Scalar Field: Online - Tikhonov

Online phase of the Tikhonov regularized Generalised Empirical Interpolation Method applied to scalar fields. Tikhonov Regularizarion allows better reconstructions when signal noise is considered.

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		
	>> /GEIM_(fieldName)
		>> /0		  
		>> /constant
		>> /(fieldName)_GEIM_Offline_files      		
		>> /system		
			controlDict
			blockMeshDict
			...
			GEIMsolverDict  <--- Dictionary needed for the input parameters	

The GEIMsolverDict must be put inside ./Study_case/GEIM_(fieldName)_s_(SensorsVariance)/system/

An example of GEIMsolverDict can be found in application/GEIM/ScalarGEIM_Online, which requires the following entries:

Online_parameters
{
	field       T;                <---- ScalarField on which GEIM is performed 
	msNumber    20;               <---- number of GEIM magic sensors to use
	foldersList  ( "Folder_3", 
			"Folder_4") ; <---- List of folder names containig the
										snapshots to be reconstructed
	noise_std   0.001;	      <---- noise Gaussian standard deviation
	N_Repeated_Experiments 10;    <---- Number of repeated "experiments" needed in order to obtain statistically relevant average reconstruction errors
}

Usage

Inside ./Study_case/GEIM_(fieldName)_s_(SensorsVariance) launch

ScalarTR-GEIM

To include folder “0” use

ScalarTR-GEIM -withZero

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
			T_TR-GEIMInterpolant  <---(fieldName) TR-GEIM interpolant obtained with mfNumber basis
			T_TR-GEIMresidual     <---(fieldName) TR-GEIM residual obtained with mfNumber basis
		>> /1	
			T_TR-GEIMInterpolant
			T_TR-GEIMresidual
		>>  ...			
				
	>> /Folder_4
		>> /0
			T_TR-GEIMInterpolant
			T_TR-GEIMresidual
		>> /1	
			T_TR-GEIMInterpolant
			T_TR-GEIMresidual
		>>  ...		
			
	>> /GEIM_T_s_0.0001		
		>> /0		        				
		>> /system			
		>> /constant
		>> /T_GEIM_Offline_files
		>> /T_GEIM_Online_files
		>> /T_TR-GEIM_files
			average_L2_relative_error_noiseStd_0.001.txt" <---- average L2 realtive error as a function of basis number with noise_std 0.001	

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

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