Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters


Authors: H. T. Banks and N. L. Gibson
Journal: Quart. Appl. Math. 64 (2006), 749-795
MSC (2000): Primary 35Q60, 93A30, 49N45
DOI: https://doi.org/10.1090/S0033-569X-06-01036-X
Published electronically: September 14, 2006
MathSciNet review: 2284469
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider electromagnetic interrogation problems for complex materials involving distributions of polarization mechanisms and also distributions for the parameters in these mechanisms. A theoretical and computational framework for such problems is given. Computational results for specific problems with multiple Debye mechanisms are given in the case of discrete, uniform, log-normal, and log-bi-Gaussian distributions.


References [Enhancements On Off] (What's this?)

  • 1. R. A. Albanese, H. T. Banks, and J. K. Raye, Nondestructive evaluation of materials using pulsed microwave interrogating signals and acoustic wave induced reflections, Inverse Problems 18 (2002), no. 6, 1935–1958. Special section on electromagnetic and ultrasonic nondestructive evaluation. MR 1955927, https://doi.org/10.1088/0266-5611/18/6/330
  • 2. Richard A. Albanese, Richard L. Medina, and John W. Penn, Mathematics, medicine and microwaves, