## THERMAL SCIENCE

International Scientific Journal

### A REGULARIZED GMRES METHOD FOR INVERSE BLACKBODY RADIATION PROBLEM

**ABSTRACT**

The inverse blackbody radiation problem is focused on determining temperature distribution of a blackbody from measured total radiated power spectrum. This problem consists of solving a first kind of Fredholm integral equation and many numerical methods have been proposed. In this paper, a regularized GMRES method is presented to solve the linear ill-posed problem caused by the discretization of such an integral equation. This method projects the orignal problem onto a lower dimensional subspaces by the Arnoldi process. Tikhonov regularization combined with GCV criterion is applied to stabilize the numerical iteration process. Three numerical examples indicate the effectiveness of the regularized GMRES method.

**KEYWORDS**

PAPER SUBMITTED: 2011-03-16

PAPER REVISED: 2012-04-19

PAPER ACCEPTED: 1970-01-01

**THERMAL SCIENCE** YEAR

**2013**, VOLUME

**17**, ISSUE

**3**, PAGES [847 - 852]

- Tikhonov A.N., Arsenin U Y, Solutions of Ill-posed Problems, New York, Wiley, 1977
- Bojarski N.N., Inverse black body radiation, IEEE Trans. Antennas and Propagation, 30(1982), 4, pp.778-780
- Chen Nanxian, Li Guangyin, Theoretical Investigation on the Inverse Black Body Radiation Problem, IEEE Trans. Antennas and Propagation, 38(1990), 8,pp.1287-1290
- Dai Xianxi, Dai Jiqiong, On Unique Existence Theorem and Exact Solution Formula af the Inverse Black-Body Radiation Problem, IEEE Trans. Antennas and Propagation, 40(1992), 3,pp.257-260
- Sun Xiaoguang, Jaggard D. L., The Inverse Blackbody Radiation Problem: A Regularization Solution, J. Appl. Phys. 62(1987), 11, pp.4382-4386
- Dou Lixin, Hodgson R J W. Application of the Regularization Methods to the Inverse Black Body Radiation Problem, IEEE Trans. Antennas and Propagation, 40(1992), 10, pp.1249-1253
- Cuihuan Li,Tingyan Xiao, Blackbody Radiation Numerical Inversion of the Fast and Stable Algorithm, Chinese Journal Of Computational Physics, 19(2002), pp.121-126
- Hung Yi Li, Solution of Inverse Blackbody Radiation Problem with Conjugate Gradient Method, IEEE Trans. Antennas and Propagation, 53(2005), 5, pp.1840-1842
- Lixin Dou,R. J. W. Hodgson., Maximum Entropy Method in Inverse Black Body Radiation Problem, J. Appl. Phys, 71(1992), 7, pp.3159-3163
- JiPing Ye, The Black-Body Radiation Inversion Problem, Its Instability and A New Universal Function Set Method, Physics Letters A, 348(2006), pp.141-146
- Wujieer,Dai Hua, Regularizing Lanczos Method for Inverse BIack Body Radiation Problem, Journal of Nanjing University of Aeronautics & Astronautics, 39(2007), 2,pp.267-272
- Saad Y., Schultz M., GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM J. Sci. Statist. Comput., 7(1986), pp.856-869
- T. K. Jensen,P. C. Hansen, Iterative Regularization With Minimum-Residual Methods, BIT Numerical Mathematics, 47(2007), pp.103-120
- D.Calvetti,B.Lewis,and L.Reichel, On the Regularizing Properties of the GMRES Method, Numerische Mathematik, 91(2002),pp.605-625
- D.Calvetti,B.Lewis, L.Reichel, GMRES,L-curves and Discrete Ill-posed Problems, BIT, 42(2002), pp.44-65
- Golub G. H.,Heath M. T., Wahba G., Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter , Technometrics, 21(1979),pp.215-223