POD from pointclouds

We will now proceed to explain how to perform POD from point clouds. In this instance, we test only for POD in serial, as to perform in parallel, a parallel reader/writer is needed.

If you have saved information in hdf5 and have habilitated mpi4py compilation of it, then you could use this code in parallel.

Import general modules

[1]:

# Import required modules
from mpi4py import MPI #equivalent to the use of MPI_init() in C
import matplotlib.pyplot as plt
import numpy as np
import h5py

# Get mpi info
comm = MPI.COMM_WORLD

# Hide the log for the notebook. Not recommended when running in clusters as it is better you see what happens
import os
os.environ["PYSEMTOOLS_HIDE_LOG"] = 'true'

Set up the input parameters

[2]:

file_sequence = [f"../4-interpolation/interpolated_fields{str(1+i).zfill(5)}.hdf5" for i in range(0, 48)]
pod_fields = ["u", "v", "w"]
mesh_fname = "../4-interpolation/points.hdf5"
mass_matrix_fname = "../4-interpolation/points.hdf5"
mass_matrix_key = "mass"
k = len(file_sequence)
p = len(file_sequence)

Call the pysemtools routines

[3]:

from pysemtools.rom.phy_pod_wrappers import pod_from_files
# Output
from pyevtk.hl import gridToVTK

pod, ioh, field3d_shape = pod_from_files(comm, file_sequence, pod_fields, mass_matrix_fname, mass_matrix_key, k, p)

Write out the data

In this case we can easily write the modes from vtk

[4]:

# Load the mesh
with h5py.File(mesh_fname, 'r') as f:
    x = f["x"][:]
    y = f["y"][:]
    z = f["z"][:]

from pysemtools.rom.phy_pod_wrappers import write_3dfield_to_file

# Load the mesh
with h5py.File(mesh_fname, 'r') as f:
    x = f["x"][:]
    y = f["y"][:]
    z = f["z"][:]

# Write out 5 modes
write_3dfield_to_file("phys_pod.vtk", x, y, z, pod, ioh, modes=[i for i in range(0, 5)], field_shape=field3d_shape, field_names=pod_fields)

# Reconstruct all the snapshots with 5 modes
write_3dfield_to_file("phys_pod.vtk", x, y, z, pod, ioh, modes=[i for i in range(0, 5)], field_shape=field3d_shape, field_names=pod_fields, snapshots=[i for i in range(0, len(file_sequence))])

Writing ./phys_pod__mode0.vtk
Writing ./phys_pod__mode1.vtk
Writing ./phys_pod__mode2.vtk
Writing ./phys_pod__mode3.vtk
Writing ./phys_pod__mode4.vtk
Writing ./phys_pod_reconstructed_data_0
Writing ./phys_pod_reconstructed_data_1
Writing ./phys_pod_reconstructed_data_2
Writing ./phys_pod_reconstructed_data_3
Writing ./phys_pod_reconstructed_data_4
Writing ./phys_pod_reconstructed_data_5
Writing ./phys_pod_reconstructed_data_6
Writing ./phys_pod_reconstructed_data_7
Writing ./phys_pod_reconstructed_data_8
Writing ./phys_pod_reconstructed_data_9
Writing ./phys_pod_reconstructed_data_10
Writing ./phys_pod_reconstructed_data_11
Writing ./phys_pod_reconstructed_data_12
Writing ./phys_pod_reconstructed_data_13
Writing ./phys_pod_reconstructed_data_14
Writing ./phys_pod_reconstructed_data_15
Writing ./phys_pod_reconstructed_data_16
Writing ./phys_pod_reconstructed_data_17
Writing ./phys_pod_reconstructed_data_18
Writing ./phys_pod_reconstructed_data_19
Writing ./phys_pod_reconstructed_data_20
Writing ./phys_pod_reconstructed_data_21
Writing ./phys_pod_reconstructed_data_22
Writing ./phys_pod_reconstructed_data_23
Writing ./phys_pod_reconstructed_data_24
Writing ./phys_pod_reconstructed_data_25
Writing ./phys_pod_reconstructed_data_26
Writing ./phys_pod_reconstructed_data_27
Writing ./phys_pod_reconstructed_data_28
Writing ./phys_pod_reconstructed_data_29
Writing ./phys_pod_reconstructed_data_30
Writing ./phys_pod_reconstructed_data_31
Writing ./phys_pod_reconstructed_data_32
Writing ./phys_pod_reconstructed_data_33
Writing ./phys_pod_reconstructed_data_34
Writing ./phys_pod_reconstructed_data_35
Writing ./phys_pod_reconstructed_data_36
Writing ./phys_pod_reconstructed_data_37
Writing ./phys_pod_reconstructed_data_38
Writing ./phys_pod_reconstructed_data_39
Writing ./phys_pod_reconstructed_data_40
Writing ./phys_pod_reconstructed_data_41
Writing ./phys_pod_reconstructed_data_42
Writing ./phys_pod_reconstructed_data_43
Writing ./phys_pod_reconstructed_data_44
Writing ./phys_pod_reconstructed_data_45
Writing ./phys_pod_reconstructed_data_46
Writing ./phys_pod_reconstructed_data_47

