Full operator preconditioning and the accuracy of solving linear systems

Abstract

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.

Complex moment-based methods for differential eigenvalue problems

“…For both problems, we set (L, M, N ) = (2,4,24) for the proposed methods and (L, N ) = (6, 24) for contFEAST. Table 4.4 gives the obtained eigenvalues of contSS-RR for the 2D Laplace eigenvalue problem.…”

“…Chebfun enables highly adaptive computation with operators and functions in the same manner as matrices and functions. This paradigm has extended numerical linear algebra techniques in finite dimensional spaces to infinite-dimensional spaces [2,32,25,31,5,24]. Under the circumstances, an operator analogue of FEAST was recently developed [8] for solving (1.1) and dealing with operators A and B without their discretization 1 .…”

Complex moment-based methods for differential eigenvalue problems

“…The orthonormalization of the columns of S (ν) 0 in each iteration may improve the numerical stability. Now, we analyze the error bound of the proposed methods with the subspace iteration technique as introduced in ( 13) and (14). We assume that B is invertible and all the eigenvalues are isolated.…”

“…, u LM } by P ( ) and P LM , respectively. Assume that P LM S (0) has full rank, where S 0 is defined in (14). Then, for each eigenfunction u i , i = 1, 2, .…”

“…Chebfun enables highly adaptive computation with operators and functions in the same manner as matrices and functions. This paradigm has extended numerical linear algebra techniques in finite dimensional spaces to infinite-dimensional spaces [9][10][11][12][13][14]. Under the circumstances, an operator analogue of FEAST was recently developed [15] for solving (1) and dealing with operators A and B without their discretization.…”