Scalar Field: Online

Online phase of the Generalised Empirical Interpolation Method applied to scalar fields (with an option to activate synthetic random noise).

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

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
}

Usage

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

ScalarGEIM_Online

To include folder “0” use

ScalarGEIM_Online -withZero

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

ScalarGEIM_Online -region <regionName>

The interpolants using msNumber magic functions will be written in the folder only by activating the option as

ScalarGEIM_Online -writeInterpolant

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

ScalarGEIM_Online -noise <value>

where is the std deviation of the noise, assumed zero-mean Gaussian.

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
			TGEIMInterpolant  <---(fieldName) GEIM interpolant obtained with mfNumber basis
			TGEIMresidual     <---(fieldName) GEIM residual obtained with mfNumber basis
		>> /1	
			TGEIMInterpolant
			TGEIMresidual
		>>  ...			
				
	>> /Folder_4
		>> /0
			TGEIMInterpolant
			TGEIMresidual
		>> /1	
			TGEIMInterpolant
			TGEIMresidual
		>> /...		
			
	>> /GEIM_T_s_0.0001		
		>> /0		        				
		>> /system			
		>> /constant
		>> /T_GEIM_Offline_files
		>> /T_GEIM_Online_files
			maximum_L2_absolute_error.txt <---- max L2 absolute error as a function of basis number
			average_L2_absolute_error.txt <---- max L2 absolute error as a function of basis number
			maximum_L2_relative_error.txt <---- max L2 relative error as a function of basis number
			average_L2_relative_error.txt <---- max L2 relative error as a function of basis number
					

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

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