Finite difference substitutes for derivatives

The modules require some effort to generate reasonably accurate derivatives or their substitutes, especially if a module contains tables, functions with discrete variables, discontinuities, and so on. Perturbation of the input to a module is the primary way in which a finite-difference substitutes for derivatives can be generated. 

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,... 

Effective computer codes for the optimization of plants using process simulators require accurate values for first-order partial derivatives. In equation-based codes, getting analytical derivatives is straightforward, but may be complicated and subject to error. Analytic differentiation ameliorates error but yields results that may involve excessive computation time. Finite-difference substitutes for analytical derivatives are simple for the user to implement, but also can involve excessive computation time. 

The second class of multivariable optimization techniques in principle requires the use of partial derivatives, although finite difference formulas can be substituted for derivatives such techniques are called indirect methods and include the following classes ... 

The NLP (nonlinear programming) methods to be discussed in this chapter differ mainly in how they generate the search directions. Some nonlinear programming methods require information about derivative values, whereas others do not use derivatives and rely solely on function evaluations. Furthermore, finite difference substitutes can be used in lieu of derivatives as explained in Section 8.10. For differentiable functions, methods that use analytical derivatives almost always use less computation time and are more accurate, even if finite difference approxima-... 

Difficulty 3 can be ameliorated by using (properly) finite difference approximation as substitutes for derivatives. To overcome difficulty 4, two classes of methods exist to modify the pure Newton s method so that it is guaranteed to converge to a local minimum from an arbitrary starting point. The first of these, called trust region methods, minimize the quadratic approximation, Equation (6.10), within an elliptical region, whose size is adjusted so that the objective improves at each iteration see Section 6.3.2. The second class, line search methods, modifies the pure Newton s method in two ways (1) instead of taking a step size of one, a line search is used and (2) if the Hessian matrix H(x ) is not positive-definite, it is replaced by a positive-definite matrix that is close to H(x ). This is motivated by the easily verified fact that, if H(x ) is positive-definite, the Newton direction... 

From numerous tests involving optimization of nonlinear functions, methods that use derivatives have been demonstrated to be more efficient than those that do not. By replacing analytical derivatives with their finite difference substitutes, you can avoid having to code formulas for derivatives. Procedures that use second-order information are more accurate and require fewer iterations than those that use only first-order information(gradients), but keep in mind that usually the second-order information may be only approximate as it is based not on second derivatives themselves but their finite difference approximations. 

As a consequence, the gradient of the objective function and the Jacobian matrix of the constraints in the nonlinear programming problem cannot be determined analytically. Finite difference substitutes as discussed in Section 8.10 had to be used. To be conservative, substitutes for derivatives were computed as suggested by Curtis and Reid (1974). They estimated the ratio /x of the truncation error to the roundoff error in the central difference formula... 

The Neumann boundaries (which involve derivatives) are converted into finite difference form and substituted into the finite difference equations for the nodes in the specified region (e.g. above the electrode surface). The Dirichlet conditions (which fix the concentration value) may be substituted directly. 

To solve such problems numerically, a popular approach is to substitute finite difference analogs for the derivatives and solve the resnlting algebraic eqnations. There are many ways in which this can be done, a few of which are summarized below ... 

We set a uniform grid of points over the domain and substitute finite difference approximations for each second derivative to obtain for each interior point (x , yj, zj ) a linear equation... 

Omitting the elevation term and substituting the centered finite differences for the partial derivatives, Eqs. (123) and (128) become... 

To solve the problem a sequential quadratic programming code was used in the outer loop of calculations. Inner loops were used to evaluate the physical properties. Forward-finite differences with a step size of h = 10 7 were used as substitute for the derivatives. Equilibrium data were taken from Holland (1963). The results shown in Table E12.1B were essentially the same as those obtained by Sargent and Gaminibandara. 

The most important feature of this dispersion-optimized FDTD method is the higher order nonstandard finite-difference schemes [6, 7] that substitute their conventional counterparts in the differentiation of Ampere s and Faraday s laws, as already described in (3.31). The proposed technique can be occasionally even 7 to 8 orders of magnitude more accurate than the fourth-order implementations of Chapter 2. Although the cost is slightly increased, the overall simulation benefits from the low resolutions and the reduced number of iterations. Thus, for spatial derivative approximation, the following two operators are defined ... 

Here (p) is a fimction of T, partial derivatives can be estimated from finite difference approximations of the measimed data, eg, d p)/dT = [J2p(Ti + i,4>i) - Y,P( ri,4>i)y -N(Ti + i - 7I)].ThevaluesofA< andATare A(p = m L and AT = mrL, where mr and m j, are the slopes of the linear gradients, known from the library preparation procedure. Making these substitutions shows that the error propagation for property p scales as... 

The method of lines lies midway between analytical and grid methods, and can also be used to solve the conservation equations. This method involves substituting finite differences for the derivatives with respect to one independent variable and retaining the derivatives with respect to the remaining variables. This approach replaces a given differential equation by a system of differential equations with a smaller number of independent variables, typically reducing a partial differential equation to a set of ordinary differential equations. 

Substituting in the approximations for the derivatives, we obtain exactly the same algebraic equations as we had for the finite difference method using the central difference approximation. 

The polarizability and first hyperpolarizability of p-nitroaniline and its methyl-substituted derivatives have been calculated using a non-iterative approximation to the coupled-perturbed Kohn-Sham equation where the first-order derivatives of the field-dependent Kohn-Sham matrix are estimated using the finite field method" . This approximation turns out to be reliable with differences with respect to the fully coupled-perturbed Kohn-Sham values smaller than 1% and 5% for a and p, respectively. The agreement with the MP2 results is also good, which enables to employ this simplified method to deduce structure-property relationships. 

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