Efficient Computation of the Sensitivity Coefficients

Therefore, efficient computation schemes of the state and sensitivity equations are of paramount importance. One such scheme can be developed based on the sequential integration of the sensitivity coefficients. The idea of decoupling the direct calculation of the sensitivity coefficients from the solution of the model equations was first introduced by Dunker (1984) for stiff chemical mechanisms 

The integration of the state equations (Equation 10.21) by the fully implicit Euler s method is based on the iterative determination of x(t1+i). Thus, having x(t,) we solve the following difference equation for x(t, i). 

Equation 10.25 is of the form AAx=b which can be solved for Ax 1 and thus x0+l)(t .,) is obtained. Normally, we converge to x(tI+]) in very few iterations. If however, convergence is not achieved or the integration error tolerances are not satisfied, the time-step is reduced and the computations are repeated. 

For the solution of Equation 10.25 the inverse of matrix A is computed by iterative techniques as opposed to direct methods often employed for matrices of low order. Since matrix A is normally very large, its inverse is more economically found by an iterative method. Many iterative methods have been published such as successive over-relaxation (SOR) and its variants, the strongly implicit procedure (SIP) and its variants, Orthomin and its variants (Stone, 1968), nested factorization (Appleyard and Chesire, 1983) and iterative D4 with minimization (Tan and Let-keman. 1982) to name a few. 

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. 

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This indicates that after an initial overhead of 0.319 model runs to set up the algorithm, an additional 0.07 of a model-run was required for the computation of the sensitivity coefficients for each additional parameter. This is about 14 times less compared to the one additional model-run required by the standard implementation of the Gauss-Newton method. Obviously these numbers serve only as a guideline however, the computational savings realized through the efficient integration of the sensitivity ODEs are expected to be very significant whenever an implicit or semi-implicit reservoir simulator is involved. 

Importantly, recognize that the sensitivity problem is a linear equation for the sensitivity coefficients regardless of whether the original problem is linear or nonlinear. Once the solution to the underlying problem is determined, the sensitivity coefficients can be computed efficiently, exploiting the inherent linearity [57,102,110,232,321], There is recent sensitivity software by Petzold that builds on the DASSL family of codes [258],... 

We have heretofore had ample discussion of linear optical properties of ZnO and related materials. In this section, the nonlinear processes in ZnO are discussed, a topic that has been investigated in some detail. The research on nonlinear optical properties of semiconductors is motivated by electro-optic devices that can be used in telecommunications and optical computing as efficient harmonic generators, optical mixers, and tunable parametric oscillators, among others. The nonlinear optical properties such as second harmonic generation (SHG), that is, (2(0i, 2(02), and the sum frequency generation (SFG), that is, (materials characterization, particularly surfaces, because the second-order susceptibility coefficient is very sensitive to the change in symmetry (178,179). The crystal should be... 

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