Probabilistic complexity analysis for linear problems in bounded domains

Many numerical problems and methods for their approximate solution can be brought into the following abstract form: We are given a bounded linear operator S E L(X, Y) between Banach spaces X and Y, the “solution operator.” That is, for x E X, Sx is the true solution of the problem. We assume that we have only partial information on the “datum” x E X, which is given by a mapping N: X + R”, the “information operator.” Finally, we have a mapping cp: N(X) + Y that represents the action of the “algorithm” by which we obtain an approximation cp(N(x)) to Sx using the information N(x). N and cp are required to belong to certain classes of not necessarily linear, not necessarily continuous mappings. This approach is developed in the monograph “Information-Based Complexity,” by Traub, Wasilkowski, and Woiniakowski, (1988), later referred to as IBC (see also Traub and Woiniakowski, 1980; Traub, Wasilkowski, and Woiniakowski, 1983). The aim of the theory is to study concrete N and cp as well as optimality over 50 or both N and p. The quality of N and (c is judged by the behavior of the error IlSx cp(N(x))lJ. In the worst case setting, one takes the supremum of the error over a