How to define a constraint function for the adjoint method

[chrisAUS]

Homework Statement

Let h be an observed value at a given time t

t = 5, 10, 20, 30, 40, 50

h = 0.72, 0.49, 0.30, 0.20, 0.16, 0.12

Let h* be a modeled estimate of h

Homework Equations

h* = [ Q / 4*Pi*T*t ] * e [ - (d^2)*S / 4*T*t ]

where the known constants are Pi, Q (= 50), and d (= 60)

and the values of the two parameters T and S are unknown.

Values of T and S can be estimated through the minimisation of the sum of squared differences between h and h* over all times.

The Attempt at a Solution

The adjoint method involves formulating a function (e.g. H) which is composed of the sum of the function of interest (i.e. in this case, h*) and a second term, which is composed of a Lagrange multiplier multiplied by a constraint function.

How do I formulate the constraint function? I'm guessing that it should involve the squared difference between h and h*, since the minimisation of (h-h*)^2 is the desired outcome. But, since there are 6 times at which h* is calculated, will I need to have 6 formulations of H (and therefore 6 different Lagrange multipliers)? Or, since the minimisation of the sum of (h-h*)^2 for all times is the overall desired outcome, do I only need to formulate one equation for H?

Another point that bothers me is this. In the examples of the use of the adjoint method that I have seen, the constraint function is formulated in terms of the parameters of interest (i.e. in this case, T and S), rather than in terms of the function of interest (i.e. in this case, h*).

Thanks in advance.

 

[chrisAUS]

Okay, after ~70 thread views but zero replies, I have had some more thoughts on formulating this problem. Perhaps this will stimulate a response.

The minimisation of the squared difference between h* and h at a given time t is the objective of interest. Therefore the equation of interest (H) is not just a function of h* but instead :

H = (h* - h)^2 where h* = [ Q / 4*Pi*T*t ] * exp [ - (d^2)*S / 4*T*t ] , as described previously.

I have seen (in Bryson & Ho 1975) the constraint function for a two parameter problem described using a linear function, e.g. for parameters x and y, f = x + my - c.

Since my problem is also a two parameter problem, could a linear constraint function also be used ?

If so, then the full equation (i.e. Z) might be :

Z = (function of interest) + lambda*(constraint function) = (h* - h)^2 + lambda*(T + mS - c)

and then this could be differentiated with respect to T and to S, and the resulting equations (along with f=T+mS-c=0) could be solved for T,S and lambda.

If anyone has experience in formulating adjoint-state equations for functions that involve the minimisation of least squares, your advice would be much appreciated !