The average a posteriori error of numerical methods

Summary

The definition of the average error of numerical methods (by example of a quadrature formula$\tilde S(f) = \sum\limits_{i = 1}^n {c_i f(a_i )}$ to approximateS(f)=∝ f d μ on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation$\tilde S$ by an averaging process over the set of possible information, which is used by$\tilde S$ (in the example of a quadrature formula,N(F)={(f(a ₁), ...,f/f∈F} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]→ℝ/f(x)−f(y)|≦|x−y|}. For arbitrary (fixed) knotsa _i we determine the optimal coefficientsc _i for the approximation$\tilde S(f) = \sum\limits_{i = 1}^n {c_i f(a_i )}$ and compute the resulting average error. The latter is minimal for the knots$a_i = \frac{{3i - 2}}{{3n - 1}}(i = 1,...,n)$. (It is well known that the maximal error is minimal for the knotsa _i $a_i = \frac{{2i - 1}}{{2n}}(i = 1,...,n)$.) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error.