On the randomized complexity of Banach space valued integration
The authors study the complexity of Banach space-valued integration in the randomized setting. Let \(Q=[0,\,1]^d\) and let \(C^r(Q,X)\) be the Banach space of all \(r\)-times continuously differentiable functions \(f:\, Q\to X\) where \(X\) is a Banach space. Let \(Sf = \int_Q f(t)\,dt\). Let \(e_n(S,B)\) denote the \(n\)th minimal error of the integration operator \(S\) on the closed unit ball \(B\subset C^r(Q,X)\) in the randomized setting, that is the minimal possible error among all randomized algorithms approximating \(S\) on \(B\) that use at most \(n\) values of the input function \(f\). The authors investigate the relation of the optimal convergence rate to the geometry of \(X\). It is shown that the minimal errors \(e_n(S,B)\) are bounded by \(c\,n^{-r/d -1+1/p}\) for all \(n\in {\mathbb N}\) with some constant \(c>0\) if and only if \(X\) is a Banach space of equal norm type \(p\in [1,\,2]\).
scientific article; zbMATH DE number 6363767