Scalar Field: Online

Online phase of the Parametrized-Backward Data-Weak formulation applied to scalar fields. There is an option to activate random noise, the regularization can be activating by using the suitable option.

Preparation

The structure of the case study folder is the following (in this example Folder3 and Folder4 are the test case folders)

>> ./Study_case
	>> /Folder_1  	
	>> /Folder_2
	>> /Folder_3  	
	>> /Folder_4
	>> /PBDW_(fieldName)_(algorithmRS)_s_(sensorsVariance)
		>> /0
		>> /constant
		>> /T_PBDW_Online_files
		>> /system
			controlDict
			blockMeshDict
			...
			PBDWsolverDict  <--- Dictionary needed for the input parameters

The PBDWsolverDict must be put inside ./Study_case/PBDW_(fieldName)_(algorithmRS)s(SensorsVariance)/system/

An example of PBDWsolverDict can be found in application/PBDW/ScalarPBDW_Online, which requires the following entries:

Online_parameters
{
	field T;				<---- ScalarField on which PBDW is performed 
	MaxSensors 20;				<---- number of PBDW magic sensors to use
	BasisNumber 10;				<---- number of PBDW basis functions to use
	sensorsFolder  "sensorsFolderName";	<----- name of the folder in which the sensors are saved
	foldersList  ( 
			"Folder_3"
			"Folder_4");		<---- List of folder names containig the snapshots to be reconstructed
}

Usage

Inside ./Study_case/PBDW_(fieldName)_(algorithmRS)s(SensorsVariance) launch

ScalarPBDW_Online

To include folder “0” use

ScalarPBDW_Online -withZero

To perform on a specified region (for multi-region cases) use

ScalarPBDW_Online -region <regionName>

Synthetic random noise can be introduced to the data term as follows

ScalarPBDW_Online -noise <value>

where is the std deviation of the noise, assumed zero-mean Gaussian. To include regularization launch

ScalarPBDW_Online -reg <value>

Results

The interpolant and the residual field, defined as

\[r_M = \left| T-\mathcal{I}_M[T]\,\right|\]

are stored in the correspondent snapshot folders

>> ./Study_case
	>> /Folder_1  	
	>> /Folder_2
	>> /Folder_3
		>> /0
			TPBDW_estimation   <---(fieldName) PBDW estimate obtained with MaxSensors and BasisNumber
			TPBDW_residual     <---(fieldName) PBDW residual obtained with MaxSensors and BasisNumber
		>> /1
			TPBDW_estimation
			TPBDW_residual
		>>  /...
	>> /Folder_4
		>> /0
			TPBDW_estimation 
			TPBDW_residual
		>> /1
			TPBDW_estimation
			TPBDW_residual
		>>  /...
	>> /PBDW_T_algorithmRS_s_sensorsVariance
		>> /0
		>> /constant
		>> /system
		>> /T_PBDW_Offline_files
		>> /T_PBDW_Online_files
			maximum_L2_relative_error.txt	<---- max L2 relative error as a function of sensor number
			average_L2_relative_error.txt	<---- ave L2 relative error as a function of sensor number
			maximum_L2_absolute_error.txt	<---- max L2 absolute error as a function of sensor number
			average_L2_absolute_error.txt	<---- ave L2 absolute error as a function of sensor number
			TmatrixA.txt			<---- matrix A_ij = (g_i, g_j) of the PBDW
			TmatrixK.txt			<---- matrix K_ij = (g_i, zeta_j) of the PBDW			

The absolute and relative error are computed as

\[E_M = || T-\mathcal{I}_M[T]||_{L^2}\qquad \epsilon_M = \frac{|| T-\mathcal{I}_M[T]||_{L^\infty}}{||T||_{L^2}}\]

recalling that the norms are defined as

\[|| T ||_{L^2(\Omega)}^2 =\int_\Omega T^2\, d\Omega\]