Fast algorithms for problems on thermal tomography

Raymond H. CHAN, Chun-pong CHEUNG, Hai-wei SUN

Research output: Book Chapters | Papers in Conference ProceedingsConference paper (refereed)Researchpeer-review

3 Citations (Scopus)


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 languageEnglish
Title of host publicationNumerical Analysis and Its Applications: 1st International Workshop, WNAA 1996, Proceedings
PublisherSpringer Berlin Heidelberg
Number of pages8
ISBN (Electronic)9783540683261
ISBN (Print)9783540625988
Publication statusPublished - 1997
Externally publishedYes
Event1st International Workshop on Numerical Analysis and its Applications, WNAA 1996 - Rousse, Bulgaria
Duration: 24 Jun 199626 Jun 1996

Publication series

NameLecture Notes in Computer Science
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference1st International Workshop on Numerical Analysis and its Applications, WNAA 1996

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1997.


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


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