Local and Global Errors

A note is in order here on errors in the numerical solution of an ode. There are (regarding errors in a certain light) two kinds of errors. One is the local error, being the error added by a single step. The solution is always carried forward to a final point in t, using a number N of steps, and at that point we have a final, or global error. Unfortunately, this is always of a lower order than the local error. 

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The explicit Euler integration method is simply the linear extrapolation from the point at f to r +, using the slope of the curve at t . Figure 7.7 shows this Euler method with the local and global errors. 

Whereas the local error is due to a single step, the global error, Cn+i, is the sum of the local error, +i, and the amplified error from previous steps. As a consequence, it is not equal to the sum of the local errors from previous steps. The local and global errors are illustrated in Figure 6.7. Note that the upper curve, y(x), is the correct (true) solution to the IVP and that the lower curve, z(x), is the correct solution of the IVP on < x <... 

In practice, a blend of local and global SA approaches may be employed. Quantitative uncertainty in the form of standard errors may be available only for some model inputs. Multiple executions of the simulation to characterize the global SA of the quantified parameters may be undertaken conditional on a series of fixed-... 

Figure 7.7 Graphical representation of Euler method, and local and global truncation error.

In the spectral SFEM, the random parameter fields are discretized by a KL or a polynomial chaos expansion, the solution is expanded with Hermite polynomials, and a Galerkin approach is applied to solve for the unknown expansion coefficients. The theoretical foundation has been laid in Deb et al. (2001) and Babuska et al. (2005), where local and global polynomial chaos expansions for linear elliptic boundary value problems with stochastic coefficients were investigated and where a priori error estimates have been proved for a fixed number of terms of the KL expansion. 

In practice, the full matrix analysis is rarely applicable because of spectral overlap and because of the global error propagation. In full matrix analysis all the elements are interconnected and the error in one volume element propagates into all cross-relaxation rates. This property is not favorable in practical situations in which a part of the spectrum may be ill-defined although a good portion of the spectrum is of a satisfactory quality. Then, the more favorable analysis is localized, i.e., errors are confined within respective cross-relaxation rates. However, such analysis is possible only on data in which spin diffusion is not dominant. 

The true (global) error of a computed solution is its deviation from the true solution of the initial value problem. Practically all present-day integrators, including this one, control the local error for each step and do not attempt to control the global error directly. 

Some can be excluded in order to limit the search space (e.g. local versus global) and the magnitude of GPT errors generated in evaluating the LPs, as explained in Section 2.2. 

FIGURE 4.2 (a) Global error versus time-step and (b) local error versus position along a line of 100 cells generated by diverse higher order (HO) PML implementations... 

The end points of any linear segment, x(ti) and x(ti+i), are either interpolated from the data or taken as actual data points as in the Boxcar method. These piecewise linear approximation techniques perform well for steady-state process data with little noise, but are inadequate for process data with important low amplitude transients and are inefficient for data with relevant high frequency features. Also, the line segments used in the approximation satisfy a local, not a global error criterion. 

To verify the accuracy in the life prediction, the experimental fatigue lives are compared with those predicted by the criterion, for a given set of data. Figures 7.17 and 7.18 show the results for global and local multiaxial data, respectively. Upper and lower bounds representative of 200% and 400% error in life assessment are plotted together with the data for an easier interpretation of the results. 

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