Differentiability error propagation

In order to compute error propagation, we must evaluate the partial derivatives with respect to both the dependent and independent variables. This will be more clearly seen by differentiating equation (4.3.24)... 

In a forward-marching method such as Euler s method, we are more interested in the total error propagation over multiple usage of the algorithm than in the local one-step error. If we let Cj be the error between the approximate solution, rcj, and exact solution, x ti) to the differential equation, then... 

This focus in this chapter is on analytical methods for measuring action spectra, with emphasis on those derived from fluence-response data. For a set of examples, the reader can consult studies of Phycomyces, not only studies of its phototropism but also of the light-growth response, carotene synthesis, and differentiation," as well as studies of other blue-light systems.The later studies on Phycomyces employed formal data analysis methods — including error analysis (and error propagation )... 

The general principle behind most commonly used back-propagation learning methods is the delta rule, by which an objective function involving squares of the output errors from the network is minimized. The delta rule requires that the sigmoidal function used at each neuron be continuously differentiable. This methods identifies an error associated with each neuron for each iteration involving a cause-effect pattern. Therefore, the error for each neuron in the output layer can be represented as ... 

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

Insisting that the propagation of error formula be used wherever appropriate in laboratory reports, we find it easier to introduce the total differential once the topic comes up during the thermodynamics portion of the course. The similarity between the propagation of error formula and the total differential provides a more intuitive model for our students. Because our order of topics delays thermodynamics to later in the semester, we have time to emphasize the more concrete example used to determine the uncertainty in a measurement. 

The use of differentials in assessing how errors in one or more properties of a system propagate through to errors in a property that is related to those properties. 

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

According to the initial optimization concept, this technique launches a set of parametric expressions for the differential operators, whose evaluation is based on proper error estimators [59]. What is actually desirable — yet analytically not feasible — for these estimators is that their dependence on frequency and propagation angle should resemble that of the true dispersion error. Hence, an expansion of these expressions in terms of basis angular functions is performed, which enables the accuracy improvement regardless the direction of propagation. Depending... 

Our problem is to take the uncertainties in x, X2, and so on, and calculate the uncertainty in y, the dependent variable. This is called the propagation of errors. If the errors are not too large, we can take an approach based on a differential calculus. The fundamental equation of differential calculus is Eq. (7.9),... 

This equation is analogous to Eq. (11.21). It will be our working equation for the propagation of errors through formulas. Since it is based on a differential formula, it becomes more nearly exact if the errors are small. 

If a formula is used to calculate values of some variable from measured values of other variables, it is necessary to propagate the errors in the measured quantities through the calculation. We provided a scheme to calculate the expected error in the dependent variable, based on the total differential of the dependent variable. 

Performing the appropriate differentiation of equation 2 and substitution into equation 23 gives the equation for the propagation of error from AH and AS , and (Tas". to give the error in AG a7, tTAc°37 (30) ... 

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