Solvable Integration Problems and Optimal Sample Size Selection

Optimal confidence for Monte Carlo integration of smooth functions

“…Our lower bounds do hold for this type of algorithms, but the upper bounds we present are based on non-adaptive methods. For simplicity, in this paper we restrict to methods with fixed cardinality n. In general, the number of function values an algorithm collects might be random and even depend on the input, see for instance [8,12,16]. Let us mention here that our auxiliary lemmas on lower bounds, Lemma 2.1 and 2.2, would then still hold with slightly worse constants.…”

“…First, we consider Hölder classes C β ([0, 1] d ) with smoothness β ∈ (0, 1], see (16). Compare also [4] for the result in terms of the root mean squared error.…”

“…where • W is the (semi-)norm of W. A method with this property of guaranteeing a small (absolute) error ε > 0 with confidence 1 − δ ∈ (0, 1) (or uncertainty δ) for inputs from the unit ball B W is called (ε, δ)-approximating in W, see also [16]. We also consider the n-th minimal probabilistic Monte Carlo error at uncertainty δ, defined by…”

“…For this reason the probabilistic criterion is frequently used in statistics, see for example [8,11,12,13]. Furthermore, there are numerical problems which can be solved with respect to the probabilistic (ε, δ)-criterion but the mean error is unbounded, see [16]. In other words, this criterion seems to be the right one for the concept of solvability.…”