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  • 1
    Electronic Resource
    Electronic Resource
    Circulant preconditioners for second order hyperbolic equations (1992)
    Springer
    BIT 32 (1992), S. 650-664 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 65N22 ; Hyperbolic equation ; circulant matrix ; condition number ; preconditioned conjugate gradient method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations in two dimensions are considered. We propose and analyze the use of circulant preconditioners for the solution of linear systems via preconditioned iterative methods such as the conjugate gradient method. Our motivation is to exploit the fast inversion of circulant systems with the Fast Fourier Transform (FFT). For second-order hyperbolic equations with initial and Dirichlet boundary conditions, we prove that the condition number of the preconditioned system is ofO(α) orO(m), where α is the quotient between the time and space steps andm is the number of interior gridpoints in each direction. The results are extended to parabolic equations. Numerical experiments also indicate that the preconditioned systems exhibit favorable clustering of eigenvalues that leads to a fast convergence rate.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01994848
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  • 2
    Electronic Resource
    Electronic Resource
    Circulant and skew-circulant preconditioners for skew-hermitian type Toeplitz systems (1991)
    Springer
    BIT 31 (1991), S. 632-646 
    Details
    ISSN: 1572-9125
    Keywords: 65F10 ; 65F15 ; Skew-Hermitian type Toeplitz matrix ; circulant matrix ; skew-circulant matrix ; preconditioned conjugate gradient method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the solutions of Toeplitz systemsA n x=b by the preconditioned conjugate gradient method. Then ×n matrixA n is of the forma 0 I+H n wherea 0 is a real number,I is the identity matrix andH n is a skew-Hermitian Toeplitz matrix. Such matrices often appear in solving discretized hyperbolic differential equations. The preconditioners we considered here are the circulant matrixC n and the skew-circulant matrixS n whereA n =1/2(C n +S n ). The convergence rate of the iterative method depends on the distribution of the singular values of the matricesC −1 n An andS −1 n A n . For Toeplitz matricesA n with entries which are Fourier coefficients of functions in the Wiener class, we show the invertibility ofC n andS n and prove that the singular values ofC −1 n A n andS −1 n A n are clustered around 1 for largen. Hence, if the conjugate gradient method is applied to solve the preconditioned systems, we expect fast convergence.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01933178
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