Mathematical Sciences: Preconditioning by Fast Transforms

点击翻译按钮获取中文摘要

The analytical problem in this proposal is the relation between a given operator and its preconditioner. Given is a differential or integral or convolution operator with general boundary conditions. The preconditioner has particularly convenient domain and boundary conditions. After discretization it can be inverted by the Fast Fourier Transform, or in the case of the Dirichlet problem by a Fast Sine Transform. In the matrix case one applies the preconditioned conjugate gradient method and the distribution of the eigenvalues is crucial. A small number of extreme eigenvalues is acceptable if all other eigenvalues are clustered - that is what appeared in numerical experiments. The conjecture on clustering depends on establishing that the eigenvectors decay exponentially, away from the boundary. The analysis should connect problems in operator theory to problems in numerical analysis, and may give a new direction in the class of equations for which preconditioning is essential. This research falls into the general area of numerical analysis, numerical computing, computational mathematics and software design. The problem of Toeplitz operators is important in signal processing. The problem of numerical computations associated with Toeplitz matrices arises in many problems of communications engineering and its solution is of considerable importance.