% Form the different components of the background error covariance matrix.
%
% [iB_,iB] = divand_background_components(s,alpha)
%
% Compute the components of the background error covariance matrix iB_ and 
% their sum based on alpha (the a-dimensional coefficients for norm, gradient, 
% laplacian,...).

function [iB_,iB] = divand_background_components(s,alpha)

WE = s.WE;
D = s.D;
coeff = s.coeff;
n = s.n;

iB_ = cell(length(alpha),1);

% constrain of total norm

iB_{1} =  (1/coeff) * (WE'*WE);


% loop over all derivatives

for j=2:length(alpha)    
    % exponent of laplacian
    k = floor((j-2)/2);
        
    if mod(j,2) == 0
        % constrain of derivative with uneven order (j-1)
        % (gradient, gradient*laplacian,...)
        % normalized by surface
                
        iB_{j} = sparse(size(D,1),size(D,1));
        
        for i=1:n
            Dx = s.WEss{i} * s.Dx{i} * D^k;            
            %Dx = s.WEs{i} * s.Dxs{i} * D^k;            
            iB_{j} = iB_{j} + Dx'*Dx;
        end                
    else
        % constrain of derivative with even order (j-1)
        % (laplacian, biharmonic,...)
        
        % normalize by surface of each cell
        % such that inner produces (i.e. WE'*WE)
        % become integrals
        % WD: units length^(n/2)
        
        WD = WE * D^(k+1);
        iB_{j} = WD'*WD;
    end
    
    iB_{j} = iB_{j}/coeff;
end


% sum all terms of iB
% iB is adimentional
iB = alpha(1) * iB_{1};
for j=2:length(alpha)    
  iB = iB + alpha(j) * iB_{j};
end

% LocalWords:  iB divand

% Copyright (C) 2014 Alexander Barth <a.barth@ulg.ac.be>
%
% This program is free software; you can redistribute it and/or modify it under
% the terms of the GNU General Public License as published by the Free Software
% Foundation; either version 2 of the License, or (at your option) any later
% version.
%
% This program is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
% FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
% details.
%
% You should have received a copy of the GNU General Public License along with
% this program; if not, see <http://www.gnu.org/licenses/>.