Randomized Complexity of Parametric Integration and the Role of Adaption II. Sobolev Spaces
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We study the complexity of randomized computation of integrals depending on a parameter, with integrands from Sobolev spaces. That is, for r,d_1,d_2∈ℕ, 1≤ p,q≤∞, D_1= [0,1]^d_1, and D_2= [0,1]^d_2 we are given f∈ W_p^r(D_1× D_2) and we seek to approximate Sf=∫_D_2f(s,t)dt (s∈ D_1), with error measured in the L_q(D_1)-norm. Our results extend previous work of Heinrich and Sindambiwe (J. Complexity, 15 (1999), 317–341) for p=q=∞ and Wiegand (Shaker Verlag, 2006) for 1≤ p=q<∞. Wiegand's analysis was carried out under the assumption that W_p^r(D_1× D_2) is continuously embedded in C(D_1× D_2) (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. The paper is based on Part I, where vector valued mean computation – the finite-dimensional counterpart of parametric integration – was studied. 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.
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