I have a heat transfer system described by, $$\{\dot{T}\} = [C^{-1}]\left([K]\{T\} + \{F\} \right)$$ where ${T}$ is a vector of the nodal temperatures of the system. From initial conditions I am able to simulate the transient heat of the system by successive iteration of: $$\{T_{i+1}\} = \{\dot{T_i}\}\Delta t + \{T_i\}$$
My problem is, I have good values for most of the constituent parameters of $[K]$ except for one, which is a function of (I assume) the form: $\frac{a}{b\cdot v^2}$. I am able to generate test data of the transient temperatures of all the nodes with a fixed $v$; however, I'm not sure how to find the parameters $a$ and $b$.
Rather than proceed with a brute force "check and see", I thought there must be some form of gradient descent or other method to fit. I'm not sure how I would generate an objective function from empirical data (do I do a fit?) as well as generate the matrix equation to be solved. To summarize, I have a single element in $[K]$ which is a function I'm trying to find and I'm not sure how to do a regression of the multiple iterations (the numerical integration) and how to characterize t