What is the most efficient way to "reverse" interpolate a 3D array?

I have a 3-D array of data: it contains Fz values for different values of alpha angle, beta angle and Mach number.

I would like to know the best way to "reverse interpolate" the value of alpha for given/target values of beta, Mach and Fz.

The best method I can think of is a 2-step interpolation process:

1) For each value of alpha, and the given target values of beta and Mach, interpolate the "beta-Mach space" to extract Fz.

2) For the extracted Fz values from step 1, interpolate this space for the target Fz to find alpha.

Here is how it is implemented (data.mat is attached):

% load interpolation data
load data

% Set up interpolation target values
Fz_target = 19.4801;
beta_target = 0;
Mach_target = 0.1;

% Step 1: interpolate "beta-Mach space" for each alpha in alpha_array
[beta_points, Mach_points] = ndgrid(beta_array, Mach_array);
Fz_space = zeros(1,alphan);

for i = 1:alphan
    Fz_space(i) = interpn(beta_points, Mach_points, reshape(Fz_basic(i,:,:), [betan, Machn]), beta_target, Mach_target);
end

% Step 2: interpolate Fz_space from step 1 for Fz_target to get alpha
alpha_extract = interpn(Fz_space, alpha_array, Fz_target);

Is there a more efficient, faster, less verbose way to do this?

If I pursue the above method, is there a way to vectorize the for loop?

Answer 1

[Jan]

Creating the grid matrices is not useful. The interpolation is about 10 times faster using the original vectors.

% Omit:
% [beta_points, Mach_points] = ndgrid(beta_array, Mach_array);

% Use the vectors instead:
% Fz_space(i) = interpn(beta_points, Mach_points, reshape(Fz_basic(i,:,:), [betan, Machn]), beta_target, Mach_target);
  Fz_space(i) = interpn(beta_array,  Mach_array,  reshape(Fz_basic(i,:,:), [betan, Machn]), beta_target, Mach_target);

Answer 2

[Star Strider]

The loop would likely not be necessary interpolationg with the scatteredInterpolant function. Pass all the ‘_target’ values as column vectors (with the rows matching the desired matching values) for any of the three values to get the fourth. The code is a bit more involved initially, however after that, the interpolations are straightforward.

This defines scatteredInterpolant functions for all the values as functions of one of the other values —

LD = load(websave('data','https://www.mathworks.com/matlabcentral/answers/uploaded_files/1291810/data.mat'))

LD = struct with fields:
       Fz_basic: [9×3×2 double]
     Mach_array: [0.1000 0.8000]
          Machn: 2
    alpha_array: [-5 -2.5000 0 2.5000 5 7.5000 10 12.5000 15]
         alphan: 9
     beta_array: [-5 0 5]
          betan: 3

alpha = LD.alpha_array;
beta = LD.beta_array;
Mach = LD.Mach_array;
Fz = LD.Fz_basic;

CheckSame = Fz(:,:,2)-Fz(:,:,1);


alphav = reshape(alpha(:)*ones(1,numel(beta)*numel(Mach)), [],1);
betav = reshape(beta(:)*ones(1,numel(alpha)*numel(Mach)),[],1);
Machv = reshape(Mach(:)*ones(1,numel(alpha)*numel(beta)),[],1);
Fzv = reshape(Fz, [], 1);

alphafcn = scatteredInterpolant(betav,Machv,Fzv,alphav);

Warning: Duplicate data points have been detected and removed - corresponding values have been averaged.

betafcn = scatteredInterpolant(alphav,Machv,Fzv,betav);

Warning: Duplicate data points have been detected and removed - corresponding values have been averaged.

Machfcn = scatteredInterpolant(alphav,betav,Fzv,Machv);

Warning: Duplicate data points have been detected and removed - corresponding values have been averaged.

Fzfcn = scatteredInterpolant(alphav,betav,Machv,Fzv);

Warning: Duplicate data points have been detected and removed - corresponding values have been averaged.

Fz_target = 19.4801;
beta_target = 0;
Mach_target = 0.1;

alpha_interp = alphafcn(beta_target, Mach_target, Fz_target)

alpha_interp = 11.9095

figure
surfc(beta, alpha, Fz(:,:,1))
hold on
surfc(beta, alpha, Fz(:,:,2))
scatter3(beta_target, alpha_interp, Fz_target, 250, 'c', 'p', 'filled')
hold off
colormap(turbo)
xlabel('\beta')
ylabel('\alpha')
zlabel('F_z')

.