Additive Schwarz Iterations for Markov Chains

Abstract

Abstract.A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations of the form Ax = b. The theory applies to singular M -matrices with one-dimensional null space and is applicable in particular to systems representing ergodic Markov chains, and to certain discretizations of partial differential equations. Additive Schwarz can be seen as a generalization of block Jacobi, where the set of indices defining the diagonal blocks have nonempty intersection; this is called the overlap. The presence of overlap is known to accelerate the convergence of the methods in the nonsingular case. By providing convergence results, as well as some characteristics of the induced splitting, we hope to encourage the use of this additional computational tool for the solution of Markov chains and other singular systems. We present several numerical examples showing that additive Schwarz performs better than block Jacobi. For completeness, a few numerical experiments with block Gauss-Seidel and multiplicative Schwarz are also included.

Parallel alternating iterative algorithms with and without overlapping on multicore architectures

“…Note that the global iteration matrix of the alternating two-stage iterative method (6) can be written as…”

Subdirect sums of doubly diagonally dominant matrices

“…This situation, as well as a more general case outlined in Theorem 3.11 later, appears in many variants of additive Schwarz preconditioning (see [2,7,8] We now have the following theorem which shows that C is a DDD matrix. Theorem 3.11.…”

Overlapping clusters for distributed computation

“…Given a domain that is decomposed into overlapping pieces, Schwarz methods solve the PDE by iterative solving the PDE on each piece and communicating between the domains via the overlapping boundary. Within the numerical linear algebra community, these ideas have been generalized to solve many linear systems Ax = b in either an additive or multiplicative Schwarz method [37,6,28]. The difference between the methods is not important for this paper, but suffice it to say that our distributed PageRank technique discussed in Section 6 is equivalent to an additive method.…”