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I have a problem where I heavily need to restrict the number of points at which I sample a function based on the values of a different function.

I have two functions:

  • $f:{\mathbb{R}\times [0,\infty)\times[0,\infty) }\rightarrow\mathbb{R}$, which is a probability density, generated on a set of data, using Gaussian kernels
  • $\sigma:\mathbb{R}^3\rightarrow\mathbb{R}$, which contains information about the quality of my model.

I try to evaluate to total quality of my model, by sampling points from the space $[a,b] \times [c,d] \times[e,f]$ and integrating over the product $f \cdot \sigma$. The intervals correspond to the natural boundaries of my data.
$f$ has at least two peaks, but could contain more interesting points, yet $f([a,b] \times [c,d] \times[e,f])$ evaluates to $0$ in most points. Now to the problem:
$f$ is relativeley easy to compute for a big amount of points, while $\sigma$ is computationally quite heavy. Therefore I try to find a way to sample $\sigma$ (or both functions) at fewer points, by using $f$ as a literal "density", sampling more points in regions of local extremes, or a higher probability density, while sampling fewer points in the regions where $f$ is $0$ or near $0$ anyway.

Is there a good/efficient way to do so? Main goal is avoiding skipping over intersting regions in $f$ while keeping the number of samples as low as possible. Bonus points: Is there MATLAB funtionality to do so, already?

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    $\begingroup$ This sounds to me like you want to do Monte Carlo integration. The way to do this is to run a Markov Chain with samples drawn from $f$, which will not contain any points where $f$ is zero and so you only have to evaluate $\sigma$ at points that actually matter. $\endgroup$ Commented Dec 21, 2022 at 2:56
  • $\begingroup$ I agree with @WolfgangBangerth. You are looking for some variant of Monte Carlo Integration where the sampling strategy adapts to a given distribution (en.wikipedia.org/wiki/VEGAS_algorithm ? ). $\endgroup$
    – MPIchael
    Commented Dec 21, 2022 at 7:47
  • $\begingroup$ My first intuition was to only sample from points where $f$ is not zero, too. Unfortunately I need $\sigma$ to calculate a prediction for my model, which should have at least some points in less dense regions. $s\sigma$ comes from a Gaussian process regression and is needed to calculate the a posteriori mean of the regression. $\endgroup$ Commented Dec 21, 2022 at 10:57
  • $\begingroup$ Would it be possible to create a 3D mesh of $[a,b] \times [c,d] \times[e,f]$, where the mesh density is adapted based on $f$ (with respect to a user-defined criterion that leaves the zone where $f$ is close to zero coarsely meshed), and then evaluated sigma in each of these adapted cells ? Maybe there are better adaptive quadrature techniques from which you can get sensible mesh points for $f$ and then use them for $\sigma$. $\endgroup$
    – Laurent90
    Commented Dec 21, 2022 at 14:21

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