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Rcoarse.m

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+function Rc = Rcoarse(x,h_c)
+%RCOARSE Summary of this function goes here
+%   coarse cheese model response
+if nargin < 2
+    h_c = 1;
+end
+l_c = x(1);
+c = x(2);
+w_c = x(3);
+Rc(1) = l_c*w_c*h_c;
+Rc(2) = (l_c-c)*w_c*h_c;
+Rc(3) = (l_c-c)*w_c*h_c;
+end
+

Rfine.m

@@ -0,0 +1,17 @@
+function R_f = Rfine(x,h_f)
+%RCOARSE Summary of this function goes here
+%   fine cheese-model response
+if nargin < 2
+    h_f = 1;
+end
+l_f = x(1);
+f1 = x(2);
+f2 = x(3);
+w_f = x(4);
+w_f1 = x(5);
+w_f2 = x(6);
+R_f(1) = (l_f*w_f*h_f);
+R_f(2) = (l_f-f1)*w_f1*h_f;
+R_f(3) = (l_f-f2)*w_f2*h_f;
+end
+

Rsurrogate.m

@@ -0,0 +1,15 @@
+function Rc = Rsurrogate(x,dR,h_c)
+%RCOARSE Summary of this function goes here
+%   coarse cheese model response
+if nargin < 3
+    h_c = 1;
+end
+l_c = x(1);
+c = x(2);
+w_c = x(3);
+Rc(1) = l_c*w_c*h_c;
+Rc(2) = (l_c-c)*w_c*h_c;
+Rc(3) = (l_c-c)*w_c*h_c;
+Rc = Rc + dR;
+end
+

moptm_coarse.m

@@ -0,0 +1,30 @@
+function [opt_lc, xe] = moptm_coarse(aim,xin,dr)
+%OPTIMCOARSE Summary of this function goes here
+%   optimize coarse model w.r.t the length parameter
+if nargin < 3
+    dr = 0;
+end
+
+if nargin == 0
+    % test-values
+    xin = [1 2 1];
+    aim = [10 10 10];
+end
+
+fun = @(l)abs((aim - Rcoarse([l,xin(2),xin(3)])+dr));
+% numel fun
+% Minimize absolute values
+%% minimax method
+options = optimoptions('fminimax','AbsoluteMaxObjectiveCount',3);
+opt_lc = fminimax(fun,xin(1),[],[],[],[],[],[],[],options);
+xe = opt_lc - aim(1);
+
+%% quasi-newton method
+% opt_lc = fminunc(fun,xin(1));
+%% simulatedannealing
+% opt_lc = simulannealbnd(fun,xin(1),[],[]);
+%% pso
+% opt_lc = particleswarm(fun,1,[],[]);
+
+end
+

