Online - Tikhonov#

Online phase of the Generalised Empirical Interpolation Method Vectorial Treatment applied to the vector field \((T,\mathbf{u},p_{rgh})\) using temperature sensors only. Tikhonov Regularizarion allows better reconstruction when signal noise is considered.

Preparation#

The structure of the case study folder is the following

>> ./Study_case
	>> /Folder_1  		  		
	>> /Folder_2
	>> /Folder_3  		  		
	>> /Folder_4
	>> /GEIM-VT_s_0.0001
		>> /0		        		
		>> /constant
		>> /GEIM-VT_Offline_files
		>> /system
			controlDict
			blockMeshDict
			...
			GEIM-VTsolverDict  <--- Dictionary needed for the input parameters

The GEIM-VTsolverDict must be put inside ./Study_case/GEIM-VT_s_(SensorsVariance)/system/

An example of GEIM-VTsolverDict can be found in application/GEIM-VT/GEIM-VT_Online, which requires the following entries:

Online_parameters
{
	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-VT_s_(SensorsVariance) launch

TR-GEIM-VT

To include folder “0” use

TR-GEIM-VT -withZero

Results#

The interpolant and the residual field, defined as

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

are stored in the correspondent snapshot folders.

>> ./Study_case
	>> /Folder_1  		  		
	>> /Folder_2
	>> /Folder_3
		>> /0
			TGEIM-VTInterpolant	<--- T GEIM-VT interpolant obtained with msNumber basis
			TGEIM-VTresidual	<--- T GEIM-VT residual obtained with msNumber basis
			p_rghGEIM-VTInterpolant	<--- p_rgh GEIM-VT interpolant obtained with msNumber basis
			p_grhGEIM-VTresidual	<--- p_rgh GEIM-VT residual obtained with msNumber basis
			UGEIM-VTInterpolant	<--- U GEIM-VT interpolant obtained with msNumber basis
			UGEIM-VTresidual	<--- U GEIM-VT residual obtained with msNumber basis
		>> /1	
			TGEIM-VTInterpolant  
			TGEIM-VTresidual    
			p_rghGEIM-VTInterpolant 
			p_grhGEIM-VTresidual   
			UGEIM-VTInterpolant 
			UGEIM-VTresidual    
		>> /...			  		
	>> /Folder_4
		>> /0
			TGEIM-VTInterpolant  
			TGEIM-VTresidual  
			...
		>> /constant
		>> /system
	>> /GEIM-VT_s_0.0001		
		>> /0		   
		>> /constant
		>> /system	
		>> /GEIM-VT_Offline_files
		>> /GEIM-VT_Online_files
			T_maximum_L2_relative_error.txt
			T_average_L2_relative_error.txt 
			p_rgh_maximum_L2_relative_error.txt
			p_rgh_average_L2_relative_error.txt 
			U_maximum_L2_relative_error.txt
			U_average_L2_relative_error.txt

The absolute and relative error are computed as

\[\epsilon_M = \frac{|| \phi-\mathcal{I}_M[\phi]||_{L^\infty}}{||\phi||_{L^2}}\qquad \qquad {\phi\in\{T, \mathbf{u}, p_{rgh}\}}\]