Save the data as numpy arrays

[5]:

from pysemtools.rom.phy_pod_wrappers import save_pod_state

save_pod_state("pod_state.hdf5", pod)

Perform tests

[6]:

from pysemtools.rom.phy_pod_wrappers import physical_space

# Load the mass matrix
with h5py.File(mass_matrix_fname, 'r') as f:
    bm_v = f["mass"][:]

# Go through the snapshots in the file sequence
total_snapshot_energy = 0
for j, fname in enumerate(file_sequence):
    print(f"Testing snapshot {j}: {fname}")

    with h5py.File(fname, 'r') as f:

        # Check one snapshot at a time
        # Here one could also just use the phys that has been previously computed.
        phys = physical_space(pod, ioh, modes=[i for i in range(0, k)], field_shape=field3d_shape, field_names=pod_fields, snapshots=[j])

        # Check if the reconstruction is accurate
        for field in pod_fields:
            passed = np.allclose(phys[j][field], f[field][:])
            print(f"Field {field} passed: {passed}")

        # Check if the energy was accurately captured
        snapshot_energy = 0
        for field in pod_fields:
            snapshot_energy += np.sum(f[field][:]**2*bm_v)
        total_snapshot_energy += snapshot_energy

        reconstruction_energy = 0
        a_i = np.diag(pod.d_1t)@pod.vt_1t[:,j]
        reconstruction_energy += np.sum(np.abs(a_i)**2)

        print(f"Snapshot energy: {snapshot_energy}")
        print(f"Reconstruction energy: {reconstruction_energy}")

        print("=======================================")

# Check if the total energy is kept in the singular values
print(f"total snapshot energy: {total_snapshot_energy}")

total_pod_energy = 0
total_pod_energy += np.sum(pod.d_1t**2)
print(f"Total POD energy: {total_pod_energy}")
print("=======================================")

