Error control

For short term simulations, accuracy requirements on the discrete solution make sense and we advocate error controlling adaptive integrators. Moreover, we have to further subdivide our classification due to the observations in Sec. 2.2 ... 

The four perspectives to be discussed in detail later in this chapter are contrasted in Table 2.1 in terms of the error control strategies that are usually employed, their main areas of application and the frequency that the approaches are applied in the CPI. 

Emphasis on the Modification of System Factors as a Major Error Reduction Strategy This emphasis replaces the reliance on rewards and pLmishment as a means of error control which characterizes the TSE approach. 

Becker, R., Rannachee, R., A feedback approach to error control infinite element methods basic analysis and examples, East-West J. Numer. Math. 4 (1996) 237-264. 

Human error analysis This method is used to identify the parts and the procedures of a process that have a higher than normal probability of human error. Control panel layout is an excellent application for human error analysis because a control panel can be designed in such a fashion that human error is inevitable. 

The simultaneous solution strategy offers several advantages over the sequential approach. A wide range of constraints may be easily incorporated and the solution of the optimization problem provides useful sensitivity information at little additional cost. On the other hand, the sequential approach is straightforward to implement and also has the advantage of well-developed error control. Error control for numerical integrators (used in the sequential approach) is relatively mature when compared, for example, to that of orthogonal collocation on finite elements (a possible technique for a simultaneous approach). 

As with all statistical methods, the mean-field estimate will have statistical error due to the finite sample size (X ), and deterministic errors due to the finite grid size (S ) and feedback of error in the coefficients of the SDEs Ui,p). Since error control is an important consideration in transported PDF simulations, we will now consider a simple example to illustrate the tradeoffs that must be made to minimize statistical error and bias. The example that we will use corresponds to (6.198), where the exact solution141 to the SDEs has the form ... 

Once the PDE has been semi-discretized (i.e., discretize the spatial derivatives but not the timelike derivatives) to form a system of ODEs, the ODEs can be solved by high-level software packages. In the standard form there are many such packages available, with relatively fewer for DAEs (see Section 15.3.3). In the method of lines, the spatial differencing must be done by the user, while time discretization and error control is handled by the ODE software. Overall, the effort to develop a new simulation is reduced, since a good deal of existing high-level software can be used. 

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

The constant-pressure formulation results in a high-index problem even with implementation of instead of A as the dependent variable. This results from the absence of any time derivatives for u. Attempts to solve the methane-ignition problem with the constant-pressure formulation were generally unsuccessful, except by significantly relaxing the error control on A and u. Even then, while the solutions appear generally correct, they exhibit unstable behavior near fast transients, particularly on A. 

This kind of an experiment is said to have type II error control, i.e. the error of not discovering the real deviation from the null hypothesis. The probability of making type II error is marked p. 

Does not require identification and measurement of any disturbance for corrective action Does not require an explicit process model Is possible to design controller to be robust to process/model errors Control action not taken until the effect of the disturbance has been felt by the system Is unsatisfactory for processes with large time constants and frequent disturbances May cause instability in the closed-loop response... 

When the quadrature of eq 2 cannot be performed analytically the integration should be carried out numerically by robust routines such as the Runge-Kutta, Adams-Moulton predictor-corrector or Bulirsch-Stoer methods with step size and error control [53, 55, 56], These routines can also be found in computer codings at Netlib and in standard books on computer codes [53]. 

All documents of the type referred to so far in Section 5.3 must be issued as controlled documents. This means that each copy is numbered and issued to a particular individual or location, A record is kept of each issue so that, when the document requires update, all copies in circulation can be updated and there is no danger of superseded versions of a document being followed in error. Controlled documents must neither be amended by unauthorised persons nor photocopied. 

We will not attempt here to give a detailed explanation of the numerical aspects (fundamentals of discretization, error estimates, and error control) of CFD since a number of excellent texts are available in the literature that deal in depth with this matter (Fletcher, 1988a,b Hirsch, 1988, 1990). First some general aspects of the numerical techniques used for solving fluid flow problems are discussed and, subsequently, a distinction is made between single-phase flows and... 

A. D. Raptis, Exponentially-fitted solutions of the eigenvalue Shrodinger equation with automatic error control, Computer Physics Communications, 1983, 427-431. 

The first requirement is generally easily met as the error in the equation solution typically becomes small compared to overall modeling error for moderate values of the error control tolerances. The tradeoff between the second and third requirements is more difficult and depends on the particular numerical characteristics of the system equations and the particular values of the optimization parameters. It is desirable to have some means of estimating and adjusting the precision error to optimize this tradeoff. This requires that the precision error be estimated, its effect on the optimization assessed, and the integrator tolerances adjusted appropriately. 

Responsible for providing reliable end-to-end communications. Includes most of the error control and flow control. 

Responsible for logical network addressing. Some error control and flow control Is performed at this level. 

This system was solved using a seventh order Runge Kutta-Vemer method, with adjustable step size and error control, which showed to be stable and last. The regression was done using a non-linear regression software, M3D [11], with the following objective function... 

In [30] we addressed explicitly the issue of error control for the discrete algorithm introduced in, [27], analyzing the quality of Fq as a statistical estimator of the free energy F. We showed that the accuracy of the discrete algorithm can be significantly improved introducing three simple modifications ... 

With the AutoAnalyzer, apart from the special requirements of multichannel analysis (Section 3.1.1.3), the confusion between standards and controls has become practically inextricable. The following quotation (L2) provides a good example Error control. Known standards accompany samples as controls, interspersed with the unknown samples at periodic intervals. In their travel through the system both are subjected to exactly the same conditions from beginning to end, and the standards act as controls in the final reading. ... 

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