| Summary: | 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.
|
|---|