Deterministic and stochastic error bounds in numerical analysis
In this book different deterministic and stochastic error bounds of numerical analysis are investigated. The author analyses approximation (App), integration or quadrature (Int) and optimization problems (with two different aspects: only sup f is requested (the \(Opt^*\) problem) or it seeks an \(x\in X\) such that sup f-f(x) is small (the Opt problem)). Different problems connected with the approximate solution of differential and integral equations can be formulated as particular cases. In the first chapter maximal errors of deterministic methods are discussed. These error bounds were compared with different stochastic error bounds for Monte Carlo methods (chapter 2). The maximal error is a very pessimistic estimate for the true error of a given function. Hence it seems to be reasonable to consider additionally the average error (chapter 3). All these error bounds are also deduced for some special problems: the linear case or for different classes of functions (Lipschitz, Sobolev or Hölder classes). In the appendix the author deals with different aspects concerning the existence and uniqueness of optimal algorithms. The book contains a lot of new results obtained by the author. Besides all this a unified and comparative study of deterministic and stochastic error bounds is presented.
scientific article; zbMATH DE number 193590