lsqcurvefit to estimate parameters in ODE15s

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Hello!
I am trying to solve a parametric ODE system. I woul like to estimate the parameters by fitting the calculated data with experimental data.
Model:
dydt(1) = - theta(2).*y(1).*y(2)-theta(3).*y(1).*(theta(4)-y(3))-theta(5)*y(1).*y(2).^2;
dydt(2) = -theta(2).*y(1).*y(2)-theta(5)*y(1).*y(2).^2;
dydt(3) = theta(3).*y(1).*(theta(4)-y(3));
dydt(4) = 0;
it is a system of differential equations with 5 parameters (theta). These are constant parameters that affect the time evolution of the set of solutions y.
What I can empirically measure is the quantity
C=theta(2)*y(1)*y(2)/max(theta(2)*y(1)*y(2));
I would like to solve the system of ODE and fit C to the measured values to obtain the best guess for the set of parameters theta.
Here my code:
%parameters initial values
theta0 = [4e17 2e-11 7e-8 5e15 9.3e-30];
% Variables
n_abs =3.3e15;
%experimental values for time (t - in seconds) and evolution (c) are pasted at the end
%of the code
[theta,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=lsqcurvefit(@(theta,t) kinetics(theta,t),theta0,t,c);
[Cfit,T] = kinetics(theta,t);%,precision);
fprintf(1,'\tRate Constants:\n')
for k1 = 1:length(theta)
fprintf(1, '\t\tTheta(%d) = %d\n', k1, theta(k1))
end
figure(1)
plot(t*1e9, c, 'p') %plot in ns
hold on
hlp = plot(T*1e9, Cfit); %plot in ns
hold off
grid
xlabel('Time')
ylabel('Concentration')
legend(hlp, 'C_1(t)', 'C_2(t)', 'C_3(t)', 'C_4(t)', 'Location','N')
set(gca,'yscale','log')
function [C,T] = kinetics(theta,t)
y0 = [3.3e15 3.3e15+theta(1) 0 0 ]; %initial conditions
[T,y] = ode15s(@odefcn, t, y0);
function dydt = odefcn(t,y)
fprintf( 'dentroOdefnc')
dydt = zeros(4,1);
dydt(1) = - theta(2).*y(1).*y(2)-theta(3).*y(1).*(theta(4)-y(3))-theta(5)*y(1).*y(2).^2;
dydt(2) = -theta(2).*y(1).*y(2)-theta(5)*y(1).*y(2).^2;
dydt(3) = theta(3).*y(1).*(theta(4)-y(3));
dydt(4) = 0;
end
C=theta(2)*y(:,1).*y(:,2)/max(theta(2)*y(:,1).*y(:,2));
end
The code is running but at the output i don't get any change in the values i initially set for theta0. Basically lsqcurvefit is not working properly.
This is what I get as output:
Local minimum possible.
lsqcurvefit stopped because the size of the current step is less than
the value of the step size tolerance.
Can you please help me?
Here the experimental data i am using for fitting
t=[4.88281749999997e-11
9.76562750000003e-11
1.46484375000000e-10
1.95312475000000e-10
2.44140675000000e-10
2.92968775000000e-10
3.41796875000000e-10
3.90624975000000e-10
4.39453175000000e-10
4.88281275000000e-10
5.37109375000000e-10
5.85937475000000e-10
6.34765675000000e-10
6.83593775000000e-10
7.32421875000000e-10
7.81249975000000e-10
8.30078175000000e-10
8.78906275000000e-10
9.27734375000000e-10
9.76562475000000e-10
1.02539067500000e-09
1.07421877500000e-09
