Fast Algorithm For 3-D Constrained Optimal Control Problem And Governed By Space-Fractional Diffusion Equations

Numerical approximation of optimal control problem is an important topic in engineering design work, and the applications of fractional diffusion equation in physics and mathematics are also very wide. Compared to the integer-order equation, fractional diffusion equations were shown to provide more accurate and appropriate description of anomalous diffusion, such as in modeling trans-port processes, turbulent flow, groundwater contaminant transport, chaotic dynamics of classical conservative system and so on. It has great significance to study the algorithm to solve the optimal control problem governed by frac-tional diffusion equation. And because of the non-local property of fractional differential operators, the finite differential methods often generate dense or full coefficient matrices, the direct algorithm for these linear systems often require computational work of O(N3) and memory of O(N2) 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 differential algorithm for the 3-D optimal control problem with pointwise constraint on control and governed by non-steady-state space-fractional diffusion equations.In Chapter 1, we introduce the background and status of the optimal control problem. The model problem that we will investigate is the following three-dimensional distributed convex optimal problem:Find u∈ K={u∈ to minimize the cost functional which is subjected to function representing the observation of the state.In Chapter 2, we propose the gradient projection algorithm and CN-WSGD difference scheme by introducing the costate equation.In Chapter 3, firstly, we apply ADI method for the state and costate e-quations. Secondly, we give PCG/PCGS algorithm for the resulting linear system. Finally, we analyse the characteristics of the obtained linear system matrix based on ADI-WSGD difference scheme. According to the Toeplitz property of the coefficient matrices, in the symmetric case, we reduce com-putational work from O(N3) to O(NlogN) by applying PCG method. In the non-symmetric case, we reduce computational work from O(N3) to O(NlogN) by applying PCGS method.In Chapter 4, we give two numerical experiments in view of symmetric case and non-symmetric case and use PCG/PCGS methods and GAUSS method to solve them respectively. Then by comparing the final convergence rates and CPU time of the methods, we show that the fast algorithm greatly improve the computational efficiency while maintain the same convergence accuracy compared with the traditional GAUSS algorithm.In Chapter 5, we give the conclusions of this paper.