Optimal algorithms for numerical integration: a survey

We discuss optimality of approximation algorithms and lower bounds on their complexity using Carl’s inequality, which states that for any two Banach spaces X and Y and any α > 0 there exists a constant γα > 0 such that sup 1≤k≤n kek(T ) ≤ γα sup 1≤k≤n kck(T ) holds for all linear and bounded operators T : X → Y . Here ek(T ) is the k-th entropy number of T and ck(T ) is the k-th Gelfand number of T . This inequality has been used to obtain lower bounds on Gelfand numbers, which in turn are a useful tool in the study of optimality of algorithms.