Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error. Often, such systems arise from the discretization of operator equations with a large number of discrete variables. In this paper we show that the accuracy can be improved significantly if the equation is transformed before discretization, a process we call full operator preconditioning (FOP). It bears many similarities with traditional preconditioning for iterative methods but, crucially, transformations are applied at the operator level. We show that while condition-number improvements from traditional preconditioning generally do not improve the accuracy of the solution, FOP can. A number of topics in numerical analysis can be interpreted as implicitly employing FOP; we highlight (i) Chebyshev interpolation in polynomial approximation, and (ii) Olver-Townsend's spectral method, both of which produce solutions of dramatically improved accuracy over a naive problem formulation. In addition, we propose a FOP preconditioner based on integration for the solution of fourth-order differential equations with the finite-element method, showing the resulting linear system is well-conditioned regardless of the discretization size, and demonstrate its error-reduction capabilities on several examples. This work shows that FOP can improve accuracy beyond the standard limit for both direct and iterative methods.
翻译:除非适用特殊条件,否则试图用标准数字方法解决线性方程式的不完善系统会导致难以控制的高数字错误。通常,这种系统产生于操作者方程式的离散性以及大量离散变量。在本文中,我们表明,如果方程式在离散前转变,准确性可以大大提高,这是一个我们称之为完全操作者先决条件的过程。它与迭代方法的传统先决条件有许多相似之处,但关键的是,在操作者一级适用变换。我们表明,虽然传统先决条件中的条件数目改进通常不会提高解决方案的准确性,但FOP可以。数字分析中的一些专题可以被解释为暗含使用FOP;我们强调(一) 切比谢夫在多音近似中互换,以及(二) Olver-Townsend的光谱法,两者都产生比天真的问题配方程式的精确性大得多的解决办法。此外,我们提议以四级差异方程式与定型方程式的整合为基础,FOP可以提高解决方案的精确性。我们强调,由此产生的直线性系统不仅展示了离层的精确性方法,而且还展示了各种离心裁制方法。