The Numerical Solution Of Stochastic Elliptic Partial Differential Equation And Optimal Control Problem

We are usually faced with a nondeterministic world in real life,such as diffusion process,fluid motion,elastic deformation,etc.These are subject to uncertainty.Stochastic partial differential equation and its optimal control problem is to take these uncertain factors into account in the mathematical model.It is generally difficult to obtain the exact solution of these problems,so it is of great research significance and application value to find their numerical solutions.At present,Monte Carlo method,stochastic Galerkin method and stochastic collocation method are commonly used to solve stochastic partial differential equations and corresponding optimal control problems.The Monte Carlo method is used to solve stochastic partial differential equations by random sampling.Every sampling in random space needs to solve the corresponding deterministic partial differential equations.This method does not depend on the dimension of the random parameter space.Furthermore,it is simple,clear and applicable,but it has the disadvantage of slow convergence.The stochastic Galerkin method solves both the probability space and the physical spatial direction that the finite dimensional space approximates the descendant into the weak form.The result is formed a huge discrete system.Its construction format and error estimation mode are relatively fixed,and its convergence speed is fast,but the shortcoming is that it will produce a crisis of dimensions with the dimension of random space increases.The stochastic collocation method solves the deterministic partial differential equation for each interpolation point in the random space,and then uses the obtained results to perform polynomial interpolation on the probability space.The method fully combines the advantages of Monte Carlo method and stochastic Galerkin method.There is no need to solve huge coupling system,and the convergence rate can be kept quickly.In this paper,the radial basis function stochastic Galerkin method is used to solve the stochastic partial differential equations and their optimal control problems.Polynomial is used to discrete of the physical spatial direction and radial basis function is used to discrete of the probability space.In Chapter 2,the prior error estimates for the radial basis function of stochastic elliptic partial differential equations with random field coefficients and source terms is given.The reliability of the theory was verified by a large number of numerical experiments,and the high efficiency and difference of Mat?ern and GIMQ radial basis function in solving stochastic elliptic partial differential equations are demonstrated.For the same number of sample points,the expected accuracy of RBF is much higher than that of Monte Carlo method.In Chapter 3,the radial basis function approximation format with the optimal control problem of elliptic partial differential equation with random field coefficients is given.A large number of numerical experiments show that the approximation effect of GIMQ radial basis function is better than that of Mat?ern radial basis function when the parameters are appropriate,and the influence of different parameters on the approximation effect of GIMQ radial basis function is also demonstrated.Radial basis function random Galerkin method can not only take the advantage of fast convergence speed of Galerkin method,but also play the advantage of radial basis function without grid division.Its discrete format does not depend on the growth of random dimensions,which could avoid the shortcomings of classical methods such as polynomials and finite elements.To a large extent,it can reduce the influence of random space dimension.It is simple in structure and easy to implement numerically.