Tensor Polynomials And Radial Basis Functions For Stochastic Optimal Control Problems

The optimal control problem of partial differential equations is widely used in physics,aerodynamics,chemical industry and other fields,such as effective cooling systems of products,optimal shape design of aviation industry products,and optimization of secondary water injection in oilfields.Many natural phenomena that can be described by partial differential equations will be affected by uncertain factors,and we need to take these uncertain factors into account,so there is a stochastic partial differential equation and its optimal control problem.Generally speaking,it is difficult to find analytical solutions to these problems,so efficient numerical solution methods are essential for their successful application.This paper mainly studies the stochastic Galerkin tensor polynomial method,stochastic Galerkin radial basis function method and Monte Carlo method for stochastic partial differential equations and its optimal control problems,and discusses both the uniform and non-uniform distributions of random variables.The main contents are:In the first chapter,we introduce the current research status of stochastic partial differential equations and its optimal control problems and the advantages and disadvantages of classical numerical solutions methods.In the second chapter,we study the efficient numerical solution method of elliptic equation with random field coefficients,and construct the solution format of stochastic Galerkin tensor polynomial method,stochastic Galerkin radial basis function method and Monte Carlo method.A large number of numerical examples are used to verify the effectiveness of the method,and show many advantages of the random Galerkin radial basis function method.Example 1 shows the effect of using tensor polynomials and radial basis functions to solve random elliptic equations when random variables obey uniform distribution and Beta distribution,respectively.The results show that the radial basis function method has more advantages in solving high-dimensional problems.Example 2 uses the random Galerkin radial basis function method and the Monte Carlo method to calculate the expectation and variance of the solution of the random elliptic equation.The results show that the accuracy of the radial basis function method is much higher than that of the Monte Carlo method,which can greatly reduce the amount of calculation.In the third chapter,we study an efficient numerical solution method for optimal control problems of elliptic equations with random field coefficients.Construct a random Galerkin tensor polynomial approximation scheme,prove the prior error estimate;give the random Galerkin radial basis function solution scheme and prior error estimate.Using the gradient projection algorithm,a large number of numerical experiments are performed on the situation where random variables obey different distributions to verify the effectiveness of the method used,and demonstrate the advantages of the random Galerkin radial basis function method in solving this type of problem.