arXiv subjects

Kensuke Aihara

Publications and source records attributed to Kensuke Aihara.

5 recordsLinked to original sources

arXiv2026

Theoretical insights on the residual transformation from bi-conjugate gradient into bi-conjugate residual via a smoothing scheme

Bi-conjugate gradient (Bi-CG) and bi-conjugate residual (Bi-CR) methods are underlying iterative solvers for linear systems with nonsymmetric matrices. Residual smoothing is a standard technique for obtaining smooth convergence behavior of residual norms; additionally, it represents the transformation between iterative methods. For example, the residuals of the CR method can be obtained by applying a smoothing scheme to those of the CG method for symmetric linear systems. Based on this relationship, the transformation from Bi-CG residuals to Bi-CR residuals using a smoothing scheme was examined in our previous study [Kawase, A., Aihara, K.: Transformation from Bi-CG into Bi-CR Using a Residual Smoothing-like Scheme. AIP Conference Proceedings (2026)]; however, we only provided heuristic and experimental observations. In the present study, we provide a detailed discussion on the theoretical aspects of these transformations. Specifically, we prove that the resulting algorithm transformed from the Bi-CG method using the residual smoothing technique has the same bi-orthogonal properties as those of the original Bi-CR method. We also present a more concise transformation algorithm and its numerical example. These analyses complement our previous study and provide theoretical validity of the residual transformation between the Bi-CG and Bi-CR methods.

math.NA↗

arXiv2025

Transformation from Bi-CG into Bi-CR using a residual smoothing-like scheme

Residual smoothing techniques, which produce a smooth convergence behavior of linear iterative solvers, also form connections between different methods. For example, minimal residual smoothing can transform the residuals of the conjugate gradient (CG) method into those of the conjugate residual (CR) method for linear systems with symmetric matrices. In this study, we investigate whether a similar relationship can be constructed between the Bi-CG and Bi-CR methods for nonsymmetric linear systems. Numerical experiments regarding the aforementioned connection demonstrate the validity of our insights.

math.NA↗

arXiv2025

Modified Armijo line search in optimization on Riemannian submanifolds with reduced computational cost

For optimization problems on Riemannian manifolds, many types of globally convergent algorithms have been proposed, and they are often equipped with the Riemannian version of the Armijo line search for global convergence. Such existing methods need to compute the value of a retraction mapping regarding the search direction several times at each iteration; this may result in high computational costs, particularly if computing the value of the retraction is expensive. To address this issue, this study focuses on Riemannian submanifolds of the Euclidean spaces and proposes a novel Riemannian line search that achieves lower computational cost by incorporating a new strategy that computes the retraction only when inevitable. A class of Riemannian optimization algorithms, including the steepest descent and Newton methods, with the new line search strategy is proposed and proved to be globally convergent. Furthermore, numerical experiments on solving optimization problems on several types of Riemannian submanifolds illustrate that the proposed methods are superior to the standard Riemannian Armijo line search-based methods.

math.OC↗

arXiv2024

Block cross-interactive residual smoothing for Lanczos-type solvers for linear systems with multiple right-hand sides

Lanczos-type solvers for large sparse linear systems often exhibit large oscillations in the residual norms. In finite precision arithmetic, large oscillations increase the residual gap (the difference between the recursively updated residual and the explicitly computed residual) and a loss of attainable accuracy of the approximations. This issue is addressed using cross-interactive residual smoothing (CIRS). This approach improves convergence behavior and reduces the residual gap. Similar to how the standard Lanczos-type solvers have been extended to global and block versions for solving systems with multiple right-hand sides, CIRS can also be extended to these versions. While we have developed a global CIRS scheme (Gl-CIRS) in our previous study [K. Aihara, A. Imakura, and K. Morikuni, SIAM J. Matrix Anal. Appl., 43 (2022), pp.1308--1330], in this study, we propose a block version (Bl-CIRS). Subsequently, we demonstrate the effectiveness of Bl-CIRS from various perspectives, such as theoretical insights into the convergence behaviors of the residual and approximation norms, numerical experiments on model problems, and a detailed rounding error analysis for the residual gap. For Bl-CIRS, orthonormalizing the columns of direction matrices is crucial in effectively reducing the residual gap. This analysis also complements our previous study and evaluates the residual gap of the block Lanczos-type solvers.

math.NA↗

arXiv2022

Cross-interactive residual smoothing for global and block Lanczos-type solvers for linear systems with multiple right-hand sides

Global and block Krylov subspace methods are efficient iterative solvers for large sparse linear systems with multiple right-hand sides. However, global or block Lanczos-type solvers often exhibit large oscillations in the residual norms and may have a large residual gap relating to the loss of attainable accuracy of the approximations. Conventional residual smoothing schemes suppress these oscillations but cannot improve the attainable accuracy, whereas a recent residual smoothing scheme enables the improvement of the attainable accuracy for single right-hand side Lanczos-type solvers. The underlying concept of this scheme is that the primary and smoothed sequences of the approximations and residuals influence one another, thereby avoiding the severe propagation of rounding errors. In the present study, we extend this cross-interactive residual smoothing to the case of solving linear systems with multiple right-hand sides. The resulting smoothed methods can reduce the residual gap with a low additional cost compared to their original counterparts. We demonstrate the effectiveness of the proposed approach through rounding error analysis and numerical experiments.

math.NA↗