A Preconditioned Minimal Residual Solver for a Class of Linear Operator Equations

Abstract

We consider the class of linear operator equations with operators admitting self-adjoint positive definite and m-accretive splitting (SAS). This splitting leads to an ADI-like iterative method which is equivalent to a fixed point problem where the operator is a 2 by 2 matrix of operators. An infinite dimensional adaptation of a minimal residual algorithm with Symmetric Gauss-Seidel and polynomial preconditioning is then applied to solve the resulting matrix operator equation. Theoretical analysis shows the convergence of the methods, and upper bounds for the decrease rate of the residual are derived. The convergence of the methods is numerically illustrated with the example of the neutron transport problem in 2-D geometry.

On GMRES for Singular EP and GP Systems

“…Furthermore, by solving min y∈R 1 βe 1 − H 2,1 y with β = 1, we have y 1 = 1/ε, i.e., y 1 has a large component. We see that x 1 = Q 1 y 1 and x 1…”

“…See [8], [9], [14] for GMRES on ill-posed linear systems, and [32] for GMRES with preconditioning. See also [15] for GMRES and [1] for GMRES with preconditioning in Hilbert spaces.…”

“…The preconditioned GMRES method applied to singular systems was considered in [32] . See also [15] for GMRES and [1] for the preconditioned GMRES method applied in Hilbert spaces.…”