Approximation errors

After application of the 6 time-stepping method (see Chapter 2, Section 2.5) and following the procedure outlined in Chapter 2, Section 2.4, a functional representing the sum of the squares of the approximation error generated by the finite element discretization of Equation (4.118) is formulated as... 

The expansion of the approximation error = Lh v — Lv in powers of h is aimed at achieving the order of approximation as high as possible. Indeed, we might have... 

For h- = h+ = h the preceding is identical with expre.ssion (7) (see Example 2). Plain calculations of the local approximation error at a point x show that... 

In Section 1.2 we have found the local approximation error taking now the form... 

In the estimation of the approximation error the well-founded choice of the norm depends on the structure of an operator and needs investigation in every particular case. A precise relationship between an operator and a norm in the process of searching the error of approximation will be established in the general case in Section 4. Its concretization for the example of interest leads naturally to the appearance of the negative norm ... 

Let us define the notion of the approximation error on an element u G B To this end, we must write down the equation for the difference Zh — Vh — W/j- Substitution of + Ufj into (21) gives... 

These assertions follow from the representation of the approximation error in the form (6) (8) and a priori estimate (12). On the basis of the estimates for T/j and obtained in Section 3.2 we find that... 

The proof of convergence of scheme (19) reduces to the estimation of a solution of problem (21) in terms of the approximation error. In the sequel we obtain such estimates using the maximum principle for domains of arbitrary shape and dimension. In an attempt to fill that gap, a non-equidistant grid... 

Furthermore, in giving the approximation error at the irregular nodes as a sum... 

Upon substituting the estimates for the approximation error obtained in Section 3 into (34) we find that... 

The statement of the difference problem and calculations of the approximation error. In this section we study the equation of vibrations of a string... 

Here the approximation error for the boundary conditions is a quantity 0(h2 + r2) if... 

An a priori characteristic of a scheme is the error of approximation. The approximation error on a function u t) for scheme (4) is known as the residual... 

Of course, the words arbitrary domain cannot be understood in a literal sense. Before giving further motivations, it is preassumed that the boundary F is smooth enough to ensure the existence of a smooth solution u = u x,t) of the original problem (l)-(2). In the accurate account of the approximation error and accuracy we always take for granted that the solution of the original problem associated w ith the governing differential equation exists and possesses all necessary derivatives which do arise in the further development. 

With the basic tools in hand, we proceed to carry out the accurate account of the approximation error j+a/2)... 

When economical schemes for multidimensional problems in mathematical physics are developed in Chapter 9, we shall need a revised concept of approximation error, thereby changing the definition of scheme. The notion of summed (in t) approximation in Section 3 of Chapter 9 is of a constructive nature, making it possible to produce economical schemes for various problems. 

No other a priori assumptions about the form or the structure of the function will be made. For a given choice of g. Kg) in Eq. (1) provides a measure of the real approximation error with respect to the data in the entire input space X. Its minimization will produce the function g (x) that is closest to G to the real function, /(x) with respect to the, weighted by the probability P(x,y) metric p.. The usual choice for p is the Euclidean distance. Then 1(g) becomes the L -metric ... 

The approximation error that stems from the finiteness of the function space, G... 

The selection to minimize absolute error [Eq. (6)] calls for optimization algorithms different from those of the standard least-squares problem. Both problems have simple and extensively documented solutions. A slight advantage of the LP solution is that it does not need to be solved for the points for which the approximation error is less than the selected error threshold. In contrast, the least squares problem has to be solved with every newly acquired piece of data. The LP problem can effectively be solved with the dual simplex algorithm, which allows the solution to proceed recursively with the gradual introduction of constraints corresponding to the new data points. 

The accuracy of the error equations (Eqs. (22) and (23)] also depends on the selected wavelet. A short and compactly supported wavelet such as the Haar wavelet provides the most accurate satisfaction of the error estimate. For longer wavelets, numerical inaccuracies are introduced in the error equations due to end effects. For wavelets that are not compactly supported, such as the Battle-Lemarie family of wavelets, the truncation of the filters contributes to the error of approximation in the reconstructed signal, resulting in a lower compression ratio for the same approximation error. 

---