The power of adaption for approximating functions with singularities

Abstract

Abstract. Consider approximating functions based on a finite number of their samples. We show that adaptive algorithms are much more powerful than nonadaptive ones when dealing with piecewise smooth functions. More specifically, let F 1 r be the class of scalar functions f : [0, T ] → R whose derivatives of order up to r are continuous at any point except for one unknown singular point. We provide an adaptive algorithm A ad n that uses at most n samples of f and whose worst case L p error (1 ≤ p < ∞) with respect to 'reasonable' function classes F 1 r ⊂ F 1 r is proportional to n −r . On the other hand, the worst case error of any nonadaptive algorithm that uses n samples is at best proportional to n −1/p .The restriction to only one singularity is necessary for superiority of adaption in the worst case setting. Fortunately, adaption regains its power in the asymptotic setting even for a very general class F ∞ r consisting of piecewise C r -smooth functions, each having a finite number of singular points. For any f ∈ F ∞ r our adaptive algorithm approximates f with error converging to zero at least as fast as n −r . We also prove that the rate of convergence for nonadaptive methods cannot be better than n −1/p , i.e., is much slower.The results mentioned above do not hold if the errors are measured in the L ∞ norm, since no algorithm produces small L ∞ errors for functions with unknown discontinuities. However, we strongly believe that the L ∞ norm is inappropriate when dealing with singular functions and that the Skorohod metric should be used instead. We show that our adaptive algorithm retains its positive properties when the approximation error is measured in the Skorohod metric. That is, the worst case error with respect to F 1 r equals Θ(n −r ), and the convergence in the asymptotic setting for F ∞ r is n −r . Numerical results confirm the theoretical properties of our algorithms.

The power of adaptive algorithms for functions with singularities

“…In this section (based on [31]), we present results pertaining to the function approximation problem. We begin with the definition of the classes of piecewise r-smooth functions.…”

“…It turns out that in the setting of this paper the use of adaptive algorithms is essential since nonadaptive algorithms generally do a very poor job. For instance, for piecewise r-smooth functions with one singular point, the errors of optimal adaptive algorithms developed in [29,31] are proportional to n −r , i.e., they are as efficient as optimal algorithms for globally r-smooth functions. This holds for both integration and L p approximation problems with 1 ≤ p < ∞.…”

“…These algorithms are simpler than those from [31] and have some additional desirable properties, using the fact that the functions are continuous. (See also the recent paper [19] where yet another approach to approximating piecewise smooth, but continuous functions is presented.)…”

“…(See also the recent paper [19] where yet another approach to approximating piecewise smooth, but continuous functions is presented.) Finally, in Section 5, we briefly discuss extensions of the results from [29,30,31] to weighted integration/approximation over R, and to noisy information.…”

“…In this survey, we give only two sample test results in Section 3.2 that were obtained for the problem of integration; see [29,31,30] for more.…”