Right preconditioned MINRES for singular systems^†

Abstract

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.

GMRES using pseudoinverse for range symmetric singular systems

“…Refer to the proof of Theorem 1 in [7]. In the present proof, the preconditioner M is an identity matrix and MINRES is replaced by GMRES.…”

“…In order to prove the theorem, we will analyse GMRES using pseudo-inverse by decomposing it into the R(A) component and the R(A) ⊥ component. Using the approach in the proof of Theorem 1 in [7], we can prove that the R(A) component x 1 k of the solution x k of GMRES using pseudo-inverse minimizes the R(A) component of b − Ax 2 when…”

“…If all computations are done in exact arithmetic, GMRES using pseudo-inverse determines a solution of min x∈R n b − Ax 2 when h k+1,k = 0. When h k+1,k = 0 holds, H k,k is singular (See [7], Theorem 1, point a, b; [8], Theorem 4). On the other hand, numerical experiments on some semi-definite inconsistent systems indicate that Ar 2 / Ab 2 becomes very small when the smallest singular value of H k+1,k is very small.…”

Corrigendum 2 to: A geometric view of Krylov subspace methods on singular systems

“…In the following, we give a complete proof of the statement. See also Sugihara et al, 4 theorem 1 for a related proof for the right-preconditioned MINRES method for symmetric singular systems.…”

GMRES on singular systems revisited

“…In the following, we give a complete proof of the statement. See also Sugihara, Hayami, Zheng [4], Theorem 1 for a related proof for the right-preconditioned MINRES method for symmetric singular systems. First, we observe the following.…”