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Volume 14, Issue 3
Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line

Raymond H. Chan & F. R. Lin

J. Comp. Math., 14 (1996), pp. 223-236.

Published online: 1996-06

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  • Abstract

We consider solving integral equations of the second kind defined on the half-line $[0,\infty)$ by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the associated integral operator. In this paper, we construct two different circulant integral operators to be used as preconditioners for the method to speed up its convergence rate. We prove that if the given integral operator is close to a convolution-type integral operator, then the preconditioned systems will have spectrum clustered around 1 and hence the preconditioned conjugate gradient method will converge superlinearly. Numerical examples are given to illustrate the fast convergence.

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@Article{JCM-14-223, author = {Raymond H. Chan and F. R. Lin}, title = {Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line}, journal = {Journal of Computational Mathematics}, year = {1996}, volume = {14}, number = {3}, pages = {223--236}, abstract = {

We consider solving integral equations of the second kind defined on the half-line $[0,\infty)$ by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the associated integral operator. In this paper, we construct two different circulant integral operators to be used as preconditioners for the method to speed up its convergence rate. We prove that if the given integral operator is close to a convolution-type integral operator, then the preconditioned systems will have spectrum clustered around 1 and hence the preconditioned conjugate gradient method will converge superlinearly. Numerical examples are given to illustrate the fast convergence.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9233.html} }
TY - JOUR T1 - Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line AU - Raymond H. Chan & F. R. Lin JO - Journal of Computational Mathematics VL - 3 SP - 223 EP - 236 PY - 1996 DA - 1996/06 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9233.html KW - AB -

We consider solving integral equations of the second kind defined on the half-line $[0,\infty)$ by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the associated integral operator. In this paper, we construct two different circulant integral operators to be used as preconditioners for the method to speed up its convergence rate. We prove that if the given integral operator is close to a convolution-type integral operator, then the preconditioned systems will have spectrum clustered around 1 and hence the preconditioned conjugate gradient method will converge superlinearly. Numerical examples are given to illustrate the fast convergence.

Raymond H. Chan and F. R. Lin. (1996). Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line. Journal of Computational Mathematics. 14 (3). 223-236. doi:
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