Bisection Rule

The bisection rule constitutes the simplest and most robust way available to determine an acceptable value for the stepping parameter p. Normally, one starts with p=1 and keeps on halving p until the objective function becomes less than that obtained in the previous iteration (Hartley, 1961). Namely we accept the first value of p that satisfies the inequality 

More elaborate techniques have been published in the literature to obtain optimal or near optimal stepping parameter values. Essentially one performs a univariate search to determine the minimum value of the objective function along the chosen direction (Ak ) by the Gauss-Newton method. 

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Determine p using the bisection rule and obtain k =k +pAk Continue until the maximum number of iterations is reached or conver-... 

The easiest way to arrive at an acceptable value of p, pa, is by employing the bisection rule as previously described. Namely, we start with p=1 and we keep on halving p until the objective function at the new parameter values becomes less than that obtained in the previous iteration, i.e., we reduce p until... 

The proposed step-size policy for differential equation systems is fairly similar to our approach for algebraic equation models. First we start with the bisection rule. We start with g=l and we keep on halving it until an acceptable value, pa, has been found, i.e., we reduce p until... 

Step 6. Use the bisection rule to determine an acceptable step-size and then update the parameter estimates. 

If we are certain that the optimum parameter estimates lie well within the constraint boundaries, the simplest way to ensure that the parameters stay within the boundaries is through the use of the bisection rule. Namely, during each iteration of the Gauss-Newton method, if anyone of the new parameter estimates lie beyond its boundaries, then vector Ak +I) is halved, until all the parameter constraints are satisfied. Once the constraints are satisfied, we proceed with the determination of the step-size that will yield a reduction in the objective function as already discussed in Chapters 4 and 6. 

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. 

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