TY - JOUR
T1 - Full operator preconditioning and the accuracy of solving linear systems
AU - Mohr, Stephan
AU - Nakatsukasa, Yuji
AU - Urzúa-Torres, Carolina
PY - 2024
Y1 - 2024
N2 - Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning. A key observation in this paper is the sometimes overlooked fact that while traditional preconditioning effectively accelerates convergence of iterative methods, it generally does not improve the accuracy of the solution. Nonetheless, it is sometimes possible to overcome this barrier: accuracy can be improved significantly if the equation is transformed before discretization, a process we refer to as full operator preconditioning (FOP). We highlight that this principle is already used in various areas, including second kind integral equations and Olver–Townsend’s spectral method. We formulate a sufficient condition under which high accuracy can be obtained by FOP. We illustrate this for a fourth order differential equation which is discretized using finite elements.
AB - Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning. A key observation in this paper is the sometimes overlooked fact that while traditional preconditioning effectively accelerates convergence of iterative methods, it generally does not improve the accuracy of the solution. Nonetheless, it is sometimes possible to overcome this barrier: accuracy can be improved significantly if the equation is transformed before discretization, a process we refer to as full operator preconditioning (FOP). We highlight that this principle is already used in various areas, including second kind integral equations and Olver–Townsend’s spectral method. We formulate a sufficient condition under which high accuracy can be obtained by FOP. We illustrate this for a fourth order differential equation which is discretized using finite elements.
KW - condition number
KW - discretization error
KW - numerical error
KW - operator preconditioning
KW - preconditioning
UR - http://www.scopus.com/inward/record.url?scp=85207978299&partnerID=8YFLogxK
U2 - 10.1093/imanum/drad104
DO - 10.1093/imanum/drad104
M3 - Article
AN - SCOPUS:85207978299
SN - 0272-4979
VL - 44
SP - 3259
EP - 3279
JO - IMA Journal of Numerical Analysis
JF - IMA Journal of Numerical Analysis
IS - 6
ER -