Using frames in GMRES-based iteration method for solving operator equations

Jamali, Hassan; Pourkani, Reza

doi:10.22103/jmmr.2023.21932.1480

Using frames in GMRES-based iteration method for solving operator equations

Document Type : Research Paper

Authors

Hassan Jamali

Reza Pourkani

Department of Mathematics, vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

10.22103/jmmr.2023.21932.1480

Abstract

‎In this paper, we delve into frame theory to create an innovative iterative method for resolving the operator equation $ Lu=f $. In this case, $ L:H\rightarrow H $, a bounded, invertible, and self-adjoint linear operator, operates within a separable Hilbert space denoted by $H$. Our methodology, which is based on the GMRES projective method, introduces an alternate search space, which brings another dimension to the problem-solving process. Our investigation continues with the assessment of convergence, where we look at the corresponding convergence rate. This rate is intricately influenced by the frame bounds, shedding light on the effectiveness of our approach. Furthermore, we investigate the ideal scenario in which the equation finds an exact solution, providing useful insights into the practical implications of our work.

Keywords

Operator equation

Frame

Preconditioning

GMRES iteration

Convergence rate

Subjects

Analysis

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