Re: integration + Levenberg-Marquardt (Numerical Recipes)


In article <d256ff43-4f70-49b8-8059-991633bf0353@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Jamie <Jamie.McQuay@xxxxxxxxxxxxxx> writes:
Hello,

I am having problems getting my fitting function derivatives correct
for the Levenburg-Marquardt fitting routine.

I have a differential in the following format:
dAB/dt = ka*A(Bo-AB) - kd*AB

additional:
A is constant
I'm fitting ka,Bo and kd

I am getting the integrated format of the formula via numerical
integration (this is a test equation, they will get more difficult).

The integration is correct when I test the resulting data but I don't
know how to properly get parameter derivatives for the fitting
function.

How do I generate the parameter derivatives (dyda in the LM fitting
function) without the integrated form of the formula (i only have it
numerically).

Thank you for any hints.

Jamie


in your example, you have the time dependent function AB, which also
depends on the real parameters ka, Bo and kd which you want ot compute
(from measured instances of AB, I guess).
one can prove, that AB depends differentiable on these parameters,
and that the derivatives, which of course also depend on t, obey
differential equations. for easier notation I name these

g1(t) = d(AB)/d(ka) (t)
g2(t) = d(AB)/d(B0) (t)
g3(t) = d(AB)/d(kb) (t)

the differential equations for the g1,g2,g3 are obtained by differentiating
the given differential equation with respect to the parameters using
the chain rule and keeping in mind that AB also is a function of them:
"d(AB)/dt = ka*A(Bo-AB) - kd*AB" =>
dg1/dt = A*(Bo-AB) - kd*g1 + ka*A*(-g1)
dg2/dt = ka*A(1-g2) - kd*g2
dg3/dt = ka*A*(-g3) - AB - kd*g3

(hope I got it right)
you add these differential equations to your original one (observe, that
also AB itselt enters) and integrate then as one system with your integrator
using the output at the grid points (t_i_given, AB_measured_i)
then you can set up the residual and the Jacobian which is required by
Levenberg Marquardt. This is much better than forcing the Levenberg
Marquardt code to do the derivative using its implemented finite
difference formula: remember that in the integrator, say a Runge Kutta code,
you will specify a "tolerance". this tolerance will yield an internally
used grid, which will, as a rule, be much finer than the grid of your
measured points. then, in finite differences, you might have a to large
change of the parameters which might change the grid . then the output
will be not a smooth function of the parameters and the finite difference
will yield nonsense. or, contrary , the discretization errors in the integrator
will be much larger than the change in the solution due to a small
change in the parameters, and again the finite diffenrce formula will yield
nonsense. with the above approach, you will get a useful jacobian
hth
peter

.

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