I want to minimize the functional $F[f(x)]$ and I'm going to try this in two different ways:
- First I am going to numerically minimize the functional $F[f(x)]$, leading to the "true solution" $f(x)$.
- Second I am going to insert a variational function $\varphi_a(x)$ into the functional, where $a$ is a variational parameter which is found by minimizing the functional $F[f(x)]$.
What I now want to know is how well my variational function $\varphi_a(x)$ approaches my correct solution $f(x)$. I was wondering if there are any general ways to do this? The goal is to eventually set up an objective criterion to compare three different variational function and pick out the best one to approximate my problem.
My first guess was to compare the norm of the difference in order to get one number that I can compare. This means that I will be comparing the number (let me call it $C$): $$ C=\sqrt{\int_D(f(x)-\varphi_a(x))^2} $$ for the three different variational functions. If the variational function $\varphi_a(x)$ and the true solution $f(x)$ coincide, $C$ will equal zero, if not $C$ will become nonzero.
Now I was wondering if this was the best way to do this, or if there are better methods available ?