An alternating iterative procedure for the Cauchy problem for the Helmholtz equation

References

• Kozlov VA, Maz’ya VG.On iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra i Analiz. 1989;192: 1207–1228 (in Russian)
   Google Scholar
• Colton D, Kress R. Inverse acoustic and electromagnetic scattering theory 2nd ed. Berlin: Springer; 1998
   Google Scholar
• JonesDS. Acoustic and electromagnetic waves. Oxford: Clarendon Press; 1986
   Google Scholar
• Regińska T, Regiński K. Approximate solution of a Cauchy problem for the Helmholtz equation. Inverse Problems. 2006;22:975–989.
   Web of Science ®Google Scholar
• Langrenne C,Garcia A. Data completion method for the characterization of sound sources. J. Acoust. Soc. Am. 2011; 130:2016–2023.
   Google Scholar
• Schuhmacher A, Hald J, Rasmussen KB, Hansen PC. Sound source reconstruction using inverse boundary element calculations. J. Acoust. Soc. Am. 2003;113:114–127.
   PubMed Web of Science ®Google Scholar
• Arendt W, Regińska T. An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains. J. Inv. Ill-Posed Problems. 2009;17:703–711.
   Web of Science ®Google Scholar
• Hadamard J. Lectures on Cauchy’s problem in linear partial differential equations. New York (NY): Dover Publications; 1952.
   Google Scholar
• John F. Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Commun. Pure Appl. Math. 1960;13:551–585.
   Web of Science ®Google Scholar
• Lavrent’ev MM,Romanov VG, Shishatskii SP. Ill-posed problems of mathematical physics and analysis. Providence (RI): American Mathematical Society; 1986.
   Google Scholar
• Sun Y, Zhan D, Ma F. A potential function method for the Cauchy problem for elliptic equation. J. Math. Anal. Appl. 2012;395:164–174.
   Web of Science ®Google Scholar
• Qin HH, Wei T. Modified regularization method for the Cauchy problem of the Helmholtz equation. Appl. Math. Model. 2009;33:2334–2348.
   Web of Science ®Google Scholar
• Qin HH, Wei T, Shi R. Modified Tikhonov regularization method for the Cauchy problems of the Helmholtz equation. J. Comput. Appl. Math. 2009;24:39–53.
   Web of Science ®Google Scholar
• Qin HH, Wei T. Two regularization methods for the Cauchy problems of the Helmholtz equation. Appl. Math. Model. 2010;34:947–967.
   Web of Science ®Google Scholar
• Regińska T, Wakulicz A. Wavelet moment method for the Cauchy problem for the Helmholtz equation. J. Comput. Appl. Math. 2009;223:218–229.
   Web of Science ®Google Scholar
• Marin L, Elliott L, Heggs PJ, Ingham DB, Lesnic D, Wen X. An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation. Comput. Meth. Appl. Mech. Eng. 2003;192:709–722.
   Web of Science ®Google Scholar
• Kozlov VA, Maz’ya VG, Fomin AV. An iterative method for solving the Cauchy problem for elliptic equations. Comput. Maths. Math. Phys. 1991;31:46–52.
   Web of Science ®Google Scholar
• Avdonin S, Kozlov V, Maxwell D, Truffer M. Iterative methods for solving a nonlinear boundary inverse problem in glaciology. J. Inv. Ill-Posed Problems. 2009;17:239–258.
   Web of Science ®Google Scholar
• Bastay G, Johansson T, Kozlov VA, Lesnic D. An alternating method for the stationary Stokes system. Z. Angew. Math. Mech. 2006;86:268–280.
   Web of Science ®Google Scholar
• Comino L, Marin L, Gallego R. An alternating iterative algorithm for the Cauchy problem in anisotropic elasticity. Eng. Anal. Boundary Elements. 2007;31:667–682.
   Web of Science ®Google Scholar
• Chapko R, Johansson BT. An alternating potential-based approach to the Cauchy problem for the Laplace equation in a planar domain with a cut. Comput. Meth. Appli.Math. 2008;8:315–335.
   Google Scholar
• Lesnic D, Elliot L, Ingham DB. An alternating boundary element method for solving numerically the Cauchy problems for the Laplace equation. Eng. Anal. Boundary Elements. 1997;20:123–133.
   Web of Science ®Google Scholar
• Lesnic D, Elliot L, Ingham DB. An alternating boundary element method for solving Cauchy problems for the biharmonic equation. Inverse Probl. Eng. 1997;5:145–168.
   Google Scholar
• Marin L, Elliott L, Ingham DB, Lesnic D. Boundary element method for the Cauchy problem in linear elasticity. Eng. Anal. Boundary Elements. 2001;25:783–793.
   Web of Science ®Google Scholar
• Maxwell D, Truffer M, Avdonin S, Stuefer M. An iterative scheme for determining glacier velocities and stresses. J. Glaciol. 2008;54:888–898.
   Web of Science ®Google Scholar
• Johansson BT, Kozlov VA. An alternating method for Helmholtz-type operators in non-homogeneous medium. IMA J. Appl. Math. 2009;74:62–73.
   Web of Science ®Google Scholar
• Mpinganzima L. An alternating iterative procedure for the Cauchy problem for the Helmholtz equation. Thesis No. 1530, Linköping; 2012.
   Google Scholar
• Lions JL, Magenes E. Non-homogeneous boundary value problems and applications. Berlin: Springer; 1972.
   Google Scholar
• Nečas J. Les Méthodes Directes en Théorie des Equations Elliptiques [Direct methods in the theory of elliptic equations]. Paris: Masson; 1967.
   Google Scholar
• Dahlberg BEJ, Kenig CE. Harmonic analysis and partial differential equations. Technological report. Göteborg: Chalmers University of Technology; 1985 Available from: http://www.math.chalmers.se/Math/Research/GeometryAnalysis/Lecturenotes/HAPE.ps [2009-01-07]
   Google Scholar