Finding best parametric function estimation for ODE of first order
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Lyudmil Yovkov
el 18 de Jul. de 2021
Editada: Lyudmil Yovkov
el 31 de Jul. de 2021
Hello to every one.
Until now I've solved differential equations numerically under given formulae for their coefficients and given initial / boundary conditions. But recently I've been dealt with the inverse problem, namely: find the best fit to the solution of the ODE regarding some unknown paramateres. The hard part here is due to the fact that we need to determine not a single value but the values of a whole parametric function.
Consider the following initial value problem for ODE of first order:
Solving the equation by separation of the variables we obtain the exact solution
Let's consider however this problem statement: given the Cauchy's problem
whose solution is known over the grid Find the estimated values of the function fitting the solution , i.e. establish that they're eventually located close enough to the curve .
I read a lot about the topic but I didn't find what I searched for. Could you give me some short explanations or guidelines on how to perform this idea in general and later using MATLAB toolboxes?
Kind regards,
Lyudmil Yovkov, PhD
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Torsten
el 18 de Jul. de 2021
Editada: Torsten
el 18 de Jul. de 2021
Search for a book on optimal control of ODEs and DAEs, e.g.
Matthias Gerdts:
Optimal control of ODEs and DAEs
De Gruyter 2011
Or start with an introductory script:
silo.tips/download/optimal-control-of-odes-introductory-lecture
Finally, you will have to use Matlab's lsqcurvefit or fmincon to determine the coefficients c_i on the grid x_i that minimize
sum_{i=0}^{i=n} (u_i - u_i(c_i))^2
where u_i(c_i) are the values for u obtained from the ODE by using the c_i as coefficients in front of du/dx.
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Torsten
el 21 de Jul. de 2021
The structure of the program will be the same. In f, you will have to call the pdesolver instead of ode15s, and the computation time will be longer.
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Alan Stevens
el 18 de Jul. de 2021
First plot the points, ui vs xi to see what sort of curve it might be. If it looks like it could be a polynomial (as in the case of u = x^2 + 1) then lookup help on polyfit.
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