arXiv subjects

Kota Sugihara

Publications and source records attributed to Kota Sugihara.

3 recordsLinked to original sources

arXiv2024

Right preconditioned GMRES for arbitrary singular systems

Brown and Walker (1997) showed that GMRES determines a least squares solution of $ A x = b $ where $ A \in {\bf R}^{n \times n} $ without breakdown for arbitrary $ b, x_0 \in {\bf R}^n $ if and only if $A$ is range-symmetric, i.e. $ {\cal R} (A^{\rm T}) = {\cal R} (A) $, where $ A $ may be singular and $ b $ may not be in the range space ${\cal R} A)$ of $A$. In this paper, we propose applying GMRES to $ A C A^{\rm T} z = b $, where $ C \in {\bf R}^{n \times n} $ is symmetric positive definite. This determines a least squares solution $ x = CA^{\rm T} z $ of $ A x = b $ without breakdown for arbitrary (singular) matrix $A \in {\bf R}^{n \times n}$ and $ b \in {\bf R}^n $. To make the method numerically stable, we propose using the pseudoinverse with an appropriate threshold parameter to suppress the influence of tiny singular values when solving the severely ill-conditioned Hessenberg systems which arise in the Arnoldi process of GMRES when solving inconsistent range-symmetric systems. Numerical experiments show that the method taking $C$ to be the identity matrix and the inverse matrix of the diagonal matrix whose diagonal elements are the diagonal of $A A^{\rm T}$ gives a least squares solution even when $A$ is not range-symmetric, including the case when $ {\rm index}(A) >1$.

math.NA↗

arXiv2022

GMRES using pseudoinverse for range symmetric singular systems

Consider solving large sparse range symmetric singular linear systems $ A {\bf x}= {\bf b} $ which arise, for instance, in the discretization of convection diffusion equations with periodic boundary conditions, and partial differential equations for electromagnetic fields using the edge-based finite element method. In theory, the Generalized Minimal Residual (GMRES) method converges to the least squares solution for inconsistent systems if the coefficient matrix $A$ is range symmetric, i.e. $ {\rm R}(A)= {\rm R}(A^{ \rm T } )$, where $ {\rm R}(A)$ is the range space of $A$. We derived the necessary and sufficient conditions for GMRES to determine a least squares solution of inconsistent and consistent range symmetric systems assuming exact arithmetic except for the computation of the elements of the Hessenberg matrix. In practice, GMRES may not converge due to numerical instability. In order to improve the convergence, we propose using the pseudoinverse for the solution of the severely ill-conditioned Hessenberg systems in GMRES. Numerical experiments on inconsistent systems indicate that the method is effective and robust. Finally, we further improve the convergence of the method by reorthogonalizing the Modified Gram-Schmidt procedure.

math.NA↗

arXiv2020

GMRES on singular systems revisited

In [Hayami K, Sugihara M. Numer Linear Algebra Appl. 2011; 18:449--469], the authors analyzed the convergence behaviour of the Generalized Minimal Residual (GMRES) method for the least squares problem $ \min_{ {\bf x} \in {\bf R}^n} {\| {\bf b} - A {\bf x} \|_2}^2$, where $ A \in {\bf R}^{n \times n}$ may be singular and $ {\bf b} \in {\bf R}^n$, by decomposing the algorithm into the range $ {\cal R}(A) $ and its orthogonal complement $ {\cal R}(A)^\perp $ components. However, we found that the proof of the fact that GMRES gives a least squares solution if $ {\cal R}(A) = {\cal R}(A^{\scriptsize T} ) $ was not complete. In this paper, we will give a complete proof.

math.NA↗