rb_nonlin_error_convergence.m

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% small script performing a reduced basis simulation with the 
% linear and nonlinear scheme and plotting the results
%
% for a single parameter a whole range of M and N is tried in the
% nonlinear simulation. The error is computed not to the detailed
% model, but to the large-N linear-evolution model as a substitute

% Bernard Haasdonk 24.6.2007

% prepare and run nonlin_evol simulation

% get_nonlin_evol_params
% params.Mmax = 100;
% detailed_data = rb_detailed_prep(params);
 clear;
% load demo_nonlin_evol_detailed_data_LE_on_RB_wo_offset;

%load demo_nonlin_evol_detailed_data3;
%load demo_nonlin_evol_detailed_data_LE_trajectory_1_1_0;
load demo_nonlin_evol_detailed_data_LE_2_beta_trajectories;

% load demo_nonlin_evol_detailed_data_LE_on_RB;
 offline_data = rb_offline_prep(detailed_data,params);

%params.N = 25; params.M = length(detailed_data.TM{1});  
%params.N = 123; params.M = length(detailed_data.TM{1});  
%params.N = 20; params.M = length(detailed_data.TM{1});  
 
 % M usually less than N !!

 % set parameter vector mu for which loop has to be performed
 params.k = 0;
 params.c_init = 1;
 params.beta = 1;
 %params.k = 0;
 %params.c_init = 0;
 %params.beta = 0;
 % => M >= 3N ist stabil
 
 % prepare linear parameters (plus some additional missing fields)
 params_lin = params;
 params_lin.rb_problem_type = 'lin_evol';
 params_lin.L_I_inv_norm_bound = 1;
 params_lin.L_E_norm_bound = 1;
 params_lin.N = size(detailed_data.RB,2);
 offline_data_lin = rb_offline_prep(detailed_data,params_lin);
 reduced_data_lin = rb_online_prep(offline_data_lin,params_lin);
 % full linear simulation: only required once, as parameter is fixed
 simulation_data_lin = rb_simulation(reduced_data_lin,params_lin);
 U_lin = rb_reconstruction(detailed_data,simulation_data_lin);
 
 Nsamples = 12; 
 Ns = round((0:(Nsamples-1)) * ...
            (size(detailed_data.RB,2)-1)/(Nsamples-1)+1);

 Msamples = 14; 
 Ms = round((0:(Msamples-1)) * ...
            (size(detailed_data.BM,2)-1)/(Msamples-1)+1);

 % storage for maximum l2-errors (i.e. linfty([0,T],l2)-norm)
 errs = zeros(Msamples,Nsamples);
 inds = zeros(Msamples,Nsamples);
 
 for Nind = 1:Nsamples
   N = Ns(Nind);
   for Mind = 1:Msamples
     M = Ms(Mind);
     
     % nonlinear simulation
     params.M = M;
     params.N = N;
     reduced_data = rb_online_prep(offline_data,params);
     simulation_data = rb_simulation(reduced_data,params);
     U = rb_reconstruction(detailed_data,simulation_data);
     
     l2_errors = fv_l2_error(U_lin,U,detailed_data.W);
     [err,ind] = max(l2_errors);
     errs(Mind,Nind) = err(1);
     inds(Mind,Nind) = ind(1);
     
   end;
 end;
 
 % define stability-region as error being smaller than
 % sqrt(diffmax * area), e.g. diffmax = 4, area = 2e-7

 stab_limit = 1e-3;
 C = ones(size(errs));
 i = find(errs<stab_limit);
 C(i) = 2;
 
 if isempty(find(C==1,1)); % i.e. all stable
   figure, surf(Ms, Ns,errs');
 else % some not stable   
   figure, surf(Ms, Ns,errs',C');
   shading interp;
 end;
% figure, pcolor(Ms, Ns,C);
 title('L-infty([0,T],L2) error of nonlinear-evolution');
 set(gca,'Zscale','log');
 xlabel('M');
 ylabel('N');
 zlabel('error');

 figure, surf(Ms, Ns,inds');
 title('time index of maximum l2-error');
 %set(gca,'Zscale','log');
 xlabel('M');
 ylabel('N');
 zlabel('time index');

% params.title = 'Solution from nonlinear rb-simulation';
% plot_element_data_sequence([],Unonlin,params);
% plot_element_data_sequence([],Unonlin(:,1:100),params);
 %plot_rb_reconstruction(detailed_data,simulation_data,params);

 %use same detailed-data, i.e. reduced basis
% lparams.title = 'Solution from linear rb-simulation';
% plot_element_data_sequence([],Ulin(:,1:100),lparams);
% plot_element_data_sequence([],Ulin,lparams);

 % plot difference
% params.title = 'Difference of linear and nonlinear';
% plot_element_data_sequence([],Ulin(:,1:100)-Unonlin(:,1:100),params);
% plot_element_data_sequence([],Ulin-Unonlin,params);

% figure,plot(l2_errors);
% title('l2-difference');


% TO BE ADJUSTED TO NEW SYNTAX
%| \docupdate