R. Chan and F. Lin, Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line, J. Comp. Math., 14 (1996), 223-236.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(j) 2. 7) where c(B (j) are the corresponding optimal circulant integral operators of B (j) We recall the following two theorems which are useful in the analysis of the spectra of the preconditioned operators (I P ) Gamma1 (I A ) Their proofs can be found in [4] and [2] respectively. THEOREM 2.1. Let K be a convolution integral operator with kernel function k( Delta) 2 L 1 (IR) L 2 (IR) Let c(K ) be the optimal circulant integral operator of K . Then for any given ffl 0, there exists a positive integer N and a 0 such that for all , ....

R. CHAN AND F. LIN, Preconditioned conjugate gradient methods for integral equations of the second kind defined on the half-line, J. Comput. Math., 14 (1996), pp. 223--236.

....proposed by T. Chan in [2] It can be defined for arbitrary matrices. T. Chan s idea of constructing optimal circulant preconditioners has been incorporated in Gohberg, Hanke and Koltracht [10] in developing optimal circulant integral operators for convolution type integral operators. Chan and Lin [5] later extended the idea to develop optimal circulant integral operators for general nonconvolution type integral operators. In this paper, we will concentrate on the use of optimal circulant integral operators for (4) The outline of the paper is as follows. In x2, we show the equivalence of the ....

R. Chan and F. Lin, Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line, J. Comp. Math., 14 (1996), 223-236.

....discrete matrices from these integral equations are Toeplitz matrices if the rectangular quadrature rule is used. For non convolution type integral equations, where the discrete matrices are no longer Toeplitz, convergence analysis of optimal circulant preconditioners has also been studied, see [6, 9]. For example, for boundary integral equations arising from potential equations, which are ill conditioned, non convolution type integral equations with condition number increasing like O(n) the preconditioned systems have been shown to be well conditioned, see Chan, Sun and Ng [9] and also x5. ....

R. Chan and F. Lin, Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line, J. Comput. Math., 14 (1996), 223--236.

....discrete matrices from these integral equations are Toeplitz matrices if the rectangular quadrature rule is used. For non convolution type integral equations, where the discrete matrices are no longer Toeplitz, convergence analysis of optimal circulant preconditioners has also been studied, see [6, 9]. For example, for boundary integral equations arising from potential equations, which are ill conditioned, non convolution type integral equations with condition number increasing like O(n) the preconditioned systems have been shown to be well conditioned, see Chan, Sun and Ng [9] and also x5. ....

R. Chan and F. Lin, Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line, J. Comput. Math., to appear.

....proposed by T. Chan in [2] It can be defined for arbitrary matrices. T. Chan s idea of constructing optimal circulant preconditioners has been incorporated in Gohberg, Hanke and Koltracht [10] in developing optimal circulant integral operators for convolution type integral operators. Chan and Lin [5] later extended the idea to develop optimal circulant integral operators for general nonconvolution type integral operators. In this paper, we will concentrate on the use of optimal circulant integral operators for (4) The outline of the paper is as follows. In x2, we show the equivalence of the ....

R. Chan and F. Lin, Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-Line, J. Comput. Math., to appear.

....function or from approximate convolution identities commonly used in Fourier analysis [190] As in the Toeplitz matrix case, there are other ways of constructing operators as preconditioners for (4.21) see Ng and Lin [150] Ng, Lin, and R. Chan [151] and R. Chan and Lin [41] In R. Chan and Lin [40], optimal and super optimal circulant integral preconditioners are constructed for general integral equations of the second kind, y(t) Z 1 0 a(t; s)y(s)ds = g(t) 0 t 1: Here a(t; s) is not necessarily a convolution kernel. When (4.21) is discretized with the rectangular quadrature rule, ....

R. Chan and F. Lin, Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-line, J. Comput. Math., to appear.

....function or from approximate convolution identities commonly used in Fourier analysis [189] As in the Toeplitz matrix case, there are other ways of constructing operators as preconditioners for (4.21) see Ng and Lin [149] Ng, Lin, and R. Chan [150] and R. Chan and Lin [40] In R. Chan and Lin [39], optimal and super optimal circulant integral preconditioners are constructed for general integral equations of the second kind, y(t) Z 1 0 a(t; s)y(s)ds = g(t) 0 t 1: Here a(t; s) is not necessarily a convolution kernel. When (4.21) is discretized with the rectangular quadrature rule, ....

R. Chan and F. Lin, Preconditioned Conjugate Gradient Methods for Integral Equations of the Second Kind Defined on the Half-line, J. Comp. Math., to appear.

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