Complexity of Sobolev imbeddings and integration in the deterministic and randomized settings

Abstract

This paper reformulates the problems of approximation of imbedding and integration from anisotropic Sobolev classes B ( W r p ([0, 1] d )) in the deterministic, randomized and average case settings, and obtains the exact orders of the n-th minimal error of these problems in all three settings. The results show that in the case of B ( W r p ([0, 1] d )) being not imbedded into the space of continuous functions C ([0, 1] d ), the randomized and average case error are essential smaller than the deterministic ones. Quantitatively, this maximal gain for imbedding problem amounts to the factor n -1+e for any e > 0, and for integration problem the gain is up to factor n -1 which is also the maximal speedup over the deterministic algorithms observed so far in natural numerical problems.