MisfitFunction

function [J,Idata,IRegC,IRegAGlen,dIduv,IBarrierC,IBarrierAGlen]=MisfitFunction(UserVar,CtrlVar,MUA,ub,vb,ud,vd,AGlen,C,Priors,Meas)

narginchk(11,11)

us=ub+ud ; vs=vb+vd; % Surface velocities

%     J=Idata+IRegC+IRegAGlen+IBarrierC+IBarrierAGlen;
%

The misfit function is

$$J=I + R + B$$

where $I$ is a data misfit term, $R$ a regularisation term, and $B$ a barrier term.

The data mistfit term is on the form

$I=\frac{1}{2\cal{A}} \int \int (u(x,y)-\hat{u}(x,y)) \, \kappa(x-x',y-y') \, (u(x',y')-\hat{u}(x',y')) dx \, dy \, dx' \, dy'$

where $\kappa$ is a (inverse) covariance kernel, $u$ are model output quantity, and $\hat{u}$ measurements of that quantity.

It is assumed that data errors are spatially uncorrelated, i.e.

$\kappa(x,y)=e^{-2} \; \delta(x,y)$

where $e$ are data errors. Therefore

$I=\frac{1}{2\cal{A}} \int (u(x,y)-\hat{u}(x,y))/e)^2 dx \, dy$

$I$ is dimentionless.

In FE context:

$I= \frac{1}{2\cal{A}} ((\mathbf{u}-\hat{\mathbf{u}})./\mathbf{e})' \, \mathbf{M} \,((\mathbf{u}-\hat{\mathbf{u}})./\mathbf{e})$

here $\mathbf{u}$ etc, are vectors and we used the ./ matlab notation to indicate element wise division.

The derivative of the data mistfit term with respect ot $u$ is

$dI/du = \frac{1}{\cal{A}} \mathbf{M} \,(\mathbf{u}-\hat{\mathbf{u}})$

The regularisation term $R$ is written in descrete form as

$R= \frac{1}{2 N} (\mathbf{c}-\hat{\mathbf{c}})' \mathbf{K}^{-1} (\mathbf{c}-\hat{\mathbf{c}})$

where $\mathbf{}$ is a covariance matrix and $N$ the number of elements in $\mathbf{c}$ . $R$ is dimentionless

$dR/dc=\mathbf{K}^{-1} (\mathbf{c}-\hat{\mathbf{c}})$

Area=TriAreaTotalFE(MUA.coordinates,MUA.connectivity);

switch lower(CtrlVar.MisfitFunction)

    case 'uvdiscrete'
        %
        %             residuals=[us ; vs]-[Meas.u;Meas.v];
        %             Idata=residuals'*(Cd\residuals)/2;
        %             dIduv=Cd\residuals;

        Idata=(us-Meas.us)'*(Meas.usCov\(us-Meas.us))/2+(vs-Meas.vs)'*(Meas.vsCov\(vs-Meas.vs))/2;
        dIdu=Meas.usCov\(us-Meas.us) ;
        dIdv=Meas.vsCov\(vs-Meas.vs);
        dIduv=[dIdu(:) ; dIdv(:)];

    case 'uvintegral'

        %             %fprintf(' misfit integral \n ')
        %             [n1,n2]=size(Cd);
        %             sqrtmCd=
        %             res=sqrtmCd*[us-Meas.u ; vs-Meas.v];
        %             ures=res(1:MUA.Nnodes) ;  vres=res(MUA.Nnodes+1:end) ;

        %           this sqrt only works for diagonal sparse matrices

%         isdiag was introduced in R2014a
         if isdiag(Meas.usCov)
             squs=1./sqrt(spdiags(Meas.usCov)); usres=squs.*(us-Meas.us);
         else
             error('MisfitFunction:Cov','Data covariance matrices but be diagonal')
         end

         if isdiag(Meas.vsCov)
             sqvs=1./sqrt(spdiags(Meas.vsCov)); vsres=squs.*(vs-Meas.vs);
         else
             error('MisfitFunction:Cov','Data covariance matrices must be diagonal')
         end

        M=MassMatrix2D1dof(MUA);
        Idata=full(usres'*M*usres/2+vsres'*M*vsres/2);
        dIdu=squs.*(M*usres);
        dIdv=sqvs.*(M*vsres);
        dIduv=[dIdu(:);dIdv(:)];

        if ~isreal(dIduv)
            save TestSave ; error('MisfitFunction:dIduvNoReal','dIduv is not real! Possibly a problem with covariance of data.')
        end

        % Two methods resulting in the same answer
        % method 1:


        % Method 2
        %           uNorm=sum(fgIntElementwise(connectivity,coordinates,nip,ures,ures,CtrlVar))/2;
        %           vNorm=sum(fgIntElementwise(connectivity,coordinates,nip,vres,vres,CtrlVar))/2;
        %           Idata2=(uNorm+vNorm)/Area;



        %                 Sx=StiffnessMatrixSx2D(coordinates,connectivity,nip,CtrlVar);
        %                 Sy=StiffnessMatrixSy2D(coordinates,connectivity,nip,CtrlVar);
        %
        %                 Errors=1./sqrt(spdiags(Priors.Cov,0)); Cres=(C-Priors.C)./Errors ;
        %                 Errors=1./sqrt(spdiags(Priors.CovAGlen,0)); Ares=(AGlen-Priors.AGlen)./Errors ;
        %
        %                 M=MassMatrix2D1dof(coordinates,connectivity,nip);
        %
        %                 IRegC=c0* Cres'*M*Cres/2+c1*Cres'*(Sx+Sy)*Cres/2;
        %                 IRegAGlen=a0* Ares'*M*Ares/2+a1* Ares'*(Sx+Sy)*Ares/2;
        %
        %                 IRegC=IRegC/Area;
        %                 IRegAGlen=IRegAGlen/Area;

        %Idata=FEmisfituv(ures,vres,coordinates,connectivity,nip);

        Idata=Idata/Area;
        dIduv=dIduv/Area;

    otherwise
        error(' what case? ' )
end

Idata=CtrlVar.MisfitMultiplier*Idata;


IRegC=Calc_IRegC(CtrlVar,Priors.CovC,C,Priors.C);
IRegAGlen=Calc_IRegdAGlen(CtrlVar,Priors.CovAGlen,AGlen,Priors.AGlen);
IBarrierC=Calc_IBarrierC(CtrlVar,C);
IBarrierAGlen=Calc_IBarrierAGlen(CtrlVar,AGlen);


% scalings
Idata=Idata*CtrlVar.AdjointfScale;
IRegC=IRegC*CtrlVar.AdjointfScale;
IRegAGlen=IRegAGlen*CtrlVar.AdjointfScale;
IBarrierC=IBarrierC*CtrlVar.AdjointfScale;
IBarrierAGlen=IBarrierAGlen*CtrlVar.AdjointfScale;


J=Idata+IRegC+IRegAGlen+IBarrierC+IBarrierAGlen;


%fprintf('MisfitFunction: J=%-g \t Idata=%-g \t IRegC=%-g \t IRegAGlen=%-g \t IBarrierC=%-g \t IBarrierAGlen=%-g \n',...
%    J,Idata,IRegC,IRegAGlen,IBarrierC,IBarrierAGlen)

end