snapshot_pod.rst

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 × N_y where N_x, N_y are the grid dimensions. Thus, by using the .SnapshotPOD class, one can reconstruct solutions much faster (:citePOD_2).

For the Snapshot POD, again a two-dimensional dataset U ∈ ℝ^n × m is constructed where m is the number of snapshots and n is the number of problem dimensions. The covariance matrix 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

CA_s = λA_s

Spatial coefficients are therefore calculated as Φ_s = U^TA_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

UQpy.dimension_reduction.pod.SnapshotPOD

Attributes

UQpy.dimension_reduction.pod.SnapshotPOD.reconstructed_solution

UQpy.dimension_reduction.pod.SnapshotPOD.reduced_solution

UQpy.dimension_reduction.pod.SnapshotPOD.U

UQpy.dimension_reduction.pod.SnapshotPOD.eigenvalues

UQpy.dimension_reduction.pod.SnapshotPOD.phi