In many applications, one needs to solve large, sparse linear systems (often with millions of unknowns) that arise from the discretization of boundary value problems. To parallelize these large computations, a natural idea is to use optimized Schwarz methods, which involves dividing the computational domain into several subdomains, solving the subproblems in parallel, and iterating to obtain a consistent solution. During this iteration, each processor must obtain information from other subdomains to ensure convergence to the global solution, and the rate of convergence is very sensitive to the type of values being transmitted. It is thus natural to ask what information should be exchanged so that the method converges optimally.
In this talk, I will first present the basic ideas behind optimized Schwarz methods and show how optimal transmission conditions are usually derived based on Fourier analysis. I will then show how matrix techniques provide an alternative way of constructing such conditions and can be used to analyze convergence of the discrete iterations directly. This approach can be used when Fourier analysis is no longer applicable, such as for domains with irregular boundaries and/or problems with discontinuous coefficients. Moreover, it reveals that an optimal algorithm can be obtained by transmitting values along a coarse-grid structure, which is not apparent from the Fourier analysis.