Stochastic Rounding in Mixed-Precision Iterative Refinement Open Access

Abstract

In this thesis, we study the use of stochastic rounding (SR) in mixed-precision iterative refinement (IR) algorithms for solving linear systems and least squares problems. Tikhonov regularization in block form is used to address ill-posed and noisy problems. In particular, SR is applied in the low-precision computation step of the mixed-precision IR algorithms, where it has been shown to yield smaller backward error bounds with high probability and is therefore less susceptible to stagnation. Numerical experiments on extremely ill-posed inverse heat problems and a large-scale PRblur Kronecker product problem are used to demonstrate its practical value. We first find that regularization is most beneficial for ill-posed and noisy problems, while otherwise increasing computational complexity with limited advantage. For the PRblur problem, we observe that mixed-precision IR using SR in the low-precision solve performs comparably to pure higher-precision computation, indicating its potential for real-world applications. These experiments also highlight current limitations, including the constraints of the Julia stochastic rounding package and the general lack of GPU support for SR. Future work may examine GPU-based implementations of SR and their impact on runtime and convergence.

Table of Contents

Contents

1 Introduction 1

1.1 Iterative Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Floating Point Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Literature Review 9

3 Rounding in Floating Point Arithmetic 13

3.1 Round-to-Nearest and Stochastic Rounding . . . . . . . . . . . . . . 13

3.2 Stochastic Rounding Packages . . . . . . . . . . . . . . . . . . . . . . 15

4 Iterative Refinement 17

4.1 IR Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.2 GMRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Numerical Experiments 22

5.1 Ill-conditioned inverse heat matrix . . . . . . . . . . . . . . . . . . . . 23

5.2 PRblur Kronecker product . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2.1 Kronecker product definition . . . . . . . . . . . . . . . . . . . 26

5.2.2 Noise free condition . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.3 Noise added conditions . . . . . . . . . . . . . . . . . . . . . . 29

5.2.4 Image display . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Concluding Remarks 35

Bibliography 38

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Iterative Refinement Stochastic Rounding
Committee Chair / Thesis Advisor	James Nagy, Emory University
Committee Members	Julianne Chung, Emory University Adam Glynn, Emory University

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