1.12304687500000e-09
1.17187497500000e-09
1.22070317500000e-09
1.26953127500000e-09
1.31835937500000e-09
1.36718747500000e-09
1.41601567500000e-09
1.46484377500000e-09
1.51367187500000e-09
1.56249997500000e-09
1.61132817500000e-09
1.66015627500000e-09
1.70898437500000e-09
1.75781247500000e-09
1.80664067500000e-09
1.85546877500000e-09
1.90429687500000e-09
1.95312497500000e-09
2.00195317500000e-09
2.05078127500000e-09
2.09960937500000e-09
2.14843747500000e-09
2.19726567500000e-09
2.24609377500000e-09
2.29492187500000e-09
2.34374997500000e-09
2.39257817500000e-09
2.44140627500000e-09
2.49023437500000e-09
2.53906247500000e-09
2.58789067500000e-09
2.63671877500000e-09
2.68554687500000e-09
2.73437497500000e-09
2.78320317500000e-09
2.83203127500000e-09
2.88085937500000e-09
2.92968747500000e-09
2.97851567500000e-09
3.02734377500000e-09
3.07617187500000e-09
3.12499997500000e-09
3.17382817500000e-09
3.22265627500000e-09
3.27148437500000e-09
3.32031247500000e-09
3.36914067500000e-09
3.41796877500000e-09
3.46679687500000e-09
3.51562497500000e-09
3.56445317500000e-09
3.61328127500000e-09
3.66210937500000e-09
3.71093747500000e-09
3.75976567500000e-09
3.80859377500000e-09
3.85742187500000e-09
3.90624997500000e-09
3.95507817500000e-09
4.00390627500000e-09
4.05273437500000e-09
4.10156247500000e-09
4.15039067500000e-09
4.19921877500000e-09
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4.83398437500000e-09
4.88281287500000e-09
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1.42578128750000e-08
1.43066408750000e-08
1.43554688750000e-08
1.44042968750000e-08
1.44531248750000e-08
1.45019528750000e-08
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1.45996098750000e-08
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1.61132808750000e-08
1.61621098750000e-08
1.62109378750000e-08
1.62597658750000e-08
1.63085938750000e-08
1.63574218750000e-08
1.64062468750000e-08
1.64550778750000e-08
1.65039058750000e-08
1.65527348750000e-08
1.66015628750000e-08
1.66503908750000e-08
1.66992188750000e-08
1.67480468750000e-08
1.67968748750000e-08
1.68457028750000e-08
1.68945308750000e-08
1.69433598750000e-08
1.69921878750000e-08
1.70410158750000e-08
1.70898468750000e-08
1.71386718750000e-08
1.71874998750000e-08
1.72363278750000e-08
1.72851558750000e-08
1.73339848750000e-08
1.73828128750000e-08
1.74316408750000e-08
1.74804688750000e-08
1.75292968750000e-08
1.75781248750000e-08
1.76269528750000e-08
1.76757808750000e-08
1.77246098750000e-08
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2.22656248750000e-08
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2.28027348750000e-08
2.28515628750000e-08
2.29003908750000e-08
2.29492188750000e-08
2.29980468750000e-08
]; %experimental data (time axis)
c=[0.978949375536276
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0.00230717894937554]; %experimental data (yaxis)

