Random Approximations Of π And Asymptotic Analysis Of A Class Of PDE Models

This dissertation consists of two parts.In the first part,we mainly study random approximations of π,which includes Chapters 2,3 and 4.In the second part,we mainly study the well-posedness and large time behavior of solutions of a class of incompressible PDE models,which contains Chapter 5 and Chapter 6.Although these two parts have certain independence,it is worth mentioning that the basic knowledge and research methods involved in them are common tools of analysis in applied mathematics research.In particular,when integrating the two important digital characteristics of expectation and variance to establish the relevant convergence results in the first part,it is indispensable to use the classic tools of measure theory such as Markov inequality,Chebyshev inequality,BorelCantelli lemma and Cramer theorem.This can be said to be in the same line with the common tools in the research of partial differential equations such as integral estimation（energy method）and Sobolev embedding relations to study the wellposedness and asymptotic behavior of the solutions.The main structure of this dissertation are as follows.Chapter 1 introduces the background and the development history of the random approximations of π,the incompressible MHD-Boussinesq equations and the magneto-micropolar equations.Some prerequisite knowledge and a brief summary of the main results of this dissertation are also presented in this chapter.In Chapters 2-4,we study the random approximations and extrapolation estimates of π,and establish the related probability convergence estimates by means of modern probability theory and computational mathematics.Since 1960s,the stochastic properties of the convex hull Kn generated by n independent points chosen at random in a set K（?）Rd have been studied extensively.In the case of a large number of independent points uniformly distributed on a unit circle in R2,due to randomness,it is intuitively clear that the corresponding random polygons should approximate the unit circle very well,and hence the semiperimeters and areas of these random polygons may be used as very natural approximations ofπ.In fact,their semiperimeters and areas converge to π with probability 1 as the number of vertices n→∞.In the special case when the vertices happen to be equally spaced on the circle,Archimedes,the famous Greek mathematician in the third century B.C.,first used the semiperimeter and area of the corresponding regular polygons to approximate π,providing a general method to estimate the value of π to any desired accuracy.Later,Chinese mathematicians Liu Hui and Zu Chongzhi also made outstanding contributions.In Chapter 2,we study random approximations of π based on these random polygons and develop various extrapolation processes to improve the convergence rates.Extrapolation methods are widely used in modern numerical analysis and scientific computing.With the help of some rough,lower-order estimates,it can greatly improve the accuracy and efficiency of approximations.In fact,the earliest applications of extrapolation techniques go back to the seventeenth century when Snellius discovered and later Huygens rigorously proved（using elaborate geometric arguments）that suitably constructed simple linear combinations of the semiperimeter and area of the corresponding regular polygons provide more refined estimates for π than those used in the classical Archimedean approach.In this chapter,by applying extrapolation techniques,especially by simultaneously considering such random n-gons and suitably constructed random 2n-gons,we can achieve the extrapolation estimates with the optimal convergence rates through well-designed linear combinations of the semiperimeters and areas of these random polygons.These extrapolation improvements are also shown to be asymptotically normal as n→∞ and satisfy the corresponding central limit theorems.Generally speaking,the central limit theorem is available for independent and identically distributed random variables.Here,because the lengths of the arc generated by random divisions of the circle are not independent,how to strictly prove the relevant central limit theorems through random divisions of the circle is emphases and difficulties of research.In Chapter 3,we aim to extend the results of Chapter 2 for the cases when the vertices on the unit circle obey uniform distributions to more general cases,especially focusing on the case of random cyclic polygons generated from multivariate symmetric Dirichlet distributions.We show that similar convergence results hold for the semiperimeters or areas of these random polygons as the number of vertices n tends to infinity.Additionally,we also present some extrapolation estimates with faster rates of convergence.Note that for the case of general asymmetric Dirichlet distributions and other multivariate distributions for the vertices,these problems are challenging,and very few results are available at this time.In Chapter 4,we further develop nonlinear extrapolation methods for approximating π.First,we construct certain nonlinear functions of the semiperimeter and area of random polygons generated by n independent points uniformly distributed on the unit circle to study their asymptotic convergence properties.Then,by deriving probabilistic asymptotic expansions for these nonlinear combinations,we carefully control the approximation errors to establish the nonlinear extrapolation estimates which converge to π almost surely as n→∞.By using Central limit theorem,Slutsky theorem,Cramer theorem and other probabilistic tools,we show that these extrapolation estimates are also asymptotically normal.In addition,we extend these nonlinear approximations results to the case of random cyclic polygons generated by symmetric Dirichlet distributions.In Chapter 5 and Chapter 6,we study the global well-posedness and large time behavior of solutions to a class of PDE models.In Chapter 5,we study the Cauchy problem for the 2D incompressible MHD-Boussinesq equations without thermal diffusion.We use the energy method to establish the global existence and uniqueness of the strong solutions for suitably regular initial data.To obtain large time decay rate of the solutions,we insert an artificial thermal damping term in the temperature equation.By applying the Fourier-splitting methods,we derive optimal large time decay rates of the solutions and their first-order derivatives.In Chapter 6,we study the initial-boundary value problem to the 2D magnetomicropolar system with zero angular viscosity in a smooth bounded domain.Specifically,we consider the Dirichlet boundary condition.We use the classic energy method to prove that there exists a unique global strong solution of the system by imposing suitable regularity assumptions on the initial data,without any compatibility condition.