Research On Optimization Algorithm Of Constraint Of Differential Equation

The numerical solution of optimization problem of differential equation constraint is a very important research field in science and engineering computation,which involves many applications in mathematics and engineering,including modern industry,medicine,optimal control,inverse problem of differential equation,ect.Because the solution of the constrained optimization problem of differential equation needs to use a variety of methods,including discrete method,optimality theory,optimization algorithm,and so on,and needs theoretical analysis and numerical solution of the problem,it is difficult,so it is very important to research and develop a mathematical method to deal with these complex problems quickly,effectively and stably.In this thesis,the research background is heat conduction equation and elliptic equation constrainted optimization problem,and the focus is on optimization algorithm and intelligent algorithm.From the general sense,the variational adjoint method and differential evolution algorithm are given to solve the constrained optimization problem of differential equation.The main research contents are as follows:(1)Aiming at the constrained optimization problem of heat conduction equation,the process of solving the classical positive problem of heat conduction equation is first given by the separation of variables method.On this basis,the adjoint equation and the gradient expression of this kind of problem are derived according to the adjoint operator and Gateaux differential.Based on the Broyden family algorithm of BFGS correction and DFP correction and their weighted linear combination,the initial condition constrained optimization problem and the point source strong constrained optimization problem of one and two dimensional heat conduction equation constrained optimization problems are given,and numerical simulations are given.(2)For the constrained optimization problem of elliptic equation,first on the basis of finite element method to solve the forward problem,based on the adjoint equation and the gradient expression of the constrained optimization problem of elliptic equation,the algorithm of elliptic equation coefficient,permeability and source term constraint optimization problem are given by using the fastest descent method.Finally,numerical simulation are carried out.(3)This thesis presents a differential evolution algorithm for constrained optimization problems of differential equations.The algorithm is a population-based optimization method,which has the advantages of fast computation speed and few control variables.It is applied to the point source strong,point source position,parameters,boundary condition,initial condition constraint optimization problem of heat conduction equation and the point source strong,point source position constraint optimization problem of elliptic equation,and the results are satisfactory.(4)For the algorithm proposed in this thesis,a numerical calculation program is written,which includes the calculation of the positive problems of differential equations and constrained optimization problems.The positive problems are solved by numerical integration method,finite difference method,finite element method,Green function method,etc.The boundary conditions,initial conditions,point source strong,parameters and source terms of onedimensional and two-dimensional differential equations are solved for constraint optimization problems.