Optimizing a function for a given set of data
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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?
, , and κ. I probably need to minimize the function, but how do I go about doing that? 5 个评论
Torsten
2023-4-27
What about U_inf and U ?
Sabal Bista
2023-4-27
I have U and 
as well
as well
Matt J
2023-4-27
If you have to minimize the function, why is it set equal to 0?
Sabal Bista
2023-4-27
移动：Matt J
2023-4-27
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.
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.
回答（2 个）
The function is a first order polynomial in Π. You can use roots to find where f(Π)=0, or just solve by hand.
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)
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2023-4-27
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