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required assistance in fitting

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Somnath Kale
Somnath Kale 2024-3-12
评论： Mathieu NOE 2024-3-25
HI
I was trying to fit my data (in attchment) with fiist column V and second column J(V) with following equation (which honestly found complex to me)
mb = 9.11E-31; e = 1.6E-19; t = 3E-9; ℏ = 1.05E-34;
with Φ1 and Φ2 as the fitting parameters
Your support in this regard is apreiable!
Best
Somnath
  2 个评论
Mathieu NOE
Mathieu NOE 2024-3-12
do we have an initial guess or range for Φ1 and Φ2 ?
Somnath Kale
Somnath Kale 2024-3-12
@Mathieu NOE values of Φ1 and Φ2 are within the range of 0.1 to 5 in eV units

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Mathieu NOE
Mathieu NOE 2024-3-12
Ran in:
so this is the poor"s man optimization (w/o any toolbox)
a brute force approach , let's create a 2 parameter grid and evaluate the function vs your data
the complex equation is simplified by taking some large blocks and define intermediate variables , instead of writing a unreadable long stuff.
we get a 2 dimensionnal error map (plotted after log conversion for better rendering) , and we pick the point of minimal error
we are fortunate that the error surface is rather smooth (banana)
for my own fun I did 2 iterations, the second is just refined grid around the first obtained minima point
% load J and V data
data = readmatrix('data.xlsx'); % first column V and second column J(V)
V = data(:,1);
J = data(:,2);
mb = 9.11E-31;
e = 1.6E-19;
t = 3E-9;
h = 1.05E-34;
A = 4*e*mb/(9*pi^2*h^3);
const_alpha = 4*t*sqrt(2*mb)/(3*h);
%Φ1 and Φ2 are within the range of 0.1 to 5 in eV units
%% First iteration
increment1 = 0.017;
increment2 = 0.019;
phi1 = (-min(0.5*V)+eps:increment1:2);
phi2 = (max(0.5*V):increment2:4);
for ci = 1:numel(phi1)
for cj = 1:numel(phi2)
P1 = phi1(ci)*e;
P2 = phi2(cj)*e;
Jm = model(const_alpha,e,A,P1,P2,V);
err2(ci,cj) = mean((J - Jm).^2);
end
end
[val,ind] = min(err2,[],'all','linear');
error_after_first_iteration = val
error_after_first_iteration = 3.9231e-07
range_factor = 100;
err2(err2>range_factor*val) = range_factor*val;
figure,imagesc(log10(err2));colorbar('vert');
colormap('jet')
% find optimum point
[r,c] = ind2sub(size(err2),ind);
P1 = phi1(r)*e;
P2 = phi2(c)*e;
Jm1 = model(const_alpha,e,A,P1,P2,V);
%% second iteration ??
clear err2
phi1 = linspace(phi1(r)*0.8,phi1(r)*1.2,100);
phi2 = linspace(phi2(c)*0.8,phi2(c)*1.2,100);
for ci = 1:numel(phi1)
for cj = 1:numel(phi2)
P1 = phi1(ci)*e;
P2 = phi2(cj)*e;
Jm = model(const_alpha,e,A,P1,P2,V);
err2(ci,cj) = mean((J - Jm).^2);
end
end
[val,ind] = min(err2,[],'all','linear');
error_after_second_iteration = val
error_after_second_iteration = 3.8205e-07
range_factor = 100;
err2(err2>range_factor*val) = range_factor*val;
figure,imagesc(log10(err2));colorbar('vert');
colormap('jet')
% find optimum point
[r,c] = ind2sub(size(err2),ind);
P1 = phi1(r)*e;
P2 = phi2(c)*e;
Jm2 = model(const_alpha,e,A,P1,P2,V);
figure,
