Differentiability estimated error

Substitution of Eq. (8.11) into Eq. (8.7) allows us to obtain an expression for the differential equation of the estimation error covariance matrix ... 

Listing 11.18 shows an example of the use of this function for the same nonlinear BV problem. This is about as simple as one can get for defining a differential equation, boundary conditions and obtaining a solution with estimated error. Note that in the xgpQ call the number of desired spatial points is omitted as the function has a default value of 2000 spatial intervals. The printed output shows that 5 Newton iterations are required for this solution. This is smaller than the 11 iterations required if the log spatial distribution is used as seen on the output line of Listing 11.17. This smaller number of Newton iterations also indicates a more... 

From this example it can be concluded that it is not necessary to convert a first order equation to a second order equation in order to use the FD solution method. One simply needs to supplement the one boundary condition with a second boundary condition that reproduces the first order differential equation. However, it may be of some interest to look at the estimated error in the solution using either... 

Figure 11.41. Estimated error in solution of first order differential equation in Listing 11.29 using FD eode.

We have in mind trajectory calculations in which the time step At is large and therefore the computed trajectory is unlikely to be the exact solution. Let Xnum. t) be the numerical solution as opposed to the true solution Xexact t)- A plausible estimate of the errors in X um t) can be obtained by plugging it back into the differential equation. 

The accuracy of any expression for first order errors can be improved as much as desired (so long as the expression is differentiable) by including additional terms of its Taylor expansion. For example, the error magnification factor in the age equation for the 230Th/238u term, (p, can be improved to include the second derivative, so the estimate of the effect of a given error in cp on the age becomes ... 

Model-fitting procedures are usually based on analytical solutions of the model however, model parameters may be estimated by fitting the differential equations describing the model. Since the numerical solution of the differential equations introduces another source of error, fitting of differential equations is usually limited to cases where nonlinearities are present. 

These differential equations depend on the entire probability density function / (x, t) for x(t). The evolution with time of the probability density function can, in principle, be solved with Kolmogorov s forward equation (Jazwinski, 1970), although this equation has been solved only in a few simple cases (Bancha-Reid, 1960). The implementation of practical algorithms for the computation of the estimate and its error covariance requires methods that do not depend on knowing p(x, t). 

Tjoa and Biegler (1991) used this formulation within a simultaneous strategy for data reconciliation and gross error detection on nonlinear systems. Albuquerque and Biegler (1996) used the same approach within the context of solving an error-in-all-variable-parameter estimation problem constrained by differential and algebraic equations. 

The average errors in evaluation of the differential heats of adsorption, as estimated by Stach et al. [152], are 1-2% only for the direct measurement and around 5% for the isosteric measurements. 

Ri is sensitive to temperature and even a relatively small error in temperature estimate can introduce a sizable discrepancy into the apparent p02 based on some PFCs. The relative error introduced into a p02 determination by a 1 °C error in temperature estimate ranges from 8 Torr/°C for PFTB [207] to 3 Torr/°C for PFOB (perflubron) [223] or 15-Crown-5-ether [218] when p02 is actually 5 Torr. HFB exhibits remarkable lack of temperature dependence and the comparative error would be 0.1 Torr/°C [224], Recognizing differential sensitivity of pairs of resonances within a single molecule to p02 and temperature, Mason et al. [207,225] patented a method to simultaneously determine both parameters by solving simultaneous equations. However, generally it is preferable for a p02 sensor to exhibit minimal response to temperature, since this is not always known precisely in vivo and temperature gradients may occur across tumors. 

One of the most frustrating problems that you may encounter using the Transient Analysis is convergence problems. When PSpice is simulating a differential equation, it calculates a data point and estimates the error associated with the calculation. If the error is larger than a specified maximum, PSpice reduces the time step and recalculates the point and the error for the new point. Reducing the time step usually reduces the error. PSpice will continue reducing the time step until the error is within acceptable limits, or until PSpice reaches the limit on the number of times it is allowed to reduce the time step. 

Stutz, J., and U. Platt, Numerical Analysis and Estimation of the Statistical Error of Differential Optical Absorption Spectroscopy Measurements with Least-Squares Methods, Appl. Opt., 35, 6041-6053 (1996),... 

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