Domain Decomposition Methods And Parallel Computing In Optimal Control And Image Processing Problems

How to use the supercomputers is a great challenge. The most popular trend is parallelism. So it is critical to develop the parallel algorithm and software in the applica-tions. In the area of the numerical computing, it is urgent to do the research on parallel algorithms. A good parallel algorithms should allow simulations for a long time, i. e. ro-bust. Further, it should show better speedup ratio and efficiency. Domain decomposition methods just satisfy these two properties. In the thesis, we apply domain decomposi-tion methods on the optimal control problems governed by partial differential equations and energy based image processing problems. To these two types problems, we do the following research.First on the optimal control problem governed by partial differential equations, we present the following research.1.1A domain decomposition method (DDM) is presented to solve the distributed op-timal control problem. The proposed algorithm, called SA-GP algorithm, consists of two iterative stages. In the inner loops the Schwarz alternating method (SA) is applied to solve the state and co-state variables, and in the outer loops the gradient projection algorithm (GP) is adopted to obtain the control variable. Convergence of iterations depends on both of the outer and the inner loops, which are coupled and affected by each other. In the classical iteration algorithms, a given tolerance would be reached after sufficiently many iteration steps, but more iterations lead to huge computational cost. For solving constrained optimal control problems, most of computational cost is used to solve PDEs. In the paper, a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost. The convergence rate of L2-error of control variable is de-rived. Also the analysis on how to choose the proposed iteration number in the inner loops is given. Some numerical experiments are performed to verify the theoretical results.1.2We consider the nonoverlapping domain decomposition algorithm for the above problems. We adopt the Robin-Type boundary transmission condition and de-compose the equivalent variational inequity into subdomain. The convergence of Robin-Type DDM is proved and the optimal convergence rate is got by choosing the optimal relaxation parameters. This is the first successful attempt to give the convergence rate analysis of applying DDMs to the non-linear systems.1.3We consider Robin-Type DDMs to the optimal control problems with more general constraints set which cannot be directly decomposed the subproblem over subdo-main. Then the optimal control problem with integral constraint is considered as an example. The convergence analysis is given and the numerical examples show the algorithm is effective.To the domain decomposition methods for the energy based image processing problem, the following work is done.2.1We concerned with overlapping domain decomposition methods (DDMs), based on successive subspace correction (SSC) and parallel subspace correction (PSC), for the Rudin, Osher and Fatemi (ROF) model in image restoration. In distinct contrast with recent attempts along this line, we work with a dual formulation of the ROF model, where one significant difficulty lies in the decomposition of the global constraint of the dual variable. We propose a stable unit decomposition which allows us to construct the SSC and PSC based DDMs. We further analyze the convergence of the proposed algorithms, and obtain the rate O(n-1) where n is the number of iterations. To the best of our knowledge, such a convergence has not claimed so far.2.2We present the domain decomposition methods(DDMs) by decomposing the origi-nal problem into the subproblem over subdomains for the nonlocal total variational image restoration, proposed by Xu, Tai and Wang [96]. For each subproblem, we adopt the Bregmanized operator splitting algorithm to solve the subproblem. Nu-merical examples imply that the algorithms we proposed for nonlocal-TV image restoration work well for both image denoising, deblurring and inpainting problem-s. Finally, parallel computing of the algorithms are realized in the cluster and good speedup ratio and efficiency is observed.