# Fast algorithms for problems on thermal tomography

Research output: Chapters, Conference Papers, Creative and Literary Works (RGC: 12, 32, 41, 45)32_Refereed conference paper (with ISBN/ISSN)peer-review

3 Scopus Citations

## Detail(s)

Original language English Numerical Analysis and Its Applications Proceedings Lubin Vulkov, Jerzy Waśniewski, Plamen Yalamov Springer-Verlag Berlin Heidelberg 90-97 9783540683261 9783540625988 Published - 1997 Yes

### Publication series

Name Lecture Notes in Computer Science 1196 0302-9743 1611-3349

### Conference

Title 1st International Workshop on Numerical Analysis and its Applications, WNAA 1996 Bulgaria Rousse 24 - 26 June 1996

## 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.

## Citation Format(s)

[ RIS ] [ BibTeX ]

Fast algorithms for problems on thermal tomography. / Chan, Raymond H.; Cheung, Chun-pong; Sun, Hai-wei.

Numerical Analysis and Its Applications: Proceedings. ed. / Lubin Vulkov; Jerzy Waśniewski; Plamen Yalamov. Springer-Verlag Berlin Heidelberg, 1997. p. 90-97 (Lecture Notes in Computer Science; Vol. 1196).

Research output: Chapters, Conference Papers, Creative and Literary Works (RGC: 12, 32, 41, 45)32_Refereed conference paper (with ISBN/ISSN)peer-review