Higham, Nicholas J. and Mary, Theo (2018) A New Preconditioner that Exploits LowRank Approximations to Factorization Error. [MIMS Preprint]
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Abstract
We consider illconditioned linear systems $Ax =$ b that are to be solved iteratively, and assume that a low accuracy LU factorization $A \approx \widehat{L}\widehat{U}$ is available for use in a preconditioner. We have observed that for illconditioned matrices $A$ arising in practice, $A^{1}$ tends to be numerically low rank, that is, it has a small number of large singular values. Importantly, the error matrix $E = \widehat{U}^{1}\widehat{L}^{1}A  I$ tends to have the same property. To understand this phenomenon we give bounds for the distance from $E$ to a lowrank matrix in terms of the corresponding distance for $A^{1}$. We then design a novel preconditioner that exploits the lowrank property of the error to accelerate the convergence of iterative methods. We apply this new preconditioner in three different contexts fitting our general framework: low floatingpoint precision (e.g., half precision) LU factorization, incomplete LU factorization, and block lowrank LU factorization. In numerical experiments with GMRESbased iterative refinement we show that our preconditioner can achieve a significant reduction in the number of iterations required to solve a variety of reallife problems.
Item Type:  MIMS Preprint 

Additional Information:  To appear in the SIAM Journal on Scientific Computing. 
Subjects:  MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis 
Divisions:  Manchester Institute for the Mathematical Sciences 
Depositing User:  Dr Theo Mary 
Date Deposited:  11 Oct 2018 09:02 
Last Modified:  11 Oct 2018 09:02 
URI:  http://eprints.maths.manchester.ac.uk/id/eprint/2667 
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A New Preconditioner that Exploits LowRank Approximations to Factorization Error. (deposited 23 Apr 2018 18:54)

A New Preconditioner that Exploits LowRank Approximations to Factorization Error. (deposited 22 Aug 2018 15:59)
 A New Preconditioner that Exploits LowRank Approximations to Factorization Error. (deposited 11 Oct 2018 09:02) [Currently Displayed]

A New Preconditioner that Exploits LowRank Approximations to Factorization Error. (deposited 22 Aug 2018 15:59)
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