Double Precision is not Necessary for LSQR for Solving Discrete Linear Ill-Posed Problems

The growing availability and usage of low precision floating point formats attracts many interests of developing lower or mixed precision algorithms for scientific computing problems. In this paper we investigate the possibility of exploiting mixed precision computing in LSQR for solving discrete li...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 98; no. 3; p. 55
Main Author: Li, Haibo
Format: Journal Article
Language:English
Published: New York Springer US 01.03.2024
Springer Nature B.V
Subjects:
Accuracy
Algorithms
Approximation
Computation
Computational Mathematics and Numerical Analysis
Floating point arithmetic
Ill posed problems
Inverse problems
Iterative methods
Iterative solution
Linear algebra
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Noise levels
Number systems
Regularization
Regularization methods
Robustness (mathematics)
Theoretical
65G50
Semi-convergence
65F10
65F22
Roundoff unit
Linear ill-posed problem
LSQR
Mixed precision
Regularization
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:The growing availability and usage of low precision floating point formats attracts many interests of developing lower or mixed precision algorithms for scientific computing problems. In this paper we investigate the possibility of exploiting mixed precision computing in LSQR for solving discrete linear ill-posed problems. Based on the commonly used regularization model for linear inverse problems, we analyze the choice of proper computing precision in the two main parts of LSQR, including the construction of Krylov subspace and updating procedure of iterative solutions. We show that, under some mild conditions, the Lanczos vectors can be computed using single precision without loss of any accuracy of the final regularized solution as long as the noise level is not extremely small. We also show that the most time consuming part for updating iterative solutions can be performed using single precision without sacrificing any accuracy. The results indicate that several highly time consuming parts of the algorithm can be implemented using lower precisions, and provide a theoretical guideline for implementing a robust and efficient mixed precision variant of LSQR for solving discrete linear ill-posed problems. Numerical experiments are made to test two mixed precision variants of LSQR and confirming our results.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-023-02447-4