The Gauss-Newton Method for PDE Models

Following the same approach as in Chapter 6 for ODE models, we linearize the output vector around the current estimate of the parameter vector kw to yield 

The output sensitivity matrix (dyT/dk)T is related to the sensitivity coefficient matrix defined as 

The relationship between (Sy/Sk,) and G is obtained by implicit differentiation of Equations 10.4 - 10.7 depending on the type of measurements we have. Namely, 

Obviously in order to implement the Gauss-Newton method we have to compute the sensitivity coefficients Gj,(t,z). In this case however, the sensitivity coefficients are obtained by solving a set of PDEs rather than ODEs. 

The governing partial differential equation for G(t,z) is obtained by differentiating both sides of Equation 10.1 with respect to k and reversing the order of differentiation. The resulting PDE for Gj,(t,z) is given by (Seinfeld and Lapidus, 1974), 

---

At this point we can summarize the steps required to implement the Gauss-Newton method for PDE models. At each iteration, given the current estimate of the parameters, ky we obtain w(t,z) and G(t,z) by solving numerically the state and sensitivity partial differential equations. Using these values we compute the model output, y(t k(i)), and the output sensitivity matrix, (5yr/5k)T for each data point i=l,...,N. Subsequently, these are used to set up matrix A and vector b. Solution of the linear equation yields Ak(jH) and hence k°M) is obtained. The bisection rule to yield an acceptable step-size at each iteration of the Gauss-Newton method should also be used. 

The solution of Equation 10.28 is obtained in one step by performing a simple matrix multiplication since the inverse of the matrix on the left hand side of Equation 10.28 is already available from the integration of the state equations. Equation 10.28 is solved for r=l,...,p and thus the whole sensitivity matrix G(tr,) is obtained as [gi(tHt), g2(t,+1),- - , gP(t,+i)]. The computational savings that are realized by the above procedure are substantial, especially when the number of unknown parameters is large (Tan and Kalogerakis, 1991). With this modification the computational requirements of the Gauss-Newton method for PDE models become reasonable and hence, the estimation method becomes implementable. 

---