GPR_ROM_prediction1.m

function [Mu_full,Var_full,X_full] = GPR_ROM_prediction1(x_test,Snapshots,Mu_t,Var_t,U_r,x_train)

%% Interpolation of ROM based on GPR (all latent states share the same hyper-parameter with the first one)
%{
Created by: Kai Cheng (kai.cheng@tum.de)
Based on: "ADAPTIVE DATA-DRIVEN PROBABILISTIC REDUCED-ORDER
MODELS FOR PARAMETERIZED DYNAMICAL SYSTEMS", submitted to SIAM journal on Scientific Computing
---------------------------------------------------------------------------
Input:
* Snapshots : Function for collecting snapshots
* x_test : Testing  parameter set
* Mu_t   : Mean of time sequence for training parameter set
* Var_t  : Variance of time sequence for training parameter set
* U_r    : Global basis
* x_train: Training parameter set
---------------------------------------------------------------------------
Output:
* Mu_full   : Prediction mean of the full order solution
* Var_full  : Prediction variance of the full order solution
* X_full    : True full order solution
%}

%% Prediction of FOM for an untried parameter 

model = Interpolation_model1(x_train,Mu_t,Var_t);

N1 = size(x_test,1); [r, N_t] = size(Mu_t{1});   N = size(Mu_t,2);

ub_input =  model.ub_input;
lb_input =  model.lb_input;

x_pre = (x_test - repmat(lb_input,N1,1))./(repmat(ub_input,N1,1)-repmat(lb_input,N1,1));

for k = 1: r 
    for i = 1:N
        y(i,:)   = Mu_t{i}(k,:);
        var(i,:) = Var_t{i}(k,:);
    end
    mu_y(k,:)  = mean(y);
    std_y(k,:) = std(y);
end

for i = 1: N1

     tic
       X_full{i} = Snapshots(x_test(i,:));
     toc

     [weight Con_var] = Kriging_weight(x_pre(i,:),model);

     Var_pred = Con_var.*std_y.^2; Mu_pred = mu_y;
   
     for j = 1:N 
         Mu_pred  = Mu_pred  + weight(j)*(Mu_t{j}-mu_y);
         Var_pred = Var_pred + weight(j)^2*Var_t{j};
     end
         
     Mu_full{i} = U_r*Mu_pred;

     % for j = 1:N_t
     %     Var_full{i}(:,j) = diag(U_r*diag(Var_pred(:,j))*U_r');
     % end

     for j = 1:N_t
       for k = 1:r
           Var_full{i}(k,j) = U_r(k,:).^2*Var_pred(:,j);
       end
     end


end