Error Analysis Of Weighted Integration For Anisotropic Besov Spaces

Multivariate numerical integration is one of the main research topics in computational mathematics. Multivariate integration for various classes of functions of d variables with arbitrary d has attracted much attention due to its applications to problems in many fields. All values of d seems of practical interest. Small d occurs in physics and engineering,and in statistics. Recently, large d (hundreds even thousands) is used in financial mathematics. For path integration, d =∞and an accurate approximation of the path integral requires approximations for d-dimensional integrals, where d can be arbitrarily large. On the other hand, we know that approximation problem of functions (or function recovery) plays a dominant role in the class of multivariate problems., and the results on approximation problems can be often used for other multivariate problems including integration. That is why the thesis studies these two multivariate problems of integration and approximation for various classes of functions of d variables.One of the main goals of multivariate numerical integration and approximation problems is to estimate the nth minimal error of the problem, which is the best possible precision that can be obtained by any algorithm using less than n information functionals for various classes of functions of d variables in worst case, average case, probabilistic and randomized settings. It is closely related to the complexity and n-widths of the approximation problem. For example, consider the worst deterministic case setting. Let F and G be (real) normed spaces over D (?) (R^d). The solution operator S: F→G is assumed to be a continuous mapping. We want to approximate S by some algorithm U =φoN, where the mapping N: F→Rⁿ is called the information operator with the form andφ: N(F)→G. In all settings, two classes of information functionals are considered: the class∧all of linear information consisting of all continuous linear functionals and class∧std of standard information consisting only of function evaluations f(x) for x∈D. That is andThe worst-case error of the algorithm U is defined as The nth minimal error is defined by where∧=∧all or∧std is the class of permissible functionals and the card(U) denotes the number of functional evaluations which is called the cardinality of U.There are a lot of papers which study the nth minimal error of multivariate integration and approximation for various function spaces in different settings. For example, Bakhvalov and Novak studied the nth minimal error of multivariate integration and approximation problem for the Holder space in the worst case and randomized setting and got the asymptotic order of the convergence rate for the nth minimal error. The nth minimal error of multivariate integration and approximation problem for Sobolev space in the worst case and randomized setting are studied in [1,2,3].Recently some people are interested in anisotropic function classes which play very important roles in the theory and application of mathematics and mathematical physics.The definition of an anisotropic function space will be given in Chapter 2. Temlyakov obtained the asymptotic order of convergence rate for the nth minimal error bounds of quaduature formula on the periodic anisotropic Sobolev class and Nikolskii class of function in the worst case setting in [4].This paper have three chapters: Chapter 1 Preliminary Knowledge. It is mainly described the develop course and the current research directions of Information-based Complexity and narrate the general theory of the Information-based Complexity and also give the problem of estimates for integration of nth minimal error in different settings, the given information classes and in the certain function spaces.Chapter 2 It is described the mainly research results of Traub,Wozniakowski,Wasilkowski, Heinrich, Novak, Fang Gensun Huang Fanglun and also give the complexity estimate of the integration problem in some function space, this chapter mainly discuss the anisotropic Besov spaces and get the results:Chapter 3 In this chapter, It is mainly described the definition and the denote of norm. and also generally discuss the question of error estimate. We mainly use the method of integration error analysis and give the results in three different case settings.In the Holder function space, there are In the Sobolev function space, there are e_n^ran(I,∧)(?)n^-r/d-1/2 2≤p＜∞．e_n^ran(I,∧)(?)n⁽(-r/d-(1-1/p)) 1≤p＜2.