plot(V,J,V,Jm1,V,Jm2)
xlabel('V');
ylabel('J(V)');
legend('raw','1st iteration','2nd iteration');
% optimal phi1 and phi2 (in eV)
phi1(r)
ans = 0.8005
phi2(c)
ans = 2.1355
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% function
function Jm = model(const_alpha,e,A,P1,P2,V)
E = sqrt(P2 - 0.5*e*V);
F = sqrt(P1 + 0.5*e*V);
G = (P2 - 0.5*e*V).^1.5;
H = (P1 + 0.5*e*V).^1.5;
alpha = const_alpha./(P1 - P2 +e*V);
N = exp(alpha.*(G - H));
D = (alpha.^2).*(E - F).^2;
B = 3/4*alpha.*e.*V;
Jm = -A*N./D.*sinh(B.*(E-F));
end
  3 个评论
显示 1更早的评论
Mathieu NOE
Mathieu NOE 2024-3-13
here, for my own fun, I used fminsearch (no toolbox required !) in the second iteration
still we need the first iteration, based on grid search principle , to find good IC for the fminsearch final optimization
again, similar results as above , the second iteration loop brings no big improvement, but it shows how to use fminsearch in this context
have fun !
error_after_first_iteration = 3.9231e-07
P1 = 0.7650
P2 = 2.1840
error_after_second_iteration = 3.8202e-07
P1 = 0.7999
P2 = 2.1360
% load J and V data
data = readmatrix('data.xlsx'); % first column V and second column J(V)
V = data(:,1);
J = data(:,2);
mb = 9.11E-31;
e = 1.6E-19;
t = 3E-9;
h = 1.05E-34;
A = 4*e*mb/(9*pi^2*h^3);
const_alpha = 4*t*sqrt(2*mb)/(3*h);
%Φ1 and Φ2 are within the range of 0.1 to 5 in eV units
%% First iteration
increment1 = 0.017;
increment2 = 0.019;
phi1 = (-min(0.5*V)+eps:increment1:2);
phi2 = (max(0.5*V):increment2:4);
for ci = 1:numel(phi1)
for cj = 1:numel(phi2)
P1 = phi1(ci)*e;
P2 = phi2(cj)*e;
Jm = model(const_alpha,e,A,P1,P2,V);
err2(ci,cj) = mean((J - Jm).^2);
end
end
[val,ind] = min(err2,[],'all','linear');
error_after_first_iteration = val
range_factor = 100;
err2(err2>range_factor*val) = range_factor*val;
figure,imagesc(log10(err2));colorbar('vert');
colormap('jet')
% find optimum point
[r,c] = ind2sub(size(err2),ind);
% optimal phi1 and phi2 (in eV)
P1 = phi1(r)
P2 = phi2(c)
Jm1 = model(const_alpha,e,A,P1*e,P2*e,V);
%% second iteration ??
% optimisation with fminsearch
% global const_alpha e A V J
IC = [phi1(r) phi2(c)];
[x,FVAL] = fminsearch(@objectivefcn1,IC);
% optimal phi1 and phi2 (in eV)
P1 = x(1)
P2 = x(2)
Jm2 = model(const_alpha,e,A,P1*e,P2*e,V);
error_after_second_iteration = FVAL
figure,
plot(V,J,V,Jm1,V,Jm2)
xlabel('V');
ylabel('J(V)');
legend('raw','1st iteration','2nd iteration');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% function
function Jm = model(const_alpha,e,A,P1,P2,V)
E = sqrt(P2 - 0.5*e*V);
F = sqrt(P1 + 0.5*e*V);
G = (P2 - 0.5*e*V).^1.5;
H = (P1 + 0.5*e*V).^1.5;
alpha = const_alpha./(P1 - P2 +e*V);
N = exp(alpha.*(G - H));
D = (alpha.^2).*(E - F).^2;
B = 3/4*alpha.*e.*V;
Jm = -A*N./D.*sinh(B.*(E-F));
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function f = objectivefcn1(x)
global const_alpha e A V J
% P1 = phi1(r)*e;
% P2 = phi2(c)*e;
P1 = x(1)*e;
P2 = x(2)*e;
Jm = model(const_alpha,e,A,P1,P2,V);
f = mean((J - Jm).^2);
end

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