# How to obtain the reduced model from a subspace projection method?

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:

1. Input: the data in the matrix $W$ consisting of the snapshots obtianed by simulating the full systems.
2. Calculate the singular value decomposition so that: $$W = UVD' .$$
3. 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.

• I'm not even going to try to explain proper orthogonal decomposition to a five-year-old. How about instead you tell us what you do know, and we take it from there? (For example, presumably you learned about subspace projection methods from somewhere.) – Christian Clason Jan 29 '16 at 15:57
• @ChristianClason Alright, I understand that the question appears vague. I'll see what more I can edit in. However, I am just learning about projection methods now, as I have only studied biology and programming. I'm reading the book Model Order Reduction - Theory, Research Aspects and Applications as we speak. Just so far I have been handed many ways to calculate POD (among others) and the basis vector, but I can't figure out how to proceed to see biologically meaningful results, if those are even obtainable. Thanks for the comment tho. – milez Jan 29 '16 at 16:53
• That would help; in particular the ODE system (without too many details) so we can use that in the explanation. In particular, it makes a huge difference whether the ODEs are linear (in which case things are straightforward) or nonlinear (in which case things get difficult)... – Christian Clason Jan 29 '16 at 17:30
• Also, the book you list is a research monograph (or rather, a collection of research/survey articles); a better starting point might be lecture notes, e.g., math.uni-konstanz.de/numerik/personen/volkwein/teaching/…, wire.tu-bs.de/OLDWEB/mameyer/cmr/cmr-main.pdf or journals.cambridge.org/action/… – Christian Clason Jan 29 '16 at 17:35
• It also makes a difference whether you want to reduce the dimension (number of equations) of the system or build a low-order model of the solution of the ODEs as a function of the parameter (both can be considered as model order reduction, but the approach is quite different). – Christian Clason Jan 29 '16 at 18:29