Generalized Inverse

Summary

In this chapter we have derived the generalized inverse formulation for the combined state and parameter estimation problem. The starting point was the Bayes’ theorem on the form (7.13) where all the data are introduced simultaneously together with an assumption of Gaussian priors. This led to the generalized inverse formulation in the form of a penalty function which is quadratic in the errors. From the generalized inverse, we derived the Euler-Lagrange equations which, in the parameter estimation case, pose a nonlinear problem even if the dynamical model is linear. We showed how we could resolve this nonlinearity by defining an iteration for the parameters to be estimated and then use the representer method to solve for the state for each iterate of the parameters.

Note that it is also possible to define a sequence of variational problems for each of (7.15-7.18) and the solution of one variational problem would then become the prior for the next. This could be a sensible approach except that the variational methods, such as the representer and adjoint methods, do not easily provide statistical information about the errors of the estimate, which is needed when the estimate is used as a prior for the next inversion. On the other hand, the genetic algorithms result in a sample of the posterior distribution, which might be used as the prior for the next inversion.