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Articles

Some new results for geometric inverse problems with the method of fundamental solutions

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Pages 131-152 | Received 17 Jul 2019, Accepted 05 Jun 2020, Published online: 27 Jun 2020

References

  • Ophir J, Céspedes I, Ponnekanti H. Elastography: A quantitative method for imaging the elasticity of biological tissues. Ultrason Imaging. 1991;13(2):111–134. doi:10.1177/016173469101300201.
  • Isakov V. Inverse problems for partial differential equations. 3rd ed. Cham: Springer International Publishing; 2017. (Applied Mathematical Sciences, 127).
  • Kavian O. Lectures on parameter identification. IRMA Lect Math Theor Phys. 2003;4:125–162.
  • Bukhgeim AL, Cheng J, Yamamoto M. Stability for an inverse boundary problem of determining a part of a boundary. Inverse Probl. 1999;15(4):1021–1032. doi:10.1088/0266-5611/15/4/312.
  • Alessandrini G, Morassi A, Rosset E. Detecting cavities by electrostatic boundary measurements. Inverse Probl. 2002;18:1333–1353. doi: 10.1088/0266-5611/18/5/308
  • Andrieux S, Abda AB, Jaoua M. On the inverse emergent plane crack problem. Math Methods Appl Sci. 1998;21:895–906. doi: 10.1002/(SICI)1099-1476(19980710)21:10<895::AID-MMA975>3.0.CO;2-1
  • Alvarez C, Conca C, Friz L, et al. Identification of immersed obstacles via boundary measurements. Inverse Probl. 2005;24:1531–1552. doi: 10.1088/0266-5611/21/5/003
  • Doubova A, Fernández-Cara E, González-Burgos M, et al. A geometric inverse problem for the Boussinesq system. Discrete Contin Dyn Syst Ser B. 2006;6:1213–1238.
  • Doubova A, Fernández-Cara E, Ortega J. On the identification of a single body immersed in a Navier-Stokes fluid. Eur J Appl Math. 2007;18:57–80. doi: 10.1017/S0956792507006821
  • Cao K, Lesnic D, Colaco MJ. Determination of thermal conductivity of inhomogeneous orthotropic materials from temperature measurements. Inverse Probl Sci Eng. 2019;27(10):1372–1398. doi:10.1080/17415977.2018.1554654.
  • Ismailov MI, Tekin I, Erkovan S. An inverse problem for finding the lowest term of a heat equation with Wentzell-Neumann boundary condition. Inverse Probl Sci Eng. 2019;27(11):1608–1634. doi:10.1080/17415977.2018.1553968.
  • Reddy SR, Dulikravich GS. Simultaneous determination of spatially varying thermal conductivity and specific heat using boundary temperature measurements. Inverse Probl Sci Eng. 2019;27(11):1635–1649. doi:10.1080/17415977.2019.1578352.
  • Alvarez C, Conca C, Lecaros R, et al. On the identification of a rigid body immersed in a fluid: A numerical approach. Eng Anal Bound Elem. 2008;32:919–925. doi: 10.1016/j.enganabound.2007.02.007
  • Abda AB, Hassine M, Jaoua M, et al. Topological sensitivity analysis for the location of small cavities in stokes flow. SIAM J Control Optim. 2009/10;48:2871–2900. doi: 10.1137/070704332
  • Karageorghis A, Lesnic D. Identification of obstacles immersed in a stationary oseen fluid via boundary measurements. Inverse Probl Sci Eng. 2019. doi:10.1080/17415977.2019.1686498.
  • Conca C, Cumsille P, Ortega J, et al. On the detection of a moving obstacle in an ideal fluid by a boundary measurement. Inverse Probl. 2008;24(5). doi: 10.1088/0266-5611/24/5/059802
  • Martínez-Castro A, Faris I, Gallego R. Identification of cavities in a three-dimensional layer by minimization of an optimal cost functional expansion. Comput Model Eng Sci. 2012;87:177–206.
  • Bonnet M, Constantinescu A. Inverse problems in elasticity. Inverse Probl. 2005;21:R1–R50. doi: 10.1088/0266-5611/21/2/R01
  • Doubova A, Fernández-Cara E. Some geometric inverse problems for the linear wave equation. Inverse Probl Imaging. 2015;9(2):371–393. doi: 10.3934/ipi.2015.9.371
