I need to estimate the value of a 2d integral
$\int_{y_{min}}^{y_{max}}dy \int_{x_{min}}^{x_{max}} dx \, f(x,y) P(x,y)$
I have the an explicit analytical form for $P(x,y)$. I have samples of the function $f(x,y)$ at a set of nodes $(x_i,y_i)$, $i=1...n$, with $n \approx 100$, and no repetitions. The points are not distributed in the $xy$ plane according to any known quadrature scheme.
Assuming the value of the integral can be represented by the quadrature formula
$I \approx \sum w_{i} f(x_i,y_i) P(x_i,y_i)$
I would like to know if there is a procedure to calculate the weights $w_{i}$ associated to the set of nodes $(x_i,y_i)$.
I know that such a procedure exists in 1d (see, e.g., 1),
but what about 2D? Any code or reference will be very much appreciated.
