On the Choice of Subspace for the Quasi-minimal Residual Method for Linear Inverse Problems Pubblico

Abstract

Inverse problems arise in various scientific and engineering applications, necessitating robust numerical methods for their solution. In this work, we investigate the effectiveness of Krylov subspace iterative methods, including GMRES, QMR, and their range-shifted variants for solving linear inverse problems. We analyze the impact of subspace selection on solution quality and stability, comparing conventional and range-shifted versions of GMRES and QMR. Our findings indicate that range shifted QMR outperforms standard QMR, and confirm the previously observed behavior that range shifted GMRES can be superior to conventional GMRES in terms of approximation efficacy. Notably, range restricted QMR demonstrates a key advantage over GMRES with respect to range restricted QMR's singular spectrum which can make the method less sensitive to errors that are naturally present making it particularly effective when the noise level in the problem is uncertain. These results provide valuable insights into selecting appropriate iterative solvers for ill-posed problems.

Table of Contents

1 Introduction

1.1 Inverse Problems

1.2 Linear Discrete Ill-posed Problems

1.3 Regularization

1.3.1 SVD of Ill-Posed Problem

1.3.2 Truncated SVD Method

2 Krylov Subspace Methods

2.1 Krylov Subspaces

2.2 Arnoldi

2.2.1 Classic Arnoldi Process

2.2.2 Arnoldi Relation

2.3 Lanczos

2.3.1 Lanczos for Symmetric Problems

2.3.2 Lanczos Bi-orthogonalization for Non-symmetric Problems

2.4 Krylov Subspace Iterative Methods

2.4.1 Generalized Minimal Residual (GMRES) Method

2.4.2 Quasi-Minimal Residual (QMR) Method

3 Range Restricted Krylov Subspace Methods

3.1 Krylov Subspace Range Restriction

3.2 The Range Restricted GMRES Method

3.3 The Range Restricted QMR Method

4 Numerical Experiments

4.1 Preliminaries

4.1.1 Termination Criterion: Discrepancy Principle

4.1.2 Solution Evaluation: Relative Residual

4.1.3 Solution Evaluation: Relative Error

4.2 1D and 2D Problems Considered

4.2.1 1D Problems

4.2.2 2D Problems

4.3 Numerical Results

4.3.1 Shifted vs. Non-Shifted: GMRES and QMR

4.3.2 QMR Shift Comparison

4.3.3 Performance Under Uncertain Error Norm Bound

4.3.4 2D Problem: Image Deblurring

5 Concluding Remarks

Bibliography

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Parola chiave	QMR (Quasi-Minimal Residual Method) Shifted QMR Regularization Krylov subspace Range restricted Krylov subspace methods Inverse Problems Ill-posed problems
Committee Chair / Thesis Advisor	James G. Nagy, Emory University
Committee Members	Lucas Onisk, Emory University Myra Woodworth-Hobbs, Emory University Sandeep Soni, Emory University

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