The Research On Tractability Of Integration Of Infinitely Differential Multivariate Functions

Multivariate problems are defined on functions space of d.There are many practical problems in which d can be very large.When d is very large,this prob-lem can hardly be solved by traditional analytic methods,and we only solve it by approximation method within allowable error ε.Tractability of multivariate prob-lems is to study how the complexity of approximating the problem depend on εand dimension d.It is one of research methods of Information-Based Complexity theory(IBC),proposed by professor Wozniakowshi from Columbia University in the mid-nineties of last century.In recent years,it has been studied by more and more scholars.Approximation problem of multivariate function integration is one of the most classical research directions in the study of tractability of multivariate problem-s.Generally,the approximation problem of infinitely differentiable multivariate function integration is mostly studied with L∞ norm.For example,the Polish mathematician Wojtaszcyzk proved that the integration problem of infinitely differ-entiable multivariate functions is not strongly tractable with L∞ norm.In recent years,some scholars have proposed the concepts of quasi-polynomial tractability and weak tractability of multivariate problems.Though some multivariate problems are not tractable or not strongly tractable,but they are quasi-polynomial tractable or weak tractable possibly.In this thesis,we mainly study the quasi-polynomial and weak tractability of approximation problem of integration for infinitely differentiable multivariate functions.We use information based complexity theory to redefine two norms for infinitely differentiable multivariate function space.Using the standard information class(function value as information),we prove that the approxima-tion problem of integration for infinitely differentiable multivariate functions is both quasi-polynomial tractable and weak tractable.In addition,we also study the os-cillation integration problem of periodic functions on Korobov space.The linear algorithm and the good lattice method are used to approximate the problem,and the error analysis is also given.The thesis consists of four chapters.In the first chapter,firstly we introduce the development of Information Based Complexity and tractability of multivariate problems,as well as their research back-ground.Secondly,we give the relevant definition and theory of functional analysis,as well as the theoretical basis of tractability of multivariate problems.In the second chapter,we renorm the space of infinitely differentiable functions,Based on the knowledge of multivariate Taylor expansions,we prove that multivari-ate integration problem is quasi-polynomial tractable in the function space Fd1,and multivariate integration problem is weak tractable in the function space Fd2.In the third chapter,we mainly study oscillatory integrals in the multivariate Korobov class Eα,d and prove that the problem is intractability(suffers from the curse of dimensionality)using standard information.In addition,we also use good lattice method to approximate this problem and obtain the upper bound of the error of the approximation algorithm.The last chapter mainly summarizes the conclusions of previous chapter,and puts forward the research contents that we will focus on in the future.