Fast reconstruction of harmonic functions from Cauchy data using the Dirichlet-to-Neumann map and integral equations

References

• Clerc, M, and Kybic, J, 2007. Cortical mapping by Laplace Cauchy transmission using a boundary element method, Inverse Probl. 12 (2007), pp. 2589–2601.
   Google Scholar
• Hasanov, A, 2009. Identification of an unknown source term in a vibrating cantilevered beam from final overdetermination, Inverse Probl. 25 (2009), 115015 (19pp).
   PubMed Web of Science ®Google Scholar
• Lavrentiev, MM, 1967. Some Improperly Posed Problems of Mathematical Physics. Berlin: Springer Verlag; 1967.
   Google Scholar
• Tarchanov, N, 1995. The Cauchy Problem for Solutions of Elliptic Equations. Berlin: Akad. Verlag; 1995.
   Google Scholar
• Yang, X, Choulli, M, and Cheng, J, 2005. An iterative BEM for the inverse problem of detecting corrosion in a pipe, Numer. Math. J. Chin. Univ. (Engl. Ser.) 14 (2005), pp. 252–266.
   Google Scholar
• Calderón, A-P, 1958. Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), pp. 16–36.
   Web of Science ®Google Scholar
• Carleman, T, 1939. Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes (French), Ark. Mat., Astr. Fys. 26 (1939), pp. 1–9.
   Google Scholar
• Cao, H, Klibanov, MV, and Pereverzev, SV, 2009. A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation, Inverse Probl. 25 (2009), pp. 1–21.
   Web of Science ®Google Scholar
• Kozlov, VA, and Maz'ya, VG, 1989. On iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra Anal. 1 (1989), pp. 144–170, (English transl.: Leningrad Math. J. 1 (1990), pp. 1207–1228).
   Google Scholar
• Kozlov, VA, Maz'ya, VG, and Fomin, AV, 1991. An iterative method for solving the Cauchy problem for elliptic equations, Zh. Vychisl. Mat. i Mat. Fiz. 31 (1991), pp. 64–74, (English transl.: U.S.S.R. Comput. Math. and Math. Phys. 31 (1991), pp. 45–52).
   Web of Science ®Google Scholar
• Avdonin, S, Kozlov, V, Maxwell, D, and Truffer, M, 2009. Iterative methods for solving a nonlinear boundary inverse problem in glaciology, J. Inverse Ill-Posed Probl. 17 (2009), pp. 239–258.
   Web of Science ®Google Scholar
• Bastay, G, Johansson, T, Kozlov, V, and Lesnic, D, 2006. An alternating method for the stationary Stokes system, ZAMM 86 (2006), pp. 268–280.
   Web of Science ®Google Scholar
• Baumeister, J, and Leitão, A, 2001. On iterative methods for solving ill-posed problems modeled by partial differential equations, J. Inverse Ill-Posed Probl. 9 (2001), pp. 13–29.
   Google Scholar
• Chapko, R, and Johansson, BT, 2008. An alternating boundary integral based method for a Cauchy problem for Laplace equation in semi-infinite domains, Inverse Probl. Imag. 3 (2008), pp. 317–333.
   Google Scholar
• Dinh Nho, Hào, and Reinhardt, HJ, 1998. Gradient methods for inverse heat conduction problems, Inverse Probl. Eng. 6 (1998), pp. 177–211.
   Google Scholar
• Engl, HW, and Leitão, A, 2001. A Mann iterative regularization method for elliptic Cauchy problems, Numer. Funct. Anal. Optim. 22 (2001), pp. 861–884.
   Web of Science ®Google Scholar
• Johansson, BT, and Lesnic, D, 2007. "A relaxation of the alternating method for the reconstruction of a stationary flow from incomplete boundary data". In: Power, H, La Rocca, A, and Baxter, SJ, eds. Advances in Boundary Integral Methods – Proceedings of the Seventh UK Conference on Boundary Integral Methods. UK: University of Nottingham; 2007. pp. 161–169.