Testing snapshot 0: ../4-interpolation/interpolated_fields00001.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 3.698958237844183e-05
Reconstruction energy: 3.698959791995519e-05
=======================================
Testing snapshot 1: ../4-interpolation/interpolated_fields00002.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 3.8898814924962444e-05
Reconstruction energy: 3.8898829611720724e-05
=======================================
Testing snapshot 2: ../4-interpolation/interpolated_fields00003.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 4.153454569587408e-05
Reconstruction energy: 4.153456122670148e-05
=======================================
Testing snapshot 3: ../4-interpolation/interpolated_fields00004.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 4.4425076415412405e-05
Reconstruction energy: 4.442509363710398e-05
=======================================
Testing snapshot 4: ../4-interpolation/interpolated_fields00005.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 4.688317219673442e-05
Reconstruction energy: 4.688319417681433e-05
=======================================
Testing snapshot 5: ../4-interpolation/interpolated_fields00006.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 4.878401448156997e-05
Reconstruction energy: 4.8784037017177103e-05
=======================================
Testing snapshot 6: ../4-interpolation/interpolated_fields00007.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 5.026805630070726e-05
Reconstruction energy: 5.026807862059571e-05
=======================================
Testing snapshot 7: ../4-interpolation/interpolated_fields00008.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 5.149475955390006e-05
Reconstruction energy: 5.149478153993227e-05
=======================================
Testing snapshot 8: ../4-interpolation/interpolated_fields00009.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 5.237488647324383e-05
Reconstruction energy: 5.237490538075368e-05
=======================================
Testing snapshot 9: ../4-interpolation/interpolated_fields00010.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 5.263423351722945e-05
Reconstruction energy: 5.263424808861043e-05
=======================================
Testing snapshot 10: ../4-interpolation/interpolated_fields00011.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 5.187294605439943e-05
Reconstruction energy: 5.187296300851487e-05
=======================================
Testing snapshot 11: ../4-interpolation/interpolated_fields00012.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 4.988241310699623e-05
Reconstruction energy: 4.98824339321701e-05
=======================================
Testing snapshot 12: ../4-interpolation/interpolated_fields00013.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 4.697643854683935e-05
Reconstruction energy: 4.697645923683903e-05
=======================================
Testing snapshot 13: ../4-interpolation/interpolated_fields00014.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 4.3863242905911426e-05
Reconstruction energy: 4.386325835715201e-05
=======================================
Testing snapshot 14: ../4-interpolation/interpolated_fields00015.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 4.04654975068024e-05
Reconstruction energy: 4.046551073804368e-05
=======================================
Testing snapshot 15: ../4-interpolation/interpolated_fields00016.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 3.708331604360593e-05
Reconstruction energy: 3.70833306496177e-05
=======================================
Testing snapshot 16: ../4-interpolation/interpolated_fields00017.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 3.361937142993228e-05
Reconstruction energy: 3.3619385479765e-05
=======================================
Testing snapshot 17: ../4-interpolation/interpolated_fields00018.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 3.042199960394419e-05
Reconstruction energy: 3.042201288840405e-05
=======================================
Testing snapshot 18: ../4-interpolation/interpolated_fields00019.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.764625401122964e-05
Reconstruction energy: 2.7646265699721833e-05
=======================================
Testing snapshot 19: ../4-interpolation/interpolated_fields00020.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.5278143358328515e-05
Reconstruction energy: 2.527815340611091e-05
=======================================
Testing snapshot 20: ../4-interpolation/interpolated_fields00021.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.3436562845034242e-05
Reconstruction energy: 2.3436571745893377e-05
=======================================
Testing snapshot 21: ../4-interpolation/interpolated_fields00022.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.2002654602291965e-05
Reconstruction energy: 2.200266188237421e-05
=======================================
Testing snapshot 22: ../4-interpolation/interpolated_fields00023.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.0920245682595327e-05
Reconstruction energy: 2.0920252797345817e-05
=======================================
Testing snapshot 23: ../4-interpolation/interpolated_fields00024.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.013654268431107e-05
Reconstruction energy: 2.0136550044303895e-05
=======================================
Testing snapshot 24: ../4-interpolation/interpolated_fields00025.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 1.9632578477660025e-05
Reconstruction energy: 1.9632586126642394e-05
=======================================
Testing snapshot 25: ../4-interpolation/interpolated_fields00026.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 1.933656650853822e-05
Reconstruction energy: 1.9336575494338164e-05
=======================================
Testing snapshot 26: ../4-interpolation/interpolated_fields00027.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 1.9193896285463495e-05
Reconstruction energy: 1.9193905632897007e-05
=======================================
Testing snapshot 27: ../4-interpolation/interpolated_fields00028.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 1.9209913696544347e-05
Reconstruction energy: 1.9209922552457982e-05
=======================================
Testing snapshot 28: ../4-interpolation/interpolated_fields00029.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 1.9433633251162603e-05
Reconstruction energy: 1.9433641564702733e-05
=======================================
Testing snapshot 29: ../4-interpolation/interpolated_fields00030.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 1.9843626264053833e-05
Reconstruction energy: 1.984363462420495e-05
=======================================
Testing snapshot 30: ../4-interpolation/interpolated_fields00031.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.0264037570696985e-05
Reconstruction energy: 2.026404726721621e-05
=======================================
Testing snapshot 31: ../4-interpolation/interpolated_fields00032.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.061492121971811e-05
Reconstruction energy: 2.0614930665485913e-05
=======================================
Testing snapshot 32: ../4-interpolation/interpolated_fields00033.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.0873392702196558e-05
Reconstruction energy: 2.0873401695293363e-05
=======================================
Testing snapshot 33: ../4-interpolation/interpolated_fields00034.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.100146581923816e-05
Reconstruction energy: 2.1001474879010305e-05
=======================================
Testing snapshot 34: ../4-interpolation/interpolated_fields00035.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.1002631484979843e-05
Reconstruction energy: 2.1002640205096e-05
=======================================
Testing snapshot 35: ../4-interpolation/interpolated_fields00036.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.0926017391762055e-05
Reconstruction energy: 2.0926025698532318e-05
=======================================
Testing snapshot 36: ../4-interpolation/interpolated_fields00037.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.0846074698639117e-05
Reconstruction energy: 2.0846082378820277e-05
=======================================
Testing snapshot 37: ../4-interpolation/interpolated_fields00038.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.0866571618859654e-05
Reconstruction energy: 2.086657965562247e-05
=======================================
Testing snapshot 38: ../4-interpolation/interpolated_fields00039.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.101439294141517e-05
Reconstruction energy: 2.1014401790563164e-05
=======================================
Testing snapshot 39: ../4-interpolation/interpolated_fields00040.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.1135889588706728e-05
Reconstruction energy: 2.1135897678024467e-05
=======================================
Testing snapshot 40: ../4-interpolation/interpolated_fields00041.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.1184459915354247e-05
Reconstruction energy: 2.1184468317008704e-05
=======================================
Testing snapshot 41: ../4-interpolation/interpolated_fields00042.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.1250999040067878e-05
Reconstruction energy: 2.1251006836458486e-05
=======================================
Testing snapshot 42: ../4-interpolation/interpolated_fields00043.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.1377532850036896e-05
Reconstruction energy: 2.1377540941763505e-05
=======================================
Testing snapshot 43: ../4-interpolation/interpolated_fields00044.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.1529543422705357e-05
Reconstruction energy: 2.1529553102361603e-05
=======================================
Testing snapshot 44: ../4-interpolation/interpolated_fields00045.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.161399267113895e-05
Reconstruction energy: 2.1614003298549693e-05
=======================================
Testing snapshot 45: ../4-interpolation/interpolated_fields00046.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.1637133597399283e-05
Reconstruction energy: 2.163714439898288e-05
=======================================
Testing snapshot 46: ../4-interpolation/interpolated_fields00047.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.1440817225395135e-05
Reconstruction energy: 2.1440827260570388e-05
=======================================
Testing snapshot 47: ../4-interpolation/interpolated_fields00048.hdf5
Field u passed: True
Field v passed: True
Field w passed: True
Snapshot energy: 2.0948984478946528e-05
Reconstruction energy: 2.0948993077000546e-05
=======================================
total snapshot energy: 0.001434071843040977
Total POD energy: 0.0014340724222272252
=======================================