Randomized complexity of parametric integration and the role of adaption II. Sobolev spaces

Abstract

We study the complexity of randomized computation of integrals depending on a parameter, with integrands from Sobolev spaces. That is, for r,d1,d2∈N, 1≤p,q≤∞, D1=[0,1]d1, and D2=[0,1]d2 we are given f∈Wpr(D1×D2) and we seek to approximate(Sf)(s)=∫D2f(s,t)dt(s∈D1), with error measured in the Lq(D1)-norm. Information is standard, that is, function values of f. Our results extend previous work of Heinrich and Sindambiwe (1999) [10] for p=q=∞ and Wiegand (2006) [15] for 1≤p=q<∞. Wiegand's analysis was carried out under the assumption that Wpr(D1×D2) is continuously embedded in C(D1×D2) (embedding condition). We also study the case that the embedding condition does not hold. For this purpose a new ingredient is developed – a stochastic discretization technique. In Part I a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting was solved. Here a further aspect of this problem is settled.