Preconditioning of elliptic problems by approximation in the transform domain |
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Authors: | Michael K Ng |
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Institution: | (1) Computer Sciences Laboratory, Research School of Information Sciences and Engineering, The Australian National University, 0200 Canberra, ACT, Australia;(2) Present address: Department of Mathematics, University of Hong Kong, Pokufulam Road, Hong Kong |
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Abstract: | Preconditioned conjugate gradient method is applied for solving linear systemsAx=b where the matrixA is the discretization matrix of second-order elliptic operators. In this paper, we consider the construction of the trnasform
based preconditioner from the viewpoint of image compression. Given a smooth image, a major portion of the energy is concentrated
in the low frequency regions after image transformation. We can view the matrixA as an image and construct the transform based preconditioner by using the low frequency components of the transformed matrix.
It is our hope that the smooth coefficients of the given elliptic operator can be approximated well by the low-rank matrix.
Numerical results are reported to show the effectiveness of the preconditioning strategy. Some theoretical results about the
properties of our proposed preconditioners and the condition number of the preconditioned matrices are discussed. |
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Keywords: | 65F10 65N22 |
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