Theoretical Analysis And Applications Of Splitting-iteration Algorithms

Parallel computation is the most popular and important method in large scale scientific and engineering computing. The understanding and efficiency analysis of the parallel meth-ods is the key point of the design of algorithms. In this paper, we deeply investigate five parallel methods:a) overlapping Schwarz waveform relaxation iteration algorithm; b) the waveform relaxation iteration algorithm; c) the parallel in time waveform relaxation itera-tion algorithm; d) Parareal algorithm; e) the relaxation Newton algorithm. The relaxation Newton algorithm is designed for solving large scale nonlinear algebraic equations. The waveform relaxation iteration algorithm, the parallel in time waveform relaxation iteration algorithm and the Parareal algorithm are introduced for concurrently solving large scale or-dinary differential equations. And the overlapping Schwarz waveform relaxation iteration algorithm is constructed for solving partial differential equations. The point in common with these five algorithms is that some physical quantity, such as time, space and system, is first partitioned and then an iterative process is invoked. Finally, The solution of the underlying problem is obtained through iterations. That is the reason why we uniformly call these five methods splitting-iteration algorithms.The relaxation Newton algorithm and the parallel in time waveform relaxation iteration algorithm are our novel work. Both algorithms are developed on the basis of the waveform relaxation iteration algorithm. In this paper, we formulate the idea of these two parallel computation methods, analyze the convergence and compare the performance of these two methods with the classical methods. After systematically researching the results obtained by the other authors, we also get some new results in the fields of the waveform relaxation iteration algorithm, the Parareal algorithm and the overlapping Schwarz waveform relax-ation iteration algorithm. In certain aspects, our results enrich and develop the existing results. The focus of this paper is on the theoretical analysis of the splitting-iteration algorithms (in the sense of stability, convergence and speed of convergence) and lots of numerical experiments are done to validate the theoretical results.