I have been looking into reducing the dimensionality of (nonlinear ordinary differential) equation systems in order to reduce online computation time. I have interpreted 'dimensions' as the number of equations or unknown variables in the systems, but it seems that this is not always the true meaning.

Now this lecture shows multiple examples of single equations or small systems of a couple of equations that are said to have hundreds or thousands of dimensions, so what does 'dimensionality' actually mean here?

For example the FitzHugh-Nagumo system supposedly has 1024 dimensions:

$$ \epsilon v_t (x,t) = \epsilon ^2 v_xx (x,t) + f(v(x,t))-w(x,t)+c \\ w_t(x,t) = bv(x,t)-\gamma w(x,t) + c $$

This just leads to so much confusion since I am not sure if any of these 'dimensionality 'reduction' methods can actually be used for reducing simulation time. Thanks for your time.

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    $\begingroup$ I think they are referring to dimension as in the number of variables used in the numerical method (e.g. number of grid points). Using model reduction allows one to solve a much smaller system of equations that still gives a "reasonable" solution to the full system. $\endgroup$ – David Feb 19 '16 at 2:16
  • $\begingroup$ @David Thank you for the comment. Do you know why they would do this, I mean what do they gain with solving a "full state space" reduced system after the real one has already been solved? I do see applications where the dynamics need to solved fast for some initial values, but the entire dimension.. $\endgroup$ – milez Feb 19 '16 at 8:02
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    $\begingroup$ In practice, you may be solving systems with far far more dimensions, where it is impractical to solve the full system, and so reducing the dimension is a way of making an approximation possible. In the example in your linked pdf, they are solving the full system to compare to the reduced system, to show it works. Check out how much time it took to solve the reduced and full systems. $\endgroup$ – David Feb 19 '16 at 21:39

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