SnapshotPOD

The Snapshot Proper Orthogonal Decomposition (POD) method is the second variant of the POD method which considers the decomposition of a dataset into deterministic temporal modes and random spatial coefficients. Essentially, this method interchanges the time and position. In most problems the number of solution snapshots \(m\) is less than the number of dimensions \(n = N_x \times N_y\) where \(N_x, N_y\) are the grid dimensions. Thus, by using the SnapshotPOD class, one can reconstruct solutions much faster (Sirovich [6]).

For the Snapshot POD, again a two-dimensional dataset \(\mathbf{U}\in \mathbb{R}^{n\times m}\) is constructed where \(m\) is the number of snapshots and \(n\) is the number of problem dimensions. The covariance matrix \(\mathbf{C_s}\), is calculated as follows

\[\mathbf{C_s} = \frac{1}{m-1} \mathbf{U} \mathbf{U}^T\]

The eigenvalue problem is solved and the temporal modes (eigenvectors) are calculated as

\[\mathbf{C} A_s = \lambda A_s\]

Spatial coefficients are therefore calculated as \(\Phi_s = \mathbf{U}^T A_s\). Finally, a predefined number of \(k\)-POD temporal modes and spatial coefficients can be considered for the reconstruction of data as follows

\[\mathbf{\tilde{u}}(\mathtt{x},t) = \sum_{i=1}^{k} A_{si}(t) \Phi_{si}(\mathtt{x})\]

SnapshotPOD Class

The SnapshotPOD class is imported using the following command:

>>> from UQpy.dimension_reduction.pod.SnapshotPOD import SnapshotPOD

One can use the following command to instantiate the class SnapshotPOD

Methods

class SnapshotPOD(solution_snapshots=None, n_modes=None, reconstruction_percentage=None)[source]

Parameters:

solution_snapshots (Union[ndarray, list, None]) –
Array or list containing the solution snapshots. If provided as an numpy.ndarray, it should be three-dimensional, where the third dimension of the array corresponds to the number of snapshots. If provided as a list, the length of the list corresponds to the number of snapshots.

If solution_snapshots is provided, the run() method will be executed automatically. If it is not provided, then the run() method must be executed manually and provided with solution_snapshots.
n_modes (Optional[int]) – Number of POD modes used to approximate the input solution. Must be less than or equal to the number of dimensions in a snapshot. Either n_modes or reconstruction_percentage must be provided, but not both.
reconstruction_percentage (Union[int, float, None]) – Specified dataset reconstruction percentage. Must be between 0 and 100. Either n_modes or reconstruction_percentage must be provided, but not both.

run(solution_snapshots)[source]: Executes proper orthogonal decomposition using the SnapshotPOD algorithm.

Attributes

SnapshotPOD.reconstructed_solution: An array containing the solution snapshots reduced in the spatial dimension.

SnapshotPOD.reduced_solution: Second order tensor containing the reconstructed solution snapshots in their initial spatial and temporal dimensions.

SnapshotPOD.U: Two dimensional dataset constructed by POD

SnapshotPOD.eigenvalues: Eigenvalues produced by the POD decomposition of the solution snapshots.

SnapshotPOD.phi: Eigenvectors produced by the POD decomposition of the solution snapshots.