Model Reduction

Our quest to accurately simulate physical phenomena often leads to large-scale dynamical systems that require large data storage and long simulation times.

Model reduction seeks models of low order (low complexity) that approximate these large-scale systems; and, require only a fraction of the storage and simulation time.

Consider the following simulation of the transient evolution of the temperature field around a cylinder immersed into a fluid (here, we only show half of the cylinder):

The simulation corresponding to the large-scale system is shown in the left-most pane. This dynamical system contains n = 878 ordinary differential equations, i.e., n = 878 states, and was obtained using the following Finite Elements discretization:

However, we can perform the same simulation much more efficiently by computing a reduced-order model that has only k = 10 states, instead of 878 states. We show this simulation in the above animated middle pane.

Notice that the two simulations are virtually the same, but, the reduced-order model requires significantly less computation time and storage, as it contains only 10 differential equations. These observations are confirmed in the right-most pane, where we notice a very small error between the full- and reduced-order models.


Details

  • The above reduced-order model was computed using the Proper Orthogonal Decomposition (POD) approach. For details, see the Matlab code attached below, or, for example, the following book (which includes other standard model reduction algorithms such as modal approximation, balanced truncation, or moment matching).
  • Details on deriving the Finite Elements model can be found here. This full-order model is part of a wider collection of benchmarks available from augustine.mit.edu.

Matlab code

The above simulations can be run in Matlab by downloading the following files.