galerkin()

galerkin()#

ParametricContinuousModel.galerkin(Vr, Wr=None)#

Construct a reduced-order model by taking the (Petrov-)Galerkin projection of each model operator.

Consider a model \(\z = \f(\q, \u)\) where

  • \(\q\in\RR^n\) is the model state,

  • \(\u\in\RR^m\) is the input, and

  • \(\z\in\RR^n\) is the model left-hand side.

Given a trial basis \(\Vr\in\RR^{n\times r}\) and a test basis \(\Wr\in\RR^{n\times r}\), the corresponding intrusive reduced-order model is the model \(\zhat = \fhat(\qhat, \u)\) where

\[\zhat = \Wr\trp\z, \qquad \fhat(\qhat,\u) = (\Wr\trp\Vr)^{-1}\Wr\trp\f(\Vr\qhat,\u).\]

Here,

  • \(\qhat\in\RR^r\) is the reduced-order state,

  • \(\u\in\RR^m\) is the input (as before), and

  • \(\zhat\in\RR^r\) is the reduced-order left-hand side.

This approach uses the low-dimensional state approximation \(\q = \Vr\qhat\). If \(\Wr = \Vr\), the result is called a Galerkin projection. If \(\Wr \neq \Vr\), it is called a Petrov-Galerkin projection.

Parameters:
Vr(n, r) ndarray

Basis for the trial space.

Wr(n, r) ndarray or None

Basis for the test space. If None, defaults to Vr.

Returns:
reduced_modelModel

Reduced-order model obtained from (Petrov-)Galerkin projection.

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