Fast Preconditioned Iterative Methods For Fractional Optimal Control Problems

The constrained fractional optimal control problem is an optimization problem constrained by the fractional differential equations(FDEs).Fractional differential equations were shown to provide an adequate and accurate description of transport processes that exhibit anomalous diffusive behavior,which cannot be modeled properly by traditional integer-order diffusion equations.The nonlocality of the fractional differential operators usually leads to a dense coefficient matrix,which brings much difficulties in numerical computations.Therefore,it is very important to study the fast method for the constrained optimization problem of fractional differential equations.The purpose of this thesis is to study the fast preconditioned iterative methods for solving constrained optimization problems of spatial fractional differential equations.With the aid of variational methods,we can transform the original problem into a coupled linear system consisting of state equation,costate equation and optimality condition.Then,the gradient projection algorithm can be utilized to solve it.As a result,we need to solve a series of discretized state equations and costate equations.The main contributions are as follows:(1)Based on the special structure of coefficient matrix,we construct a class of approximate inverse preconditioner and prove that it is a good approximation of the inverse of coefficient matrix.(2)To reduce the computational cost of computing the inverse matrix involved in the preconditioner,we use circulant matrix to approximate Toeplitz matrix.It is proved that the difference between new approximate inverse preconditioner and inverse of coefficient matrix can be written as sum of a small norm matrix and a low rank matrix.(3)To further improve the computational efficiency,we utilize the interpolation method to construct an interpolation approximation inverse preconditioner and prove the difference between inverse of coefficient matrix and it can also be written as sum of a small norm matrix and a low rank matrix.(4)Numerical tests are carried out to show the performance of the new proposed preconditioner.