multiplecheese_arrsm.m

@@ -0,0 +1,158 @@
+% AGGRESSIVE-RESPONSE RESIDUAL SPACE MAPPING ALGORITHM
+% MULTIPLE-CHEESE CUTTER ILLUSTRATION
+%% House keeping
+clc; close all;
+clearvars;
+co = [4 0]; fo =[4 4];
+for ic = 1:length(co)
+    %% Inits
+    % desired fine model volume response
+    Raim = 10;
+    % candidate designable parameters
+    % initial guess; length and width
+    % n.b: if c ~= f then there is a misalignment
+    % to get the root solution, first calculate without misalignment, c == f
+    % then calculate with misalignment.
+    l=1; c = co(ic); w_c = 1;
+    l_c = moptm_coarse(Raim,[l, c, w_c]);
+    %
+    l_f = l_c; f1=fo(ic); f2=fo(ic); w_f=1; w_f1=1; w_f2=1;
+
+    %% Initial Model Guess
+    %coarse model
+    R_c = Rcoarse([l_c, c, w_c]);
+    %fine model
+    R_f = Rfine([l_f, f1, f2, w_f, w_f1, w_f2]);
+    Raim = [Raim Raim Raim];
+    % display
+    fprintf('l:%g\n', l_c)
+    fprintf('w:%g\n', w_c)
+    fprintf('R_c: %g\n',R_c)
+    fprintf('R_f: %g\n',R_f)
+    fprintf('Fine aim: %g\n',Raim)
+
+    id = 1;
+
+    % Parameter Extraction -> new coarse length mapping
+    rng default % For reproducibility
+    fun_x = @(x)norm(R_f-Rcoarse([l_c+x, c, w_c])); % cost function
+    options = optimoptions(@fminunc, 'Algorithm', 'quasi-newton');
+    x = fminunc(fun_x,1,options);
+    %
+    l = l_c + x;
+    f = l-l_c; % error vector
+
+    % Mapping Jacobian
+    h = Inf;
+    B = eye(1,1); % unit mapping
+    % init. surrogate and residual response
+    R_s = [0 0 0]; dR = 0;
+
+    chck = norm(Raim - R_f); % [] store error
+    Rf = R_f; % [] store fine model response
+    itTab = []; % store parameters per iteration
+    %%
+    while norm(h) > 1e-4
+        %% Prediction, Evaluate fine length
+        % Inverse mapping of coarse length;
+        h = -(f) / B; % quasi-newton step in fine space
+        l_f = l_f + h; % update
+        %% Verification
+        % coarse model
+        R_c = Rcoarse([l_c,c,w_c]);
+        % fine model
+        R_f = Rfine([l_f, f1, f2, w_f, w_f1, w_f2]);
+        Rf = [Rf; R_f];
+        % save tab
+        itTab= [itTab; [id f B h l_c R_c R_s l_f R_f (chck(id)*100/(sum(Raim)/3))]];
+        %% Next Iterate Prediction
+        fun_x = @(x)norm(R_f-Rcoarse([l_c+x, c w_c])); % cost function
+        options = optimoptions(@fminunc, 'Algorithm', 'quasi-newton');
+        x = fminunc(fun_x,1,options); % alternative: fminunc, but slower convergence
+        %
+        l = l_c + x;
+        f = l-l_c; % update error vector
+
+        if norm(f) < 1e-4
+            % update response residual
+            dR = R_f - R_c;
+            % optimise coarse length-> minimise surrogate
