A Fast Algorithm For Optimal Control Governed By Non-steady-state Space-Fractional Diffusion Equations

Finite element approximation of optimal control problem is an importan-t topic in engineering design work, and the applications of fractional diffu-sion equation in physics and mathematics are also very wide. Compared to the integer-order equation, fractional diffusion equations were shown to pro-vide more accurate and appropriate description of anomalous diffusion,such as in modeling transport processes, turbulent flow,groundwater contaminant transport,chaotic dynamics of classical conservative system and so on. So the fast algorithm research to solve the optimal control problem governed by frac-tional diffusion equation is of great significance. And because of the non-local property of fractional differential operators, the finite element methods often generate dense or full coefficient matrices, consequently,the direct algorith-m for these linear systems often require memory of O(N2) and computational work of O(N3) for where N is the number of grid points. Therefore the research of fast algorithm to solve these linear systems is significant.In this paper, we develop a fast gradient algorithm for the optimal control problem governed by non-steady-state space-fractional diffusion equations.In chapter1, the model problem that we will investigate is the following one-dimensional distributed convex optimal problem: subject to where u(x) is the control function,y(x) is the state function, y(x) is a known function.then we proof the existence and uniqueness of solutions of the prob-lem and give the equivalent form of optimal control problem.In chapter2,we present the backward difference-finite element approx-imation scheme and a priori error estimates of this problem,then estimate results as follows can be obtained:In chapter3,we similarly give the central difference-finite element ap-proximation scheme and a priori error estimates of this problem,thcn estimate results as follows can be obtained:In chapter4, we analyze the characteristics of the obtained linear system matrix based on the two different finite element approximation schemes.ac-cording to the Toeplitz property of the coefficient matrices, in the symmetric case, we reduce computational work from O(N3) to O(N log2N) by applying the PCG and supcrfast algorithm while in the non-symmetric case, we can get the same effect by using the PCGS algorithm.In chapter5, giving two numerical experiments in view of symmetric case and non-symmetric case, we use PCG (PCGS),supcrfast and Guass algorithm to solve these two different finite element approximation schemes,Then by comparing the final convergence and CPU time of the methods,we show that the two fast algorithms greatly improve the computational efficiency while maintain the same precision convergence effect compared with the traditional Guass algorithm.In chapter6, we give the conclusions of this paper.