We consider solving large sparse symmetric singular linear systems. We first introduce an algorithm for right preconditioned minimum residual (MINRES) and prove that its iterates converge to the preconditioner weighted least squares solution without breakdown for an arbitrary right-hand-side vector and an arbitrary initial vector even if the linear system is singular and inconsistent. For the special case when the system is consistent, we prove that the iterates converge to a min-norm solution with respect to the preconditioner if the initial vector is in the range space of the right preconditioned coefficient matrix. Furthermore, we propose a right preconditioned MINRES using symmetric successive over-relaxation (SSOR) with Eisenstat's trick. Some numerical experiments on semidefinite systems in electromagnetic analysis and so forth indicate that the method is efficient and robust. Finally, we show that the residual norm can be further reduced by restarting the iterations.
KEYWORDSEisenstat's trick, Krylov subspace methods, MINRES method, right preconditioning, SSOR preconditioner, symmetric singular systems † In memory of Professor Masaaki Sugihara Numer Linear Algebra Appl. 2020;27:e2277. wileyonlinelibrary.com/journal/nla Another example for (1), (2) comes from the analysis of static magnetic fields. 25 See other works also. 20-24Here, ⃗ J is the external current density. When this partial differential equation is discretized by the edge-based finite element method and ⃗ A m is the unknown variable, the coefficient matrix is SPSD. If ⃗ J does not satisfy ∇· ⃗ J = 0 in Ω, then b of (1), (2) is not in R(A), and the systems is inconsistent. For such an inconsistent system, CG or the preconditioned conjugate gradient (PCG) method does not converge.In order that (preconditioned) CG converge to a solution, one could make the system consistent by projecting b to R(A). However, in general, this may be infeasible if R(A) is not given explicitly. Therefore, we focus on (preconditioned) MINRES, which converges even if the system is inconsistent.We note that there are existing state-of-the-art solvers for H(curl) problems, which are highly scalable and run on many processors. 20,26-28 The purpose of this paper is to propose an efficient preconditioner for general consistent or inconsistent positive semidefinite systems of linear equations, for which the above applications 1 and 2 are examples giving rise to such systems.
In [Hayami K, Sugihara M. Numer Linear Algebra Appl. 2011; 18:449-469], the authors analyzed the convergence behavior of the generalized minimal residual (GMRES) method for the least squares problem min x∈R n ||b − Ax|| 2 2 , where A ∈ R n × n may be singular and b ∈ R n , by decomposing the algorithm into the range (A) and its orthogonal complement (A) ⟂ components. However, we found that the proof of the fact that GMRES gives a least squares solution if (A) = (A T ) was not complete. In this article, we will give a complete proof.
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