Reduced-order modeling refers to building low-dimensional models
for large- and infinite-dimensional dynamical systems. "Low" can typically
be on the order of 10-100, and "large" can be as large as 10^6 or more.
The premise is that there must exist underlying low-dimensional dynamics
for the problem and hopefully something like an inertial manifold.
The computer reduced-order models have wide-ranging applications
from
- prediction: reduced-order models have their roots in weather
forecasting. In fact, the well-known Lorenz equations were
developed as a reduced-order model for Benard convection. Today,
reduced-order models play a role in determining initial conditions
for forecasting and climate models using a procedure known as
4DVar (look
here for more
details and links).
- control: where the reduced-order model is used to generate
feedback control laws or state estimators for complex systems.
Recently, reduced-order models have been studied for their potential
in solving the Hamilton-Jacobi equations arising in nonlinear
feedback control (look
here
for more details and links).
- optimization: where the reduced-order model is used as a
surrogate model in the optimization in problems such as
PDE-constrained optimization. (Initial work in this area is being
conducted by Ekkehard Sachs.
For nonlinear problems, the current strategy of choice is projection.
Thus, reduced-order models are built in two parts. The first is to
construct a suitable basis on which to represent the solution. The
second is to project the differential equations governing the dynamics
onto this basis. For the latter part, Galerkin projection is typically
used. Most of the "art" in this field is in building suitable bases
and developing fast, reliable algorithms to do this.
Our research group aims to study the entire reduced-order modeling
problem (explore improvements in both pieces of the problem).
Here are movies of flow snapshots (solutions courtesy
Alexander Hay and Dominique Pelletier, Ecole Polytechnique de Montreal).
Note that a mesh containing over 7152 elements, 21626 nodes and over
50000 variables were used to generate this solution. Note the von
Karman vortex street at this modest Reynolds number of 100.
Usnap.mov or Usnap.avi
Vsnap.mov or Vsnap.avi
The basic idea is to generate low-dimensional global basis functions
for this solution. In this example, we use the singular value decomposition;
also known as proper orthogonal decomposition (POD), principle component
analysis (PCA), Karhunen-Loeve expansion (KLE), empirical orthogonal
functions (EOFs), to generate a 12 degree of freedom model (6 basis
functions for each component of velocity).
Basis for horizontal velocity component, u
Using this basis, we evolve a 12th order nonlinear differential
equation to obtain the following simulations. The movie of the
horizontal velocity component follows.
Same time window (one period of oscillation)
U_POD.mov
Extrapolation over several periods U_PODextrap.mov
or U_PODextrap.avi.
Dynamically Optimal Basis
Functions
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