Algorithms and complexity for functions on general domains

Abstract

Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the approximation or integration of functions defined on D d ⊂ R d and only assume that D d is a bounded Lipschitz domain. Some results are even more general. We study three different concepts to measure the complexity: order of convergence, asymptotic constant, and explicit uniform bounds, i.e., bounds that hold for all n (number of pieces of information) and all (normalized) domains.It is known for many problems that the order of convergence of optimal algorithms does not depend on the domain D d ⊂ R d . We present examples for which the following statements are true:1. Also the asymptotic constant does not depend on the shape of D d or the imposed boundary values, it only depends on the volume of the domain.2. There are explicit and uniform lower (or upper, respectively) bounds for the error that are only slightly smaller (or larger, respectively) than the asymptotic error bound. *Much is known about the order of convergence of the numbers e n , in particular for simple domains D d , such as the cube or the torus. General bounded Lipschitz domains are studied in [7,10,11,12,26,27,29,38,39,40,42,43]. In many cases, the optimal order of the e n is of the form e n ≍ n −α (log n) β , where α and β do not depend on the domain D d . It is interesting to know also the exact asymptotic constant C := lim n→∞ e n n α (log n) −β , if it exists. The value of C is known only in rare cases (unless d = 1, we do not discuss the univariate case in detail), and usually only for very special domains,

How anisotropic mixed smoothness affects the decay of singular numbers for Sobolev embeddings

“…However, only [40] is really close to our setting. Closer to us are the papers by Cobos, Kühn, Sickel [7] (L ∞ approximation), by Krieg [26] (dominating mixed smoothness, periodic and nonperiodic), by Wang et al [3,4,22] (anisotropic Sobolev spaces, Sobolev spaces on the sphere) as well as the papers by Mieth [34] and Novak [37] (approximation on general domains).…”

Recovery of Sobolev functions restricted to iid sampling

“…We conjecture that all these asymptotic constants exist and possibly they do not depend on the shape of Ω, only on its volume. The asymptotic constants are known only in very rare cases, see [26], though. Here we present an example from [12] that nicely shows the quality of iid samples.…”