I have a function $c ( I (\vec{r}) )$. Not a constant, c doesn't denote a constant. So $c$ is a function of $I$ which is a function of $\vec{r}$.

This is hard to sample and I have sampled it for 10,000 values of $I$.

I need to integrate $c$ across all the space, i.e. a 3Dimensional integral: $d^3 r$.

$$\int\int\int c(I(\vec{r})) d^3 r$$

I want to use the already sampled 10,000 $c$'s.

Is there a method to numerically integrate my $c$ using those samples I already have? The values of $I$ at which $c$ is sampled are equally spaced in logspace. $c$ dies (goes to 0) apart from a very small region in the $d^3 r$ space.

I only have $c(I)$ and not $c(\vec{r})$. I can create a routine which outputs $I$ from a vector $\vec{r}$ if needed. Again, the sampling of $c(I)$ is hard and I cannot sample $c(\vec{r})$, i.e. I cannot directly sample $c$ from a vector $\vec{r}$, but only from a value of $I$.

A photo of how c looks like:c versus I

Thank you!

  • $\begingroup$ To clarify: You have $c_i = c(I_i)$ and you know for each $I_i$ what the value of $\vec{r}_i$ was? $\endgroup$ – user619894 Mar 8 at 13:08
  • $\begingroup$ The $\vec{r}$ never entered into the problem when sampling the $c_i$ values. This sampling was done using a code which takes as input $I$ and outputs $c$. I don't know what the value of $\vec{r}$ was for each of those $I_i$. $\vec{r}$ only comes into play at the integration part. So I don't know what the value of $\vec{r}_i$ was for my sampled $c_i$ values $\endgroup$ – velenos14 Mar 8 at 13:10
  • $\begingroup$ I have added a photo to show how the c looks like. $\endgroup$ – velenos14 Mar 8 at 13:17
  • $\begingroup$ So to make this clearer. For some specific values of $\vec{r}$, I have the value of $c$. For other values of $\vec{r}$, I don't have the value of $c$. How can I use only those specific values of $\vec{r}$ I have the $c$ value at to integrate $c$ across the volume space? $\endgroup$ – velenos14 Mar 8 at 13:26
  • $\begingroup$ What are these "specific" values? the ones for which you have sampled? A subset of them? $\endgroup$ – user619894 Mar 8 at 13:34

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