Inverse Transformation from Response Surface

[Omega0]

TL;DR

If I get global parameters from data which can be approximated with a response surface, how do I interpolate data from it.

Let us say we have data which is for simplicity in N tables. All the tables have the same number of rows and columns. The columns $A_i$ have for all tables the same meaning (say measured quantaties like pressure, temperature) where the first 3 columns is the position in space. Again for simplicity, let the quantities for all tables be identified in the same space coordinates. This means we have N tables with the same structure but the values themselves are different. The main reson for the difference is that I change the physical conditions: We vary parameters $b_k$ (like angles etc.) called independent parameters. This could be a setup for a Design of Experiment study. So, we have a parametric study.
Now we calculate somehow integral quantities $I_m$ for all the tables. Those quantities are called our dependent parameters. We now analyze if we can see a dependency between $I_m$ and $b_k$. Here we will have a bunch of methods $M_j$. Let us say $M_1$ is a quadratic response surface for the dependent parameter $I_1$ and the both independent parameters $b_1$ and $b_2$. This means basically we believe that a second order interpolation is representing the dependancy good enough. Having done this we get a smooth function $I_1(b_1,b_2)$ where $b_1$ and $b_2$ are real numbers. With other words: We get results for arbitrary independent values (for better imagination, those are typically values between two discrete really existing independent parameters).

Okay, now me question: For such an arbitrary pair of independent values I would like to calculate or recreate the table from which the dependent values came. The table would represent an interpolation of the really excisting values in the N tables.

I know that this is a standard thing in engineering but I am not sure how it works. I would guess in the dark that I take the unique N vectors of my tables and run an optimizer to find the distribution of weight for the N vectors to get the $I_1$. Next step would be to use the weighting for the tables to get a new table, representing the interpolation. This would be pretty simple if we use linear optimization but we have to take into account that my response surface is non-linear. In arbitrary cases the function could be of higher order or some way more fancy approximation.
How do I get my table? Thanks.

EDIT: I think that it doesn't matter how complex the function is because I am just interested in one point after each other or with other words: I solve a linear optimization for any point I am interested in, right? It seems like calculus in arbitrary dimensions.

 

[Stephen Tashi]

I suggest you make another attempt to state you question. Your description of the data isn't clear and you haven't stated a well defined mathematical question. Perhaps a specific example would be the simplest way to explain things.