Jacobian calculation

In this paper, after the description of the general features for the retrieval model (Sect. 2) and the optimisations implemented in both the forward model and Jacobian calculation for matching the run-time requirements (Sect. 3 and 4), we will focus on the validation tests that have been performed on the code (Sect.5) and on the accuracy and run-time performances of the retrieval algorithm (Sect. 6 and Sect. 7). 

If the cost of Jacobian calculation becomes prohibitive, e.g. in the case of numerical evaluation, it may be interesting to use a direct iteration. The equation f(x) = 0 is transformed to... 

The Jacobian must be updated when the convergence efficiency of the Newton method decays. It is usually recalculated after an assigned nimiber of iterations. In such a case, it is preferable to link this update to the problem dimension. In fact, the advantages to keeping the Jacobian constant increase with the size of the problem. With small-dimension problems, it can be disadvantageous to keep the Jacobian constant since the increase in the number of iterations can be larger than the computations saved to decrease the nmnber of Jacobian calculations. 

Appendix B Calculation of the Jacobian Matrix of an Averaging Function... 

DERIV. Calculates the inverse of the Jacobian matrix used in isoparametric transformations. 

Unlike the other alternative methods, analytical expressions of partial derivatives are required and the Jacobian must be evaluated in the Newton-Raphson method. These requirements sometimes prove to be the undoing when the method is applied to complicated equations. Brown (B12) has developed a modification to the Newton-Raphson method, which requires only some of the partial derivatives to be calculated. We have tested Brown s method on our sample problems but have found that it actually required more computing time than the unmodified Newton-Raphson method. 

As we explained earlier, the difficulty in this formulation is that generalized coordinates appear explicitly in the form of the Jacobian J, which may be difficult to calculate in many cases. It is therefore desirable to find an expression in which the explicit knowledge of the generalized coordinates is not required. This is done in Sect. 4.4. We end this section with a proof of (4.14). 

A more efficient way of solving the DFT equations is via a Newton-Raphson (NR) procedure as outlined here for a fluid between two surfaces. In this case one starts with an initial guess for the density profile. The self-consistent fields are then calculated and the next guess for density profile is obtained through a single-chain simulation. The difference from the Picard iteration method is that an NR procedure is used to estimate the new guess from the density profile from the old one and the one monitored in the single-chain simulation. This requires the computation of a Jacobian matrix in the course of the simulation, as described below. 

Calculations of relative free energies of binding often involve the alteration of bond lengths in the course of an alchemical simulation. When the bond lengths are subject to constraints, a correction is needed for variation of the Jacobian factor in the expression for the free energy. Although a number of expressions for the correction formula have been described in the literature, the correct expressions are those presented by Boresch and Karplus.21... 

Like Newton s method, the Newton-Raphson procedure has just a few steps. Given an estimate of the root to a system of equations, we calculate the residual for each equation. We check to see if each residual is negligibly small. If not, we calculate the Jacobian matrix and solve the linear Equation 4.19 for the correction vector. We update the estimated root with the correction vector,... 

In this section we consider how Newton-Raphson iteration can be applied to solve the governing equations listed in Section 4.1. There are three steps to setting up the iteration (1) reducing the complexity of the problem by reserving the equations that can be solved linearly, (2) computing the residuals, and (3) calculating the Jacobian matrix. Because reserving the equations with linear solutions reduces the number of basis entries carried in the iteration, the solution technique described here is known as the reduced basis method. ... 

Fig. 4.3. Calculation of the entries in the Jacobian matrix on a vector-parallel computer, using a concurrent-outer, vector-inner (COVI) scheme. Each summation in the Jacobian can be calculated as a vector pipeline as separate processors calculate the entries in parallel.

Such a scheme is sometimes called a soft Newton-Raphson formulation because the partial derivatives in the Jacobian matrix are incomplete. We could, in principle, use a hard formulation in which the Jacobian accounts for the devia-tives dy/dm,i and daw/dm,i. The hard formulation sometimes converges in fewer iterations, but in tests, the advantage was more than offset by the extra effort in computing the Jacobian. The soft method also allows us to keep the method for calculating activity coefficients (see Chapter 8) separate from the Newton-Raphson formulation, which simplifies programming. 

To do so, we calculate the Jacobian matrix, which is composed of the partial derivatives of the residual functions with respect to the unknown variables. Differentiating the mass action equations for aqueous species Aj (Eqn. 4.2), we note that,... 

At each step in the Newton-Raphson iteration, we evaluate the residual functions and Jacobian matrix. We then calculate a correction vector as the solution to the matrix equation... 

For modular-based process simulators, the determination of derivatives is not so straightforward. One way to get partial derivations of the module function(s) is by perturbation of the inputs of the modules in sequence to calculate finite-difference substitutes for derivatives for the tom variables. To calculate the Jacobian via this strategy, you have to simulate each module (C + 2) nT + nF + 1 times in sequence, where C is the number of chemical species, nT is the number of tom streams, and nF is the number of residual degrees of freedom. The procedure is as follows. Start with a tear stream. Back up along the calculation loop until an unperturbed independent variable xI t in a module is encountered. Perturb the independent variable,... 

Another way to calculate the partial derivatives is possible. Figure 15.12 represents a typical module. If a module is simulated individually rather than in sequence after each unknown input variable is perturbed by a small amount, to calculate the Jacobian matrix, (C + 2)nci + ndi + 1 simulations will be required for the ith module, where nci = number of interconnecting streams to module i and ndi = number of unspecified equipment parameters for module /. This method of calculation of the Jacobian matrix is usually referred to as full-block perturbation. 

Theta functions are special functions related to Jacobian elliptic functions (Morse and Feshbach, 1953 Widder, 1975) with special properties that make then extremely useful to calculate solutions to diffusion problems for small values of time. Three of the four theta functions will be used in the present context... 

The Jacobian can be written as the product of a diagonal matrix, containing the inverse component concentrations times the matrix J > where J 1 -=J 1 diag c). The shifts are then calculated as ... 

Finally, the calculation of the elements of the Jacobian J (actually J ) according to equation (3.42) ... 

As we have demonstrated, systems of non-linear equations with several unknowns are difficult to resolve. The task of developing a general program that can cope with all eventualities is huge. We are only offering a very minimal program that specifically analyses the system of equations (3.70). Instead of the two variables x and y we use a vector x with two elements similarly, we use a vector z instead of zl and z2. The elements of the required Jacobian J can be given explicitly (see Two Equations. m). The shift vector delta x is calculated as in equation (3.38). 

Next, several deliberations with respect to the calculation of the Jacobian are appropriate. 

For reaction mechanisms that have explicit solutions to the set of differential equations, it is always also possible to define the derivatives dC /dp explicitly. In such cases the Jacobian can be calculated in explicit equations and time consuming numerical differentiations are not required. The equations are rather complex, although implementation in Matlab is straightforward. The calculation of numerical derivatives is always possible and for mechanisms that require numerical integration, it is the only option. 

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