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Conjugate Gradient Type Methods for Ill-Posed Problems

Martin Hanke
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A01=Martin Hanke
Author_Martin Hanke
Banach Steinhaus Theorem
Category=PBKJ
Compact Operator
Conjugate Gradient
Conjugate Gradient Method
Conjugate Gradient Type Methods
Discrepancy Principle
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Equidistant Mesh
Finite Precision Arithmetic
Fixed Regularization Parameter
Fredholm Integral Equations
Hilbert Schmidt Operators
Ill Posed Problems
Invariant Subspace
Iterative Regularization Methods
Jacobi Polynomials
Kernel Polynomials
Krylov Subspace
Krylov Subspace Method
Mr Iterate
Noise Samples
Orthogonal Polynomials
Point Spread Function
Posteriori Error Estimate
Stopping Rule
Tikhonov Regularization

Product details

  • ISBN 9780582273702
  • Weight: 430g
  • Dimensions: 174 x 246mm
  • Publication Date: 26 Apr 1995
  • Publisher: Taylor & Francis Ltd
  • Publication City/Country: GB
  • Product Form: Hardback
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Description
The conjugate gradient method is a powerful tool for the iterative solution of self-adjoint operator equations in Hilbert space.This volume summarizes and extends the developments of the past decade concerning the applicability of the conjugate gradient method (and some of its variants) to ill posed problems and their regularization. Such problems occur in applications from almost all natural and technical sciences, including astronomical and geophysical imaging, signal analysis, computerized tomography, inverse heat transfer problems, and many more This Research Note presents a unifying analysis of an entire family of conjugate gradient type methods. Most of the results are as yet unpublished, or obscured in the Russian literature. Beginning with the original results by Nemirovskii and others for minimal residual type methods, equally sharp convergence results are then derived with a different technique for the classical Hestenes-Stiefel algorithm. In the final chapter some of these results are extended to selfadjoint indefinite operator equations. The main tool for the analysis is the connection of conjugate gradient type methods to real orthogonal polynomials, and elementary properties of these polynomials. These prerequisites are provided in a first chapter. Applications to image reconstruction and inverse heat transfer problems are pointed out, and exemplarily numerical results are shown for these applications.
About Martin Hanke
Hanke\, Martin

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