How to perform the online phase?

The BuoyantCavity problem is parameter-dependent and before performing the online phase some changes in the directory should be made.

The structure of the folder would look like

BuoyantCavity
    |---> allrun.py
    |---> BaseCase.py
    |---> TrainSet
    |    |
    |    |---> Case_000_*
    |    |---> Case_001_*
    |    |---> ...
    |    |---> POD_T
    |    |---> POD_U
    |    |---> EIM_T
    |    |---> ...
    |---> TestSet
    |    |
    |    |---> Case_000_*
    |    |---> Case_001_*
    |    |---> ...

Change the name of TrainSet to ROM (or any other name that you like), create then in ROM a new folder named TrainSet and move all the train snapshots into this: these operations can be done in the terminal with the path set in BuoyantCavity

mv TrainSet ROM
cd ROM
mkdir TrainSet
mv Case_00* TrainSet/.

Then, we can also move the test snapshots into ROM

mv ../TestSet .

Therefore, the structure of the repository will look like

BuoyantCavity
    |---> allrun.py
    |---> BaseCase.py
    |---> ROM
    |    |
    |    |---> TrainSet
    |    |    |
    |    |    |---> Case_000_*
    |    |    |---> Case_001_*
    |    |    |---> ...
    |    |---> TestSet
    |    |    |
    |    |    |---> Case_000_*
    |    |    |---> Case_001_*
    |    |    |---> ...
    |    |---> POD_T
    |    |---> POD_U
    |    |---> EIM_T
    |    |---> ...

POD

Change directory to POD_T and make sure that the file test_folders.txt generated with the test snapshots is in POD_T/system. The POD-Online solver takes as input the test snapshots and projects them into the reduced space, spanned by the POD, and computes the reconstruction error (average and maximum). To execute the solver for the scalar field \(T\), the correspondent dictionary must be set up as follows

Online_parameters
{
	field      T;			
	BasisNumber   40;		
	foldersList  (#include "test_folders.txt") ;	
}

Now in the terminal (path: BuoyantCavity/ROM/POD_T) execute the following command

ScalarPOD_Online

For the vector field \(\mathbf{u}\), change the field name and use the VectorialPOD_Online solver.

The output of these solvers is a series of text files into <field_name>_reconstruction_POD_files with the average and maximum absolute and relative error.

EIM

Change directory to EIM_T and make sure that the file test_folders.txt generated with the test snapshots is in EIM_T/system. The EIM-Online solver takes as input the test snapshots and use their information at the magic points location to find the interpolant, then computes the reconstruction error (average and maximum). To execute the solver for the scalar field \(T\), the correspondent dictionary must be set up as follows

Online_parameters
{
	field      T;		
	mfNumber   40;	
	foldersList  (#include "test_folders.txt");
}

Now in the terminal (path: BuoyantCavity/ROM/EIM_T) execute the following command

ScalarEIM_Online

For the vector field \(\mathbf{u}\), change the field name and use the VectorialEIM_Online solver.

GEIM - clean data

Change directory to GEIM_s2_0.0001 and make sure that the file test_folders.txt generated with the test snapshots is in GEIM_s2_0.0001/system. The GEIM-Online solver takes as input the test snapshots and evaluates them at the sensors location through the magic sensors and this information is used to find the interpolant. Then, the solver computes the reconstruction error (average and maximum). To execute the solver for the scalar field \(T\), the correspondent dictionary must be set up as follows

Online_parameters
{
	field        T;			
	msNumber     40;		
	foldersList  (#include "test_folders.txt") ;	
}

Now in the terminal (path: BuoyantCavity/ROM/GEIM_s2_0.0001) execute the following command

ScalarGEIM_Online

In the offline phase, we have generated two additional folders considering \(s^2= 0.0004\) and \(s^2=0.0016\), repeat the procedure to get the online reconstruction.