+            l_c = optim_sug(Raim,[l_c, c, w_c],dR);
+            f = l - l_c; % update error vector
+            % Inverse mapping of coarse length;
+            h = -(f) ./ B; % quasi-newton step in fine space
+            l_f = l_f + h;
+            %
+            R_s = Rsurrogate([l_c, c, w_c],dR);
+            B = eye(1,1); % constrain B to an identity mapping
+        else
+            % broyden rank-one update
+            B = B + ((f.*h')/(h'.*h));
+        end
+        %
+        % display
+        fprintf('\nIteration.%g\n', id)
+        fprintf('f: %g\n',f)
+        fprintf('l:%g\n', l_c)
+        fprintf('R_c: %g\n',R_c)
+        fprintf('R_s: %g\n',R_s)
+        fprintf('lf:%g\n', l_f)
+        fprintf('R_f: %g\n',R_f)
+        fprintf('Fine aim: %g\n',Raim)
+        %
+        chck = [chck; norm(Raim - R_f)]; %#ok<*AGROW>
+        itTab= [itTab; [id f B h l_c R_c R_s l_f R_f (chck(id)*100/(sum(Raim)/3))]];
+        % stop if limit attractor reached
+        % not converging
+        if (id >= 2)
+            if abs(chck(id) - chck(id-1)) <= 1e-6
+                break;
+            end
+        end
+        id = id + 1;
+    end
+    fprintf('Fine Model Parameters are: l=%g, w=%g, h=%g, volume=%g\n',...
+        l_f,w_f,1,R_f(1))
+    fprintf('Fine Model Parameters are: l=%g, w=%g, h=%g, volume=%g\n',...
+        l_f,w_f1,1,R_f(2))
+    fprintf('Fine Model Parameters are: l=%g, w=%g, h=%g, volume=%g\n',...
+        l_f,w_f2,1,R_f(3))
+    if ic == 1
+        chck_opt = chck;
+        l_opt = l_f;
+    end
+end
+
+%% Visualization
+figure(1);
+% subplot 1
+subplot(211)
+plot(1:numel(chck),chck,'-.sr','LineWidth',1.25)
+grid on;
+xlabel('Iteration','Interpreter','latex')
+ylabel('Error, $$||R_{f}-{R_{f}}^{\ast}||$$',...
+    'FontSize',12,'Interpreter','latex')
+title('Multiple Cheese Cutter:Response Residual Aggressive Space Mapping Optimization',...
+    'FontSize',10,'Interpreter','latex')
+axis([1,inf,min(chck)-0.01,inf])
+% subplot 2
+subplot(212)
+% figure(2);
+for ix = 1:id
+    %     l_err(ix) = norm(l_opt-itTab(ix,3));
+    l_err(ix) = norm(chck_opt(end)- chck(ix));%#ok<SAGROW>
+end
+plot(l_err,'Marker','.','MarkerSize',10,'LineWidth',1.25)
+% plot(Rf,'-.o','LineWidth',1.25)
+grid on;
+xlabel('Iteration','Interpreter','latex')
+ylabel('Error, $$||l_{f}-{l_{f}}^{\ast}||$$',...
+    'FontSize',12,'Interpreter','latex')
+% ylabel('$$R_{f}$$',...
+%     'FontSize',12,'Interpreter','latex')
+title('Multiple Cheese Cutter: Fine-Model Response',...
+    'FontSize',10,'Interpreter','latex')
+axis([1,inf,min(l_err)-0.01,inf])
+% legend({'Cheese1','Cheese2','Cheese3'})
+
+

