I have a system of ordinary differential equations (ODEs). It is a large system that has dozens of equations and hundreds of parameters. I wish to reduce its size so it becomes computationally more efficient to simulate, and I'm willing to tolerate some error in the reduced model. So the end result should be a system of ODEs that has a smaller number of equations or parameters.
I can supposedly use methods of subspace projection to obtain a reduced model. For example Principal component analysis / Proper orthogonal decomposition (POD) with the method of snapshots. With this method a basis is obtained.
This leads to the following POD algorithm:
- Input: the data in the matrix $W$ consisting of the snapshots obtianed by simulating the full systems.
- Calculate the singular value decomposition so that: $$W = UVD' .$$
- Output: The first $l$ columns of $U$ form the reduced subspace and the basis is then $\Phi = [u_1, ... , u_l]$.
If the coefficients of my linear ODE system are in $A$, I can use $\Phi' A \Phi$ to project the system to a reduced subspace and simulate it there.
Having obtained the simulation results $Y$ in the reduced space, I can do $\Phi Y$ to project the results back to the original space.
Now if $A$ models a biological system with meaningful equations, the same cannot be said for $\Phi' A \Phi$. But is there any way $\Phi$ can be used to remove equations or variables from $A$ to obtain a simpler 'real world' system?
Edit: I'm not very capable of reading math, so if possible, please explain like I am five.