Differentiation Algorithms

In Sec. 3.6.4.1, smoothing and filtering were treated in detail. This was necessary because these operations represent the basis for the following differentiation methods. The quality of the set of data, for instance, the SNR, determines the quality as well as the suitability of derivatives. This point must be repeatedly emphasized because it is usually responsible for unsatisfactory results and artifacts. 

There are many ways to generate derivatives by numerical computations. The most important digital methods are described below. 

In 1953 Morrison [67] was the first to use this simple mode of differentiation by sub-stracting ordinate data in small intervalls of equal length along the abscissa (AA or A, which is proportional to AA). In reality, the difference quotient and not the differential quotient is computed, but — if the steps on the x-axis are small enough — the results are nearly the same. Using the difference quotient method, the slope of the signal can be calculated according to Equation (3-30)  

Similar to the smoothing width in polynomial smoothing operations, the differentiation width h = Ax = AA) has an influence on the noise and therefore also on the SNR. The greater the differentiation width (h), the smaller the noise and the higher the SNR (Fig. 3-52). In analogy to the smoothing ratio r, the differentiation ratio f can be defined as the ratio of the differentiation width to the half width of a band  

SW smoothing width (number of smoothing points) FWHM half width (see Sec. 2.3) 

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Fig. 5.10-2 The redox buffer strength of an aqueous vanadium solution of 0.01 M analytical concentration at pH = 0, calculated from (5.10-3) or by differentiation of the progress curve of Fig. 5.9-3. Again the two agree to within the computational accuracy of the differentiation algorithm used.

Luca, L. De and Musmanno, R., 1997, A Parallel Automatic Differentiation Algorithm for Simulation Models, Simulation Practice and Theory, 5,235-252. 

Extending time scales of Molecular Dynamics simulations is therefore one of the prime challenges of computational biophysics and attracted considerable attention [2-5]. Most efforts focus on improving algorithms for solving the initial value differential equations, which are in many cases, the Newton s equations of motion. 

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

The reaction rate equations give differential equations that can be solved with methods such as the Runge-Kutta [14] integration or the Gear algorithm [15]. 

With these reaction rate constants, differential reaction rate equations can be constructed for the individual reaction steps of the scheme shown in Figure 10.3-12. Integration of these differential rate equations by the Gear algorithm [15] allows the calculation of the concentration of the various species contained in Figure 10.3-12 over time. This is. shown in Figure 10.3-14. 

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-HaU, Englewood Cliffs, NJ, 1971 Gear Algorithm, QCPE Program No. QCMP022. 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

In general, the utilization of integral models requires more elaborate algorithms than the differential viscoelastic equations. Furthermore, models based on the differential constitutive equations can be more readily applied under general concUtions. 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

With the introduction of Gear s algorithm (25) for integration of stiff differential equations, the complete set of continuity equations describing the evolution of radical and molecular species can be solved even with a personal computer. Many models incorporating radical reactions have been pubHshed. 

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

Alternative algorithms employ global optimization methods such as simulated annealing that can explore the set of all possible reaction pathways [35]. In the MaxFlux method it is helpful to vary the value of [3 (temperamre) that appears in the differential cost function from an initially low [3 (high temperature), where the effective surface is smooth, to a high [3 (the reaction temperature of interest), where the reaction surface is more rugged. 

The software in the Turbolog DSP eontroller ineludes a eontrol algorithm ealled SHEDCON, whieh eontrols the regenerator differential pressure under all operating eonditions. This software was designed in 1984 speeifieally for ECC applieations and was improved for the refinery projeet deseribed here. 

The sigmoid aetivation funetion is popular for neural network applieations sinee it is differentiable and monotonie, both of whieh are a requirement for the baek-propagation algorithm. The equation for a sigmoid funetion is... 

Normally, the reactor temperature and the stripper level controllers regulate he movement of the regenerated and spent catalyst slide valves, le algorithm of these controllers can drive the valves either fully Of [ or fully closed if the controller set-point is unobtainable. It is ext nely important that a positive and stable pressure differential be mail ined across both the regenerated and spent catalyst slide valves. r safety, a low differential pressure controller overrides the tempera re/level controllers should these valves open too much. The shutdov is usually set at 2 psi (14 Kp). 

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