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  • 1
    Electronic Resource
    Electronic Resource
    The instability of some gradient methods for ill-posed problems (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 129-134 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65J10 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary For the solution of linear ill-posed problems some gradient methods like conjugate gradients and steepest descent have been examined previously in the literature. It is shown that even though these methods converge in the case of exact data their instability makes it impossible to base a-priori parameter choice regularization methods upon them.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385614
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    Paper (German National Licenses)
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  • 2
    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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  • 3
    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)
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  • 4
    Book
    Book
    Numerische Mathematik kompakt : Grundlagenwissen für Studium und Praxis (2000)
    Braunschweig u.a. :Vieweg,
    Details
    Title: Numerische Mathematik kompakt : Grundlagenwissen für Studium und Praxis
    Author: Plato, Robert
    Publisher: Braunschweig u.a. :Vieweg,
    Year of publication: 2000
    Pages: 360 S.
    Type of Medium: Book
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