   Google Scholar
• Jourhmane, M, and Nachaoui, A, 1999. An alternating method for an inverse Cauchy problem, Numer. Algorithms 21 (1999), pp. 247–260.
   Web of Science ®Google Scholar
• Jourhmane, M, and Nachaoui, A, 2002. Convergence of an alternating method to solve the Cauchy problem for Poission's equation, Appl. Anal. 81 (2002), pp. 1065–1083.
   Google Scholar
• Jourhmane, M, Lesnic, D, and Mera, NS, 2004. Relaxation procedures for an iterative algorithm for solving the Cauchy problem for the Laplace equation, Eng. Anal. Boundary Elements 28 (2004), pp. 655–665.
   Web of Science ®Google Scholar
• Lesnic, D, Elliott, L, and Ingham, DB, 1997. An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation, Eng. Anal. Boundary Elements 20 (1997), pp. 123–133.
   Web of Science ®Google Scholar
• Marin, L, Elliott, L, Ingham, DB, and Lesnic, D, 2001. Boundary element method for the Cauchy problem in linear elasticity, Eng. Anal. Boundary Elements 25 (2001), pp. 783–793.
   Web of Science ®Google Scholar
• Maxwell, D, Truffer, M, Avdonin, S, and Stuefer, M, 2008. Determining glacier velocities and stresses with inverse methods: an iterative scheme, J. Glaciol. 54 (2008), pp. 888–898.
   Web of Science ®Google Scholar
• Mera, NS, Elliott, L, Ingham, DB, and Lesnic, D, 2000. The boundary element solution of the Cauchy steady heat conduction problem in an anisotropic medium, Int. J. Numer. Method Eng. 49 (2000), pp. 481–499.
   Web of Science ®Google Scholar
• Dinh Nho, Hào, and Lesnic, D, 2000. The Cauchy problem for Laplace's equation via the conjugate gradient method, IMA J. Appl. Math. 65 (2000), pp. 199–217.
   Web of Science ®Google Scholar
• Dinh Nho, Hào, Johansson, BT, Lesnic, D, and Pham Minh, Hien, 2010. A variational method and approximations of a Cauchy problem for elliptic equations, J. Algorithms Comput. Technol. 4 (2010), pp. 89–119.
   Google Scholar
• Helsing, J, and Johansson, BT, 2010. Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques, Inverse Probl. Sci. Eng. 18 (2010), pp. 381–399.
   Web of Science ®Google Scholar
• Helsing, J, 2009. Faster convergence and higher accuracy for the Dirichlet–Neumann map, J. Comput. Phys. 228 (2009), pp. 2578–2576.
   Web of Science ®Google Scholar
• Helsing, J, 2009. Integral equation methods for elliptic problems with boundary conditions of mixed type, J. Comput. Phys. 228 (2009), pp. 8892–8907.
   Web of Science ®Google Scholar
• Zeb, A, Elliott, L, Ingham, DB, and Lesnic, D, 1997. "Solution of the Cauchy problem for Laplace equation". In: Elliott, L, Ingham, DB, and Lesnic, D, eds. First UK Conference on Boundary Integral Methods. Leeds: Leeds University Press; 1997. pp. 297–307.
   Google Scholar
• Lions, J-L, and Magenes, E, 1972. Non-Homogeneous Boundary Value Problems and Applications, Vol. I. (Translated from the French, Die Grundlehren der Mathematischen Wissenschaften, Band 181). New York: Springer-Verlag; 1972.
   Google Scholar
• Agoshkow, VI, 1988. "Poincaré–Steklov's operators and domain decomposition methods in finite dimensional spaces". In: Glowinski, R, et al., eds. First International Symposium on Domain Decomposition Methods for Partial Differential Equations. Philadelphia: SIAM; 1988. pp. 73–112.
   Google Scholar
• Schmidt, G, 1994. Boundary element discretization of Poincaré–Steklov operators, Numer. Math. 69 (1994), pp. 83–101.
   Web of Science ®Google Scholar