Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-10T05:28:26.084Z Has data issue: false hasContentIssue false
  • English
  • Français

An Inverse Diffraction Problem: Shape Reconstruction

Part of: Numerical analysis in abstract spaces

Published online by Cambridge University Press:  10 November 2015

Yanfeng Kong ,
Zhenping Li  and
Xiangtuan Xiong
Yanfeng Kong
Affiliation:
Department of Mathematics, Northwest Normal University, Gansu, China
Zhenping Li
Affiliation:
Department of Mathematics, Luoyang Institute of Science and Technology, Henan, China
Xiangtuan Xiong*
Affiliation:
Department of Mathematics, Northwest Normal University, Gansu, China
*
*Corresponding author. Email address:xiongxt@gmail.com (X. Xiong)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

An inverse diffraction problem is considered. Both classical Tikhonov regularisation and a slow-evolution-from-the-continuation-boundary (SECB) method are used to solve the ill-posed problem. Regularisation error estimates for the two methods are compared, and the SECB method is seen to be an improvement on the classical Tikhonov method. Two numerical examples demonstrate their feasibility and efficiency.

Keywords

Inverse diffraction problemill-posed problemsTikhonov regularisationstability estimateerror estimateSECB

MSC classification

Secondary: 65J20: Improperly posed problems; regularization

Information

Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 4 , November 2015 , pp. 342 - 360
Copyright
Copyright © Global-Science Press 2015 