multiplecheese_asm.m

@@ -0,0 +1,137 @@
+% AGGRESSIVE SPACE MAPPING ALGORITHM
+% MULTIPLE-CHEESE CUTTER ILLUSTRATION
+%% House keeping
+clc; close all;
+clearvars;
+co = [4 0]; fo =[4 4];
+for ic = 1:length(co)
+    %% Inits
+    % desired fine model volume response
+    Raim = 10;
+    % candidate designable parameters
+    % initial guess; length and width
+    % n.b: if c ~= f then there is a misalignment
+    % to get the root solution, first calculate without misalignment, c == f
+    % then calculate with misalignment.
+    l=1; c = co(ic); w_c = 1;
+    l_c = moptm_coarse(Raim,[l, c, w_c]);
+    %
+    l_f = l_c; f1=fo(ic); f2=fo(ic); w_f=1; w_f1=1; w_f2=1;
+
+    %% Initial Model Guess
+    %coarse model
+    R_c = Rcoarse([l_c, c, w_c]);
+    %fine model
+    R_f = Rfine([l_f, f1, f2, w_f, w_f1, w_f2]);
+    Raim = [Raim Raim Raim];
+    % display
+    fprintf('l:%g\n', l_c)
+    fprintf('w:%g\n', w_c)
+    fprintf('R_c: %g\n',R_c)
+    fprintf('R_f: %g\n',R_f)
+    fprintf('Fine aim: %g\n',Raim)
+
+    id = 1;
+
+    % Parameter Extraction -> new coarse length mapping
+    rng default % For reproducibility
+    fun_x = @(x)norm(R_f-Rcoarse([l_c+x, c, w_c])); % cost function
+    options = optimoptions(@fminunc, 'Algorithm', 'quasi-newton');
+    x = fminunc(fun_x,1,options);
+    %
+    l = l_c + x;
+    f = l-l_c; % error vector
+
+    % Mapping Jacobian
+    B = eye(1,1); % unit mapping
+
+    chck = norm(Raim - R_f); % [] store error
+    Rf = R_f; % [] store fine model response
+    itTab = []; % store parameters per iteration
+    %%
+    while norm(f) > 1
+        %% Prediction, Evaluate fine length
+        % Inverse mapping of coarse length;
+        h = -(f) ./ B; % quasi-newton step in fine space
+        l_f = l_f + h; % update
+        %% Verification
+        % coarse model
+        R_c = Rcoarse([l_c,c,w_c]);
+        % fine model
+        R_f = Rfine([l_f, f1, f2, w_f, w_f1, w_f2]);
+        % display
+        fprintf('\nIteration.%g\n', id)
+        fprintf('f: %g\n',f)
+        fprintf('lc:%g\n', l_c)
+        fprintf('lf:%g\n', l_f)
+        fprintf('R_c: %g\n',R_c)
+        fprintf('R_f: %g\n',R_f)
+        fprintf('Fine aim: %g\n',Raim)
+        itTab= [itTab; [id f B h l_c R_c l_f R_f (chck(id)*100/(sum(Raim)/3))]];
+        %% Next Iterate Prediction
+        fun_x = @(x)norm(R_f-Rcoarse([l_c+x, c w_c])); % cost function
+        options = optimoptions(@fminunc, 'Algorithm', 'quasi-newton');
+        x = fminunc(fun_x,1,options); % alternative: fminunc, but slower convergence
+        %
+        l = l_c + x;
+        f = l-l_c; % update error vector
+        l_c = l;
+        % broyden rank-one update
+        B = B + ((f.*h')/(h'.*h));
+
+        %
+        chck = [chck; norm(Raim - R_f)]; %#ok<*AGROW>
+        itTab= [itTab; [id f B h l_c R_c l_f R_f (chck(id)*100/(sum(Raim)/3))]];
+        % stop if limit attractor reached
+        % not converging
+        if (id >= 2)
+            if abs(chck(id) - chck(id-1)) <= 1e-6
+                break;
+            end
+        end
+        id = id + 1;
+    end
+    fprintf('Fine Model Parameters are: l=%g, w=%g, h=%g, volume=%g\n',...
+        l_f,w_f,1,R_f(1))
+    fprintf('Fine Model Parameters are: l=%g, w=%g, h=%g, volume=%g\n',...
+        l_f,w_f1,1,R_f(2))
+    fprintf('Fine Model Parameters are: l=%g, w=%g, h=%g, volume=%g\n',...
+        l_f,w_f2,1,R_f(3))
+    if ic == 1
+        chck_opt = chck;
+        l_opt = l_f;
+    end
+end
+
+%% Visualization
+figure(1);
+% subplot 1
+subplot(211)
+plot(1:numel(chck),chck,'-.sr','LineWidth',1.25)
+grid on;
+xlabel('Iteration','Interpreter','latex')
+ylabel('Error, $$||R_{f}-{R_{f}}^{\ast}||$$',...
+    'FontSize',12,'Interpreter','latex')
+title('Multiple Cheese Cutter: Aggressive Space Mapping Optimization',...
+    'FontSize',10,'Interpreter','latex')
+axis([1,inf,min(chck)-0.01,inf])
+% subplot 2
+subplot(212)
+% figure(2);
+for ix = 1:id
+    %     l_err(ix) = norm(l_opt-itTab(ix,3));
+    l_err(ix) = norm(chck_opt(end)- chck(ix));%#ok<SAGROW>
+end
+plot(l_err,'Marker','.','MarkerSize',10,'LineWidth',1.25)
+% plot(Rf,'-.o','LineWidth',1.25)
+grid on;
+xlabel('Iteration','Interpreter','latex')
+ylabel('Error, $$||l_{f}-{l_{f}}^{\ast}||$$',...
+    'FontSize',12,'Interpreter','latex')
+% ylabel('$$R_{f}$$',...
+%     'FontSize',12,'Interpreter','latex')
+title('Multiple Cheese Cutter: Fine-Model Response',...
+    'FontSize',10,'Interpreter','latex')
+axis([1,inf,min(l_err)-0.01,inf])
+% legend({'Cheese1','Cheese2','Cheese3'})
+