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  • 1
    Electronic Resource
    Electronic Resource
    On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations (1996)
    Springer
    Numerische Mathematik 75 (1996), S. 99-120 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65J20, 65R30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle (being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev. A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050232
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    The conjugate gradient method for linear ill-posed problems with operator perturbations (1999)
    Springer
    Numerical algorithms 20 (1999), S. 1-22 
    Details
    ISSN: 1572-9265
    Keywords: ill-posed problems ; first kind integral equations ; conjugate gradient-type methods ; minimal error method ; regularization schemes ; discrepancy principle ; parameter estimation problems ; 65J20 ; 65R30 ; 45E99 ; 65N38
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider an ill-posed problem Ta = f* in Hilbert spaces and suppose that the linear bounded operator T is approximately available, with a known estimate for the operator perturbation at the solution. As a numerical scheme the CGNR-method is considered, that is, the classical method of conjugate gradients by Hestenes and Stiefel applied to the associated normal equations. Two a posteriori stopping rules are introduced, and convergence results are provided for the corresponding approximations, respectively. As a specific application, a parameter estimation problem is considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019139414435
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    Paper (German National Licenses)
    Fulltext
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