Finding four input combinations with the same output value by interpolation method, and find the one that minimizes the cost function.

A(x,y,z,w) is 4-D dobule data, for example A(1,1,1,1) is determined as the specific value. each dimension length is 60, so the number of data is 60^4.

B(x,y,z,w) is also 4-D double data like A.

I want to find a solution, (x_sol,y_sol,z_sol,w_sol), that minimizes B(x,y,z,w) among the combinations of (x,y,z,w) that satisfy an arbitrary A*=A(x,y,z,w).

I've been able to find solutions using meshgrid and interp2 for data with 2-D variables, but I'm having trouble with data with 4-D variables.

Below shows the desired solution for A and B with 2-D variables. I want to solve the same problem for A and B extended to 4-D.

X_interp = 0 : 0.01 : 29.99; % tighter spaced linear array for interpolation

[X_mesh,Y_mesh] = meshgrid(X,Y);

min_B=zeros(1,length(Astar_arr)-1);

for i = 1:length(Astar_arr)-1

XY_temp = contour(X,Y,A',[Astar_arr(i) Astar_arr(i)],ShowText="on");

XY=XY_temp(:,2:end);

Y_interp = interp1(XY(1,:),XY(2,:),X_interp);

XY_interp(:,i) = Y_interp;

B_interp(:,i) = interp2(X_mesh,Y_mesh,B',X_interp,Y_interp);

idx_min_B(:,i)=find(B_interp(:,i)==min(B_interp(:,i)));

min_B(i)=min(B_interp(:,i));

end

How can I solve this problem?

Answer 1

[John D'Errico]

Um, your technique will not really solve the problem. It approximates the solution. Regardless, you will now have difficulties extending it to higher dimensions.

What you did was to generate a contour path for each value of A*. So you got a list of points that form a piecewise linear approximation to the solution A(x,y)==A*. Those points are just a subset of the locus of all such values.

Then you chose the point from that set that minimized B(x,y). Again, it is not really a true solution, but an approximation.

But to extend that idea to 4 independent variables, you need to solve for a 4-d iso-surface.