Optimizing a function for a given set of data

I need to optimize the Krogstad's Velocity Deficit Law equation to find the value of Π. The equation is given as:

I have the data for , , and κ. I probably need to minimize the function, but how do I go about doing that?

[Torsten]

What about U_inf and U ?

[Sabal Bista]

I have U and as well

[Matt J]

If you have to minimize the function, why is it set equal to 0?

[Sabal Bista]

That's what's been confusing me. It says they have done the optimized the function by minimizing it if you look at the highlighted text.

[Torsten]

That's what's been confusing me. It says they have done the optimized the function by minimizing it if you look at the highlighted text.

I guess you have vectors (say with n elements) of experimental data for y and U, and you have values for U_tau, U_inf, kappa and delta.

Then you cannot find PI that satisfies all n equations simultaneously, but you have to minimize

F(PI) = sum_{i=1}^{i=n} f(PI,Ui,yi)^2

And this optimum value for PI is given by the formula I gave you below.

Answer 1

[Matt J]

The function is a first order polynomial in Π. You can use roots to find where f(Π)=0, or just solve by hand.

Answer 2

[Torsten]

Arrange your equation as

F(PI) = PI * a + b = 0

where a, b are column vectors depending on U_inf, U, U_tau, kappa, y and delta.

The optimal estimate for PI is then given by

PIopt = - (a.'*b) / (a.'*a) 

[Sabal Bista]

Thank you. I'll try it