Intrusive Model Reduction

POD

Proper orthogonal decomposition originated from the analysis of turbulent flows in aerodynamics, and it has become one of the most widespread projection-based model reduction methods. POD reduces the model by projecting it onto a reduced subspace defined to be the span of basis vectors that optimally represent a set of simulation or experimental data. See the original literatures on POD [3], [4], [5].

In POD, we begin by collecting snapshots of state trajectory time series data by simulating the original full model ODE with $K$ timesteps. We define the state snapshot data matrix as follows:

\[ \mathbf{X} = \begin{bmatrix} | & | & & | \\ \boldsymbol{\mathbf x}(t_1) & \mathbf{x}(t_2) & \cdots & \mathbf{x}(t_K) \\ | & | & & | \\ \end{bmatrix} \in \mathbb{R}^{n\times K} ~.\]

More generally, the state snapshot matrix can contain state data from multiple simulations, e.g., from different initial conditions or using different parameters. Let $\mathbf{X} = \mathbf{V\Sigma W}^\top$ denote the singular value decomposition of the state snapshot. To reduce the dimension of the large-scale model, we denote by $\mathbf{V}_r\in \mathbb R^{n\times r}$ the first $r \ll n$ columns of $\mathbf V$; this is called the POD basis. Then, we approximate the state $\mathbf{x}$ in the subspace spanned by the POD basis, $\mathbf x \approx \mathbf V_r \hat{\mathbf x}$ where $\hat{\mathbf x}\in\mathbb{R}^r$ is called the reduced state. If we substitute this approximation into a linear-quadratic system and enforce the Galerkin orthogonality condition that the approximation residual be orthogonal to the span of $\mathbf V_r$, we arrive at a POD-Galerkin reduced model of the form

\[ \dot{\hat{\mathbf x}}(t) = \mathbf{\hat A}\hat{\mathbf x}(t) + \hat{\mathbf{H}}(\hat{\mathbf{x}}(t) \otimes \hat{\mathbf{x}}(t)),\]

where the reduced operators are $\mathbf{\hat{A}} = \mathbf{V}^\top_r \mathbf{AV}_r \in \mathbb{R}^{r\times r}$ and $\hat{\mathbf{H}} = \mathbf{V}^\top_r \mathbf{H}(\mathbf{V}_r \otimes \mathbf{V}_r) \in \mathbb R^{r\times r^2}$.

Implementation

The implementation of this corresponds to the following function:

pod(op, Vr, sys_struct; nonredundant_operators)