# Multivariate numerical integration with a non-uniform grid

I want to approximate the integral:

$$I = \int f(\boldsymbol{x})d\boldsymbol{x}$$

where $\boldsymbol{x}$ is $d$-dimensional. I have a set of non-equally spaced points $\boldsymbol{x}_1, \dots, \boldsymbol{x_n}$, and the corresponding values $f(\boldsymbol{x}_1), \dots, f(\boldsymbol{x_n}$), and I would like to use them to approximate the integral $I$. Actually the integral is over $R^d$, but I know that $f(\boldsymbol{x})$ goes to zero quite rapidly outside my sample $\boldsymbol{x}_1, \dots, \boldsymbol{x_n}$.

My question is the following: are there rules to approximate the integral using the non-equally spaced points? I've found simple rules for the univariate case, but for multiple dimensions. Thanks

Note: the points $\boldsymbol{x}_1, \dots, \boldsymbol{x_n}$ are given and I can't choose them.

• What dimension is your integral (line, surface, volume), and what the bounds? – meawoppl Jun 17 '13 at 18:27
• I have between 3 and 7 dimensions, depending on the problem. The integral is over $R^d$, but the sample covers most of the mass, so I could assumes that the area outside the sample doesn't contribute to the integral. – Jugurtha Jun 18 '13 at 8:43
• I might be tempted to try doing some form of Kernel Density Estimate to get your sampling to act uniform, then move forward from there. – meawoppl Jun 22 '13 at 21:37
• I guess the problem with KDE is that the quality of the estimates degrade very rapidly as $d$ increases. – Jugurtha Jun 22 '13 at 23:30
• What is your total point count? – meawoppl Jun 23 '13 at 17:49

If the number of points is small and you are in a low-dimensional situation, then you can think of this as trying to find a polynomial that interpolates $f(x)$ and the weights are then computed so that the integral over $f(x)$ is chosen as the integral over the approximating polynomial.