On Method of Fundamental Solution for Nondestructive Testing

Nondestructive testing in engineering is an inverse Cauchy problem for Laplace equation. In this paper the problem of nondestructive testing is expressed by a Laplace-s equation with third-kind boundary conditions. In order to find unknown values on the boundary, the method of fundamental solution is introduced and realized. Because of the ill-posedness of studied problems, the TSVD regularization technique in combination with L-curve criteria and Generalized Cross Validation criteria is employed. Numerical results are shown that the TSVD method combined with L-curve criteria is more efficient than the TSVD method combined with GCV criteria. The abstract goes here.

Authors:

• Jieer Wu
• Zheshu Ma

Keywords:

• ill-posed
• TSVD
• Laplace's equation
• inverse problem
• L-curve
• Generalized Cross Validation.

References:

[1] Santosa.F. Applications Of Electrical Impedance Tomography To Nondestructive
Evalution, Preprint, CTC95TR210 Cornell Theory Center.1995.
[2] Dario Fasino, Gabriele Inglese,Discrete Methods In The Study of An
Inverse Problem for Laplace-S Equation, IMA Journat of Numerical
Analysis,vol.19, 1999,pp.105-118.
[3] Inglese,G., Santosa,F. An inverse problem in corrosion detection. Pubblicazioni
dell-Istituto di Analisi. Globale ed Applicazioni del CNR Firenze
no 75,1995.
[4] Inglese,G. An inverse problem in corrosion detection. Inv. Prob. vol.13,
1997, pp. 977-994.
[5] Bryan,K., Caudill,L.F., An inverse problem in themal imaging. SIAM J.
Appl.Math.,vol.56, 1996 , pp.715-735.
[6] J.Cheng and M.Yamamoto, Unique continuation on a line for harmonic
functions. Inverse Problems, vol. 14, 1998, pp.869-882.
[7] J.Cheng and M.Yamamoto, Unique continuation on a line for harmonic
functions. Inverse Problems, vol. 14, 1998, pp.869-882.
[8] P.C.Hansen, Analysis Of Discrete Ill-Posed Problems By Means Of The
L-Curve, Siam Review vol.34, 1992, pp.561-580.
[9] Tikhonov, AN, and Arsenin VY, Solution of ill-posed problems, Wiley,
New York,1977.
[10] Hansen P C, Truncated SVD solutions to discrete ill posed problems
with ill-determined numerical rank, SIAM J Sci Statist Comput , vol.11,
1990, pp.503-518.
[11] G.H.Golub, M.T.Heath, and G.Wahba, Generalized Cross-Validation As
A Method For Choosing A Good Ridge Parameter, Technometrics, vol.21,
1979,pp215-223.
[12] Hansen P C, Analysis of discrete ill-posed problems by means of the
l-curve, SIAM Review, vol. 34 , 1992,pp. 561-580.
[13] A. Bjork, E. Grimme, P. Van Dooren, An implicit shift bidiagonalization
algorithm for ill-posed problems, BIT vol.34 ,