How can I optimize 2 parameters of my ODE system (with 3 deferential equation) to minimize distance between theorical and experimental value

Hello,

I want to develope a model in mathlab but i don't know how can i find best K and De (constant parameter) from my models to minimaze the sum of distance betwen theorical and expermental value of C.

Thanks you advance for your help.

Answer 1

[Bjorn Gustavsson]

The way you've written the equations above only gives you 2 differential equations, in your odethreevariable function you code it as if the first equation was:

Either your odethreevariables function or your equations are correct, you'll have to decide which it is.

To solve this type of problems you can do something like:

K_De_0 = [1 1]; % Or a more appropriate start-guess
K_De_optimal = fminsearch(@(K_De) sum((C_B_obs-C_B_fcn(tC,K_De)).^2),K_De_0);

That is a standard least-square fit of your function C_B_fcn with parameters K_De evaluated at times tC (that should be the instances in time for which you have your estimates of C_B). The only thing you'll need to do is to integrate your ODEs inside C_B_fcn to get the C_B-curve, something like this:

function C_B = C_B_fcn(tC,K_De)

ICs=zeros(3,1);
ICs(1,1) = 0.0035; 
ICs(2,1) = 0;
[t,X]=ode45(@(t,X) ode2variable(t,X,K_De),tC,ICs);
C_B = X(:,2);

end

function [dX_dt]=ode2variable(t,X,K_De)

% EDIT:
K = K_De(1);
De = K_De(2);

rc = X(1);
Cb = X(1);
rg = 0.0035;
Y0 = 182;
a = 190.47;

Jb = 1/((rg^2/(rc^2*(K*Y0)))+((rg*(rg-rc))/(rc*De)));
dX_dt = [-((Jb*(rg^2))/(Y0*rc^2)) ; Jb*a]; % ODE (system of 2 diferential equations)

end

This is not tested but should at least give you a starting point.

HTH

[Jeremy Brasseur]

Hi thank you for answers,

I did a mistake like you explain : i have only 2 differentials equations.

I tried the script with 2 fonctions that you write but the run let a mistake message.

I see that K and De are conciderat like 1 paramter on the script (K_De), so for the first function, mathwork don't find K and/or De in "Jb = 1/((rg^2/(rc^2*(K*Y0)))+((rg*(rg-rc))/(rc*De)));". And it didn t find tC.

Do you know how can i find the best combination of K and De to optimaze this models?

Sorry, i m realy a beginer in mathwork x)

[Bjorn Gustavsson]

Sorry about that. In the ode2variable function we have to extract the K and De parameters from the K_De input-parameter. I've now changed that. You will have to set tC and C_B_obs from your experimental observations in experimental data.xlsx. Once you've done that you have to supply some initial guesses for the best K and De parameters and put them in a variable K_De_0. After that you should be able to run the fminsearch-optimization that will find you the best K and De values.

My suggestion is that you first make sure that you can integrate the ODE (if it is in the ode2variable.m) you should be able to run the:

[t,X]=ode45(@(t,X) ode2variable(t,X,K_De_0),tC,ICs);

If that works you should be able to then run the fminsearch-optimisation as above. It will automatically find the best K and De values, provided the initial guess is reasonably close. What "reasonably close" is depends on the problem you have. For a 2-parameter minimization like here you can think of the optimisation as a hiker rambling around in a topography trying to find the lowest point - if the "terrain" is simple enough it will succeed finding a global minimum, if the "terrain" is comlpex one might get stuck in a local minima.

[Bjorn Gustavsson]

Did you get it to work?

[Jeremy Brasseur]

Yes perfectly ! I found De et K (Ke optimale =4.06824920654296e-07 et De optimal=5.72655059814452e-11) based on the K_De_0 = [10^-8 10^-12]. The answert is exactly that i wanted (similar at the literatur) :)

Thank you very much for your help :D