Accepted Answer

Star Strider
Star Strider on 27 May 2021
The problem is in the definition of ‘theta0’ and not including the initial conditions in ‘theta0’.
Changing those produces —
T1 = readtable('Antonella_Treglia_Data.txt');
t = T1.t;
c = T1.c;
%parameters initial values
theta0 = [4e17 2e-11 7e-8 5e15 9.3e-30];
theta0 = [theta0 3.3e15 3.3e15+theta0(1) rand(1,2) ];
theta0 = rand(9,1);
% Variables
n_abs =3.3e15;
%experimental values for time (t - in seconds) and evolution (c) are pasted at the end
%of the code
[theta,Rsdnrm,Rsd,ExFlg,OptmInfo,Lmda,Jmat]=lsqcurvefit(@(theta,t) kinetics(theta,t),theta0,t,c,zeros(1,9));
dentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdent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efncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefnc...
[Cfit,T] = kinetics(theta,t);%,precision);
dentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefncdentroOdefnc
fprintf(1,'\tRate Constants:\n')
Rate Constants:
for k1 = 1:length(theta)
fprintf(1, '\t\tTheta(%d) = %10.5f\n', k1, theta(k1))
end
Theta(1) = 0.36428 Theta(2) = 61.16478 Theta(3) = 0.36475 Theta(4) = 1.75649 Theta(5) = 515.96813 Theta(6) = 35.60667 Theta(7) = 728.17186 Theta(8) = 1.73429 Theta(9) = 0.68600
fprintf('\nNOTE — The last four ‘theta’ values are the initial conditions.\n\n')
NOTE — The last four ‘theta’ values are the initial conditions.
figure(1)
plot(t*1e9, c, 'p') %plot in ns
hold on
hlp = plot(T*1e9, Cfit, 'LineWidth',2); %plot in ns
hold off
grid
xlabel('Time')
ylabel('Concentration')
legend(hlp, 'C_1(t)')%, 'C_2(t)', 'C_3(t)', 'C_4(t)', 'Location','N')
set(gca,'yscale','log')
function [C,T] = kinetics(theta,t)
% y0 = [3.3e15 3.3e15+theta(1) 0 0 ]; %initial conditions
y0 = theta(6:end);
[T,y] = ode15s(@odefcn, t, y0);
function dydt = odefcn(t,y)
fprintf( 'dentroOdefnc')
dydt = zeros(4,1);
dydt(1) = - theta(2).*y(1).*y(2)-theta(3).*y(1).*(theta(4)-y(3))-theta(5)*y(1).*y(2).^2;
dydt(2) = -theta(2).*y(1).*y(2)-theta(5)*y(1).*y(2).^2;
dydt(3) = theta(3).*y(1).*(theta(4)-y(3));
dydt(4) = 0;
end
C=theta(2)*y(:,1).*y(:,2)/max(theta(2)*y(:,1).*y(:,2));
end
One set of parameters that worked are:
% Parameters = [0.9; 22; -48; 15; 455]
% InitialConditions = [38; 776; -32; 0.7]
Constraining the parameters so they are all greater than 0 is an option with lsqcurvefit, and was added in this code.
Doing that produces:
% Parameters = [0.4; 61; 0.4; 1.8; 516]
% InitialConditions = [36; 728; 1.7; 0.7]
With an equally (visually) acceptable fit.
NOTE — I copied the posted data to a text file so that I could easily run the code here. The file is attached.
.
  6 Comments
Star Strider
Star Strider on 27 May 2021
@Antonella Treglia — The parameters you chose are so far from the parameters that are appropriate for this model that they do not change because starting with them, lsqcurvefit is searching the parameter space for a gradient to descend toward a global minimum and not finding it.
This:
Warning: Failure at t=4.882817e-11. Unable to meet integration tolerances without reducing the step size below the smallest value allowed
indicates that the differential equation integration is enountering a singularity () and stopping. That also indicates a problem with the parameter scaling.
.

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More Answers (1)

Torsten
Torsten on 27 May 2021
function C = kinetics(theta,t)
instead of
function [C,T] = kinetics(theta,t)
Define stronger RelTol and AbsTol for ODE15S and lsqcurvefit (around 1e-8)
Can you include a plot in which we see the t-c-curve for the experiment together with the t-c_simulated curve with the starting parameters for theta ?
  3 Comments
Antonella Treglia
Antonella Treglia on 27 May 2021
I set:
options = optimoptions('lsqcurvefit','StepTolerance',1e-12,'MaxFunEvals',5e3,'MaxIterations',5e3);
Opt = odeset('RelTol',1e-8,'AbsTol',1e-8);
but continues to give me
Warning: Failure at t=4.882817e-11. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.033976e-25) at time t.
It stops at the first value of the time vector t

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