POD on data

In this example, the Direct Proper Orthogonal Decomposition (POD) method is used to decompose a set of data and extract basis functions which can be used for the reconstruction of the solution.

Import the necessary libraries. Here we import standard libraries such as numpy, matplotlib, and we also import the POD class from UQpy.

from UQpy.dimension_reduction import DirectPOD
import time
import numpy as np

Input dataset in the form of a second order tensor consists of three snapshot solutions. Using the POD method we reconstruct the input dataset by keeping a predefined number of modes.

Data = np.zeros((3, 5, 3))

Data[:, :, 0] = [
    [0.9073, 1.7842, 2.1236, 1.1323, 1.6545],
    [0.8924, 1.7753, -0.6631, 0.5654, 2.1235],
    [2.1488, 4.2495, 1.8260, 0.3423, 4.9801]]

Data[:, :, 1] = [
    [0.7158, 1.6970, -0.0740, 5.478, 1.0987],
    [-0.4898, -1.5077, 1.9103, 6.7121, 0.5334],
    [0.3054, 0.3207, 2.1335, 1.1082, 5.5435]]

Data[:, :, 2] = [
    [-0.3698, 0.0151, 1.4429, 2.5463, 6.9871],
    [2.4288, 4.0337, -1.7495, -0.5012, 1.7654],
    [2.3753, 4.7146, -0.2716, 1.6543, 0.9121]]

The Direct POD method is used to compute the reconstructed solutions and reduced-order solutions in the spatial dimension of the data. Full reconstruction is achieved when the number of modes chosen, equals the number of dimensions.

start_time = time.time()

pod = DirectPOD(solution_snapshots=Data, n_modes=1)
Data_reconstr = pod.reconstructed_solution

Print the reconstructed dataset.

print('Reconstructed snapshot no.1:')
print(Data_reconstr[:, :, 0])

if np.allclose(Data, Data_reconstr[0]) == True:
    print('Input data and reconstructed data are identical.')

elapsed_time = time.time() - start_time
time.strftime("%H:%M:%S", time.gmtime(elapsed_time))
print('Elapsed time: ', round(elapsed_time, 8))