frequently. Now, evaluation $f$ is expensive, but the exact value $f$ is not so important; an approximation would be enough.
So, my idea is to approximate the value of $f$ at any $x$ somehow in terms of the values of $f$ at the $k$-nearest neighbors (or inside a ball around $x$? I don't know what would be better and am open for suggestions) in the Sobol sequence. This way, I can compute the values of $f$ only once at the whole Sobol sequence and then don't need to evaluate $f$ anymore.
The question is: No matter how I selected a suitable subsequence $(x_i)_{i\in I}$ of the Sobol sequence to approximate $f$, how should I average them to find an approximation of $f(x)$?
I think it should be a weighted sum depending on the distances $w_i:=\|x_i-x\|$. So, maybe $$f(x)\approx\sum_{i\in I}e^{-\left(\frac{w_i}\sigma\right)^2}f(x_i)\tag1?$$ If so, how should I choose $\sigma$? I guess somehow such that the approximation is almost exact for $x=x_i$. Maybe depending on the distances between the $x_i$?