Quadrature integration of spherical function

[coderdave]

I have a function that is parametrized by a direction and returns a signal strength,

$$F(\theta, \phi) \rightarrow R$$

It's a very smooth function and I have been using Monte Carlo to do the integration. I just pick a random direction, sample, and average later. It's worked but since its smooth I figured I could use a quadrature style integration to see if it converges faster.

I just don't know how to set up my integrand to do that because all texts I've read on it usually talk about quadrature in euclidean coordinates.

Thank you,
-= Dave

 

[jvicens]

Hello Dave,

Thank you for sharing your approach to integrate your parametrized function. It sounds like you have already made some progress using Monte Carlo integration, which is a commonly used method for numerical integration. However, as you mentioned, quadrature integration may be a more efficient approach for your smooth function.

Quadrature integration, also known as numerical integration, is a method for approximating the definite integral of a function using a finite number of evaluations. This approach can be applied to functions in any coordinate system, including parametrized functions like yours. To set up the integrand for quadrature integration, you will need to discretize your function and then use a numerical integration formula, such as the trapezoidal rule or Simpson's rule, to approximate the integral.

To discretize your function, you will need to divide the range of your parameters (in this case, \theta and \phi) into a finite number of intervals. Then, within each interval, you can evaluate your function at a set of sample points and use the numerical integration formula to approximate the integral within that interval. Finally, you can sum up the approximations from each interval to get an overall approximation for the integral of your function.

There are many resources available online that explain the process of setting up the integrand for quadrature integration in more detail. I would suggest looking for resources specifically on numerical integration in parametrized functions to get a better understanding of how to apply this method to your function.

I hope this helps and good luck with your integration!