# Interpreting numerical double integration as a matrix multiplication

I have a convex optimization problem of finding a function Q(x,y) as below:

Minimize $\int{k(x,y)Q(x,y)dxdy}$ subject to a list of constraints which are not relevant to the question, so I'm skipping them. The function $k(x,y)$ is known and my aim is to find the form of $Q(x,y)$. I was thinking if I could represent this double integration as a matrix multiplication and I convexly minimize over the unknown matrix of $Q(i,j)$ values, then I can curve fit or do something from the values available. My question is how do I represent a double integral as a matrix multiplication? Say, the limits are (-10,10) for both variables' integration.

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## 1 Answer

Okay found it out myself. It could be done the following way: Compute $k(x,y)$ for each incrementation and make a matrix $K$ which contain the values of $k(x,y)$ all through the interval for specified x,y limits. $Q(x,y)$ becomes a matrix itself which is to the optimization variable. Now the objective is:

Minimize $\int\int k(x,y)Q(x,y) dx dy$ = minimize sum(sum(K.* Q)) where {.*} means scalar multiplication of the matrices. sum(sum()) simply adds all the values in the matrix, which is exactly the numerical integration result.

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