The power of adaptive algorithms for functions with singularities

Abstract

This is an overview of recent results on complexity and optimality of adaptive algorithms for integrating and approximating scalar piecewise r-smooth functions with unknown singular points. We provide adaptive algorithms that use at most n function samples and have the worst case errors proportional to n −r for functions with at most one unknown singularity. This is a tremendous improvement over nonadaptive algorithms whose worst case errors are at best proportional to n −1 for integration and n −1/p for the L p approximation problem. For functions with multiple singular points the adaptive algorithms cease to dominate the nonadaptive ones in the worst case setting. Fortunately, they regain their superiority in the asymptotic setting. Indeed, they yield convergence of order n −r for piecewise r-smooth functions with an arbitrary (unknown but finite) number of singularities. None of these results hold for the L ∞ approximation. However, they hold for the Skorohod metric, which we argue to be more appropriate than L ∞ for dealing with discontinuous functions. Numerical test results and possible extensions are also discussed. (2000). 65Y20, 65D05, 65D30, 41A10, 41A25, 41A55. Mathematics Subject Classification

Some Results on the Complexity of Numerical Integration

“…c) More on the optimality of linear algorithms and on the power of adaption can be found in [15,79,89,113,114]. There are important classes of functions that are not balanced and convex, and where Theorem 0 can not be applied, see also [13,95].…”

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“…Other approaches were suggested by Markakis and Barack [11], where the authors revise the Lagrange interpolation formula to approximate univariate discontinuous functions, and by Plaskota et al ( [14], [15]), where the authors suggest using adaptive methods for this approximation. Batenkov et al ([4], [5], [6]) address a similar problem, the reconstruction of a piecewise smooth function from its integral measurements.…”

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