  • Doubova A, Fernández-Cara E. Some geometric inverse problems for the lamé system with applications in elastography. Appl Math Optim. 2018;76(1):1–21.
  • Hecht F. New development in freefem++. J Numer Math. 2012;20:251–266. doi: 10.1515/jnum-2012-0013
  • Kupradze VD, Aleksidze MA. The method of functional equations for the approximate solution of certain boundary value problems. USSR Comput Math Math Phys. 1964;4:82–126. doi: 10.1016/0041-5553(64)90006-0
  • Karageorghis A, Lesnic D, Marin L. A survey of applications of the mfs to inverse problems. Inverse Probl Sci Eng. 2011;19:309–336. doi: 10.1080/17415977.2011.551830
  • Golberg MA. The method of fundamental solutions for poisson's equations. Eng Anal Bound Elem. 1995;16:205–213. doi: 10.1016/0955-7997(95)00062-3
  • Gu MH, Fan CM, Young DL. The method of fundamental solutions for the multi-dimensional wave equations. J Marine Sci Tech. 2011;19(6):586–595.
  • Chen CW, Young DL, Tsai CC, et al. The method of fundamental solutions for inverse 2D stokes problems. Comput Mech. 2005;37(1):2–14. doi:10.1007/s00466-005-0692-3.
  • Karageorghis A, Lesnic D. The pressure-streamfunction MFS formulation for the detection of an obstacle immersed in a two-dimensional Stokes flow. Adv Appl Math Mech. 2010;2(2):183–199. doi: 10.4208/aamm.09-m0962
  • Martins N, Silvestre A. An iterative mfs approach for the detection of immersed obstacles. Eng Anal Bound Elem. 2008;32(6):517–524. doi: 10.1016/j.enganabound.2007.10.011
  • Nath D, Kalra MS, Munshi P. One-stage method of fundamental and particular solutions (MFS-MPS) for the steady Navier-Stokes equations in a lid-driven cavity. Eng Anal Bound Elem. 2015;58:39–47. doi:10.1016/j.enganabound.2015.03.003.
  • Borman D, Ingham DB, Johansson BT, et al. The method of fundamental solutions for detection of cavities in EIT. J Integral Equ Appl. 2009;21(3):381–404. doi:10.1216/JIE-2009-21-3-383.
  • Betcke T. The generalized singular value decomposition and the method of particular solutions. SIAM J Sci Comput. 2008;30:1278–1295. doi: 10.1137/060651057
  • Betcke T, Trefethen LN. Reviving the method of particular solutions. SIAM Rev. 2005;47:469–491. doi: 10.1137/S0036144503437336
  • Liu TY, Chen CS, Karageorghis A. Conformal mapping for the efficient solution of poisson problems with the Kansa-RBF method. J Sci Comput. 2017;71:1035–1061. doi: 10.1007/s10915-016-0330-6
  • Karageorghis A, Lesnic D, Marin L. The method of fundamental solutions for three-dimensional inverse geometric elasticity problems. Comput Struct. 2016;166:51–59. doi: 10.1016/j.compstruc.2016.01.010
  • Balakrishnan K, Ramachandran PA. The method of fundamental solutions for linear diffusion-reaction equations. Math Comput Model. 2000;31:221–237. doi: 10.1016/S0895-7177(99)00233-2
  • Loulou T, Scott E. An inverse heat conduction problem with heat flux measurements. Int J Numer Methods Eng. 2006;67(11):1587–1616. doi: 10.1002/nme.1674
  • Souza CFL, Costa M, Colaco MJ, et al. Inverse determination of blood perfusion coefficient by using different deterministic and heuristic techniques. J Braz Soc Mech Sci Eng. July 2014;36(1):193–206. doi: 10.1007/s40430-013-0065-3
  • Trucu D., Ingham D. B, Lesnic D. Inverse space-dependent perfusion coefficient identification. J Phys Conf Ser. 2008;135:012098. doi: 10.1088/1742-6596/135/1/012098
  • Trucu D. Inverse problems for blood perfusion identification. University of Leeds; 2009.
  • Grabski JK, Lesnic D, Johansson BT. Identification of a time-dependent bio-heat blood perfusion coefficient. Int Comm Heat Mass Transfer. July 2016;75:218–222. doi: 10.1016/j.icheatmasstransfer.2015.12.028

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