## Abstract

In this paper, we study an ill-posed, nonlinear inverse problem in heat conduction and hydrology applications. In [2], the problem is linearized to give a linear integral equation, which is then solved by the Tikhonov method with the identity as the regularization operator. We prove in this paper that the resulting equation is well-condition and has clustered spectrum. Hence if the conjugate gradient method is used to solve the equation, we expect superlinear convergence. However, we note that the identity operator does not give good solution to the original equation in general. Therefore in this paper, we use the Laplacian operator as the regularization operator instead. With the Laplacian operator, the regularized equation is ill-conditioned and hence a preconditioner is required to speed up the convergence rate if the equation is solved by the conjugate gradient method. We here propose to use the Laplacian operator itself as preconditioner. This preconditioner can be inverted easily by fast sine-transforms and we prove that the resulting preconditioned system is well-conditioned and has clustered spectrum too. Hence the conjugate gradient method converges superlinearly for the preconditioned system. Numerical results are given to illustrate the fast convergence.

Original language | English |
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Title of host publication | Numerical Analysis and Its Applications: 1st International Workshop, WNAA 1996, Proceedings |

Editors | Lubin VULKOV, Plamen YALAMOV, Jerzy WAGNIEWSKI |

Publisher | Springer Berlin Heidelberg |

Pages | 90-97 |

Number of pages | 8 |

ISBN (Electronic) | 9783540683261 |

ISBN (Print) | 9783540625988 |

DOIs | |

Publication status | Published - 1997 |

Externally published | Yes |

Event | 1st International Workshop on Numerical Analysis and its Applications, WNAA 1996 - Rousse, Bulgaria Duration: 24 Jun 1996 → 26 Jun 1996 |

### Publication series

Name | Lecture Notes in Computer Science |
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Volume | 1196 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 1st International Workshop on Numerical Analysis and its Applications, WNAA 1996 |
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Country/Territory | Bulgaria |

City | Rousse |

Period | 24/06/96 → 26/06/96 |

### Bibliographical note

Publisher Copyright:© Springer-Verlag Berlin Heidelberg 1997.

## Keywords

- Convergence Rate
- Laplacian Operator
- Identity Operator
- Conjugate Gradient Method
- Regularization Method