PBDW - clean data

Change directory to PBDW_T_WeakGreedy_s_0.0004 and make sure that the file test_folders.txt generated with the test snapshots is in PBDW_T_WeakGreedy_s_0.0004/system. The PBDW-Online solver takes as input the test snapshots and evaluates them at the sensors location through the basis sensors and this information is used to find the reconstruction. Then, the solver computes the reconstruction error (average and maximum). To execute the solver for the scalar field \(T\), the correspondent dictionary must be set up as follows

Online_parameters
{
	field T;				 
	MaxSensors 40;			
	BasisNumber 20;				
	sensorsFolder  "PBDW_T_WeakGreedy_s_0.0004";	
	foldersList  (#include "test_folders.txt");		
}

The input sensorsFolder is used to tell the solver from where the sensors have to be loaded. Now in the terminal (path: BuoyantCavity/ROM/PBDW_T_WeakGreedy_s_0.0001) execute the following command

ScalarPBDW_Online

GEIM - noisy data

Change directory to GEIM_s2_0.0004 and make sure that the file test_folders.txt generated with the test snapshots is in GEIM_s2_0.0004/system. The main difference with respect to the standard solver is the addiction of noise to the measurement vector \(\mathbf{y}\in\mathbb{R}^M\) represented by

\[\begin{equation*} y_m = v_m(u(\cdot;\,\boldsymbol{\mu})) + \epsilon_m \end{equation*}\]

where \(\epsilon_m\) models random noise as a random variable, i.i. with a zero-mean Gaussian distribution \(\sim \mathcal{N}(0,\sigma^2)\). To execute the solver for the scalar field \(T\), the correspondent dictionary must be set up as follows (same as clean data)

Online_parameters
{
	field        T;			
	msNumber     40;		
	foldersList  (#include "test_folders.txt") ;	
}

Now in the terminal (path: BuoyantCavity/ROM/GEIM_s2_0.0004) execute the following command

ScalarGEIM_Online -noise 0.01

to have \(\sigma = 0.01\).

PBDW - noisy data

Change directory to PBDW_T_WeakGreedy_s_0.0004 and make sure that the file test_folders.txt generated with the test snapshots is in PBDW_T_WeakGreedy_s_0.0004/system. The main difference with respect to the standard solver is the addiction of noise to the measurement vector \(\mathbf{y}\in\mathbb{R}^M\) represented by

\[\begin{equation*} y_m = v_m(u(\cdot;\,\boldsymbol{\mu})) + \epsilon_m \end{equation*}\]

where \(\epsilon_m\) models random noise as a random variable, i.i. with a zero-mean Gaussian distribution \(\sim \mathcal{N}(0,\sigma^2)\). To execute the solver for the scalar field \(T\), the correspondent dictionary must be set up as follows (same as clean data)

Online_parameters
{
	field T;				 
	MaxSensors 40;			
	BasisNumber 20;				
	sensorsFolder  "PBDW_T_WeakGreedy_s_0.0004";	
	foldersList  (#include "test_folders.txt");		
}

The input sensorsFolder is used to tell the solver from where the sensors have to be loaded. Now in the terminal (path: BuoyantCavity/ROM/PBDW_T_WeakGreedy_s_0.0001) execute the following command

ScalarPBDW_Online -noise 0.01

to have \(\sigma = 0.01\).

TR-GEIM - noisy data

Change directory to GEIM_s2_0.0004 and make sure that the file test_folders.txt generated with the test snapshots is in GEIM_s2_0.0004/system. This solver is needed in presence of noisy data. To execute the solver for the scalar field \(T\), the correspondent dictionary must be set up as follows

Online_parameters
{
	field       T;                 
	msNumber    40;               
	foldersList  (#include "test_folders.txt") ; 
	noise_std   0.01;	      
	N_Repeated_Experiments 5;    
}
in which `noise_std` is the standard deviation $\sigma$ of the random noise and `N_Repeated_Experiments` tells the solver to compute the interpolant more times to have statically robust results.

Now in the terminal (path: BuoyantCavity/ROM/GEIM_s2_0.0004) execute the following command

ScalarTRGEIM

A new folder named T_TR-GEIM_files is created to store the results.

GEIM-VT

Change directory to GEIM-VT_s_0.0004 and make sure that the file test_folders.txt generated with the test snapshots is in GEIM-VT_s_0.0004/system. To execute the solver, the correspondent dictionary must be set up as follows

Online_parameters
{
	msNumber 40;         
	foldersList (#include "test_folders.txt") ;                 
} 

Now in the terminal (path: BuoyantCavity/ROM/GEIM-VT_s2_0.0004) execute the following command

GEIM-VT_Online