References

[1]Sondhi, M., Reconstruction of objects from their sound-diffraction patterns, J. Acoust. Soc. Am. 46, 1158–1164 (1969).Google Scholar
[2]Bertero, M. and De, C. M., Stability problems in inverse diffraction, IEEE Trans. Antennas and Propagation AP–29, 368–372 (1981).Google Scholar
[3]Magnanini, R. and Papi, G., An inverse problem for the Helmholtz equation, Inverse Problems 1, 357–370 (1985).CrossRefGoogle Scholar
[4]Engl, H. W., Hanke, M. and Neubauer, A., Regularisation of Inverse Problems, Kluwer Academic, Dordrecht/Boston/London (1996).CrossRefGoogle Scholar
[5]Sun, Y., Zhan, D., Ma, F., A potential function method for the Cauchy problem for elliptic operators, J. Math. Anal. Appl. 395, 164–174 (2012).Google Scholar
[6]Qin, H. H., Wei, T. and Shi, R., Modified Tikhonov regularisation method for the Cauchy problems of the Helmholtz equation, J. Comp. Appl. Math. 224, 39–53 (2009).CrossRefGoogle Scholar
[7]Xiong, X. T. and Fu, C. L., Two approximate methods of a Cauchy problem for the Helmholtz equation, Comp. Appl. Math. 26, 285–307 (2007).CrossRefGoogle Scholar
[8]Feng, X. L., Fu, C. L. and Cheng, H., A regularisation method for solving the Cauchy problem for the Helmholtz equation, Appl. Math. Mod. 35, 3301–3315 (2011).CrossRefGoogle Scholar
[9]Qin, H. H. and Wei, T., Two regularisation methods for the Cauchy problems of the Helmholtz equation, Appl. Math. Mod. 34, 947–967 (2010).Google Scholar
[10]Xiong, X. T., A regularisation method for a Cauchy problem of Helmholtz equation, J. Comp. Appl. Math. 233, 1723–1732 (2010).CrossRefGoogle Scholar
[11]Regińska, T. and Regiński, K., Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Problems 22, 975–989 (2006).CrossRefGoogle Scholar
[12]Regińska, T. and Wakulicz, A., Wavelet moment method for the Cauchy problem for the Helmholtz equation, J. Comp. Appl. Math. 223, 218–229 (2009).Google Scholar
[13]Wei, T., Hon, Y. C. and Ling, L., Method of fundamental solutions with regularisation techniques for Cauchy problems of elliptic operators, Eng. Anal. Bound. Elem. 31, 373–385 (2007).CrossRefGoogle Scholar
[14]Marin, L., Boundary element-minimal error method for the Cauchy problem associated with Helmholtz-type equations, Comp. Mech. 44, 205–219 (2009).CrossRefGoogle Scholar
[15]Marin, L., Elliott, L., Heggs, P. J., Ingham, D. B., Lesnic, D. and Wen, X., Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations, Comp. Mech. 31, 367–377 (2003).CrossRefGoogle Scholar
[16]Marin, L., Elliott, L., Heggs, P. J., Ingham, D. B., Lesnic, D. and Wen, X., An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comp. Meth. Appl. Mech. Eng. 192, 709–722 (2003).CrossRefGoogle Scholar
[17]Kozlov, V. A., Maz'ya, V. G. and Fomin, A. V., An iterative method for solving the Cauchy problem for elliptic equations, Comp. Math. Math. Phys. 31, 46–52 (1991).Google Scholar
[18]Berntsson, F., Kozlov, V. A., Mpinganzima, L. and Turesson, B. O., An alternating iterative procedure for the Cauchy problem for the Helmholtz equation, Inverse Problems Sci. Eng. 22, 45–62 (2014).CrossRefGoogle Scholar
[19]Hohage, T., Regularisation of exponentially ill-posed problems, Num. Funct. Anal. Optim. 21, 439–464 (2000).Google Scholar
[20]Tautenhahn, U., Optimality for linear ill-posed problems under general source conditions, Num. Funct. Anal. Optim. 19, 377–398 (1998).Google Scholar
[21]Tautenhahn, U., Optimal stable approximations for the sideways heat equation, J. Inv. Ill-Posed Problems 5, 287–307 (1997).Google Scholar
[22]Tautenhahn, U., Optimal stable solution of Cauchy problems for elliptic equations, Z. Anal. Anw. 15, 961–984 (1996).Google Scholar
[23]Tautenhahn, U. and Gorenflo, R., On optimal regularisation methods for fractional differentiation, Z. Anal. Anw. 18, 449–467 (1999).CrossRefGoogle Scholar
[24]Tautenhahn, U. and Schröter, T., On optimal regularisation methods for the backward heat equation, Z. Anal. Anw. 15, 475–493 (1996).Google Scholar
[25]Nair, M. T. and Tautenhahn, U., Lavrentiev regularisation for linear ill-posed problems under general source conditions, Z. Anal. Anw. 23, 167–185 (2004).Google Scholar
[26]Nair, M. T., Linear operator equations: Approximation and Regularisation, World Scientific, Singapore (2009).Google Scholar
[27]Carasso, A. S., Overcoming Hölder continuity in ill-posed continuation problems, SIAM J. Num. Anal. 31, 1535–1557 (1994).Google Scholar
[28]Carasso, A. S., Error bounds in nonsmooth image deblurring, SIAM J. Math. Anal. 28, 656–668 (1997).Google Scholar
[29]Carasso, A. S., Logarithmic convexity and the “slow evolution” constraint in ill-posed initial value problems, SIAM J. Math. Anal. 30, 479–496 (1999).Google Scholar
[30]Carasso, A. S., Linear and nonlinear image deblurring: A documented study, SIAM J. Num. Anal. 36, 1659–1689 (1999).Google Scholar
[31]Carasso, A. S., Direct blind deconvolution, SIAM J. Appl. Math. 61, 1980–2007 (2001).CrossRefGoogle Scholar
[32]Carasso, A. S., Bochner subordination, Logarithmic diffusion equations, and blind deconvolution of Hubble space telescope imagery and other scientific data, SIAM J. Imaging Sc. 3, 954–980 (2010).Google Scholar
[33]Lee, J. and Sheen, D., F John's stability conditions versus A Carasso's SECB constraint for backward parabolic problems, Inverse Problems 25, 055001 (2009).Google Scholar
[34]Cheng, J. and Yamamoto, M., One new strategy for a priori choice of regularizing parameters in Tikhonov's regularisation, Inverse Problems 16, L31–L38 (2000).Google Scholar
[35]Hofmann, B., Mathé, P. and Schieck, M., Modulus of continuity for conditionally stable ill-posed problems in Hilbert apace, J. Inv. Ill-posed Problems 16, 567–585 (2008).Google Scholar
[36]Kabanikhin, S. I. and Schieck, M., Impact of conditional stability: Convergence rates for general linear regularisation methods, J. Inv. Ill-posed Problems 16, 267–282 (2008).Google Scholar
[37]Schieck, M., Modulus of continuity and conditional stability for linear regularization schemes, J. Inv. Ill-posed Problems 17, 85–89 (2009).Google Scholar
[38]Tikhonov, A. N. and Arsenin, V. Y., Solutions of Ill-Posed Problems, Winston and Sons, Washington (1977).Google Scholar