Background
I'm currently working with a system that has a 4-dimensional function. Currently, an algorithm is used to speed up calculation of the final value via interpolation, and two of the independent variables are used to key off lookup tables based on the remaining 2. For example, I have 4 independent variables $\alpha_i, \alpha_e, n,$ and $a_l$. There exists a function $\eta = f(\alpha_i,\alpha_e,n,a_l)$. Due to the complex nature of this function and the need to get a decent estimation in quick time, lookup tables are used. By fixing $\alpha_i$ and $\alpha_e$, tables are then built essentially providing $\eta_{\alpha_i,\alpha_e} = f(n,a_l)$.
Several such tables are built, and then a weighted interpolation is built between them for the final answer.
I.e. given 4 such tables, $\eta_{-10,0},\eta_{-20,10}, \eta_{-30,20}, \eta_{-40,30}$ and a given $(\alpha_i,\alpha_e)$, we calculate the final value based on $$\eta = w_1.\eta_{-10,0}(n,a_l) + w_2.\eta_{-20,10}(n,a_l) + w_3.\eta_{-30,20}(n,a_l) + w_4.\eta_{-40,30}(n,a_l)$$
Example
Given a basis for our lookups: $$(\alpha_i,\alpha_e) = (-10,0);(-20,10);(-30,20);(-40,30)$$
Assuming an $(\alpha_i,\alpha_e) = (-15, 27)$.
We calculate the residuals as follows: $$r_1 = (\alpha_i - -10, \alpha_e - 0) = (-5, 27)\\ r_2 = (\alpha_i - -20, \alpha_e - 10) = (5, 17)\\ r_3 = (\alpha_i - -30, \alpha_e - 20) = (15, 7)\\ r_4 = (\alpha_i - -40, \alpha_e - 30) = (25, -3)$$
Now we calculate intermediate weights: $$v_1 = \frac{1}{r_{1,\alpha_i}^2 + r_{1,\alpha_e}^2} = 0.00132625994\\ v_2 = \frac{1}{r_{2,\alpha_i}^2 + r_{2,\alpha_e}^2} = 0.00318471337\\ v_3 = \frac{1}{r_{3,\alpha_i}^2 + r_{3,\alpha_e}^2} = 0.00364963503\\ v_4 = \frac{1}{r_{4,\alpha_i}^2 + r_{4,\alpha_e}^2} = 0.00157728706$$
The final weights are then: $$w_1 = \frac{v_1}{\sum v_i} \approx 0.136 \\ w_2 = \frac{v_2}{\sum v_i} \approx 0.327 \\ w_3 = \frac{v_3}{\sum v_i} \approx 0.375 \\ w_4 = \frac{v_4}{\sum v_i} \approx 0.162 $$
Question
Does this particular method of finding weights for the interpolation algorithm have a name? I did not design it, but I must now work with it and would like to learn more about the math behind it. The calculation of the weights seems like some sort of projection of a given vector onto a linear combination of the basis vectors using square errors, but I don't recognize it.
My goal is to know how to pick good basis vectors to give a nice interpolation knowing I'll operating over a certain range of $\alpha_i$ and $\alpha_e$, and more importantly to understand more about the properties of this particular interpolation algorithm.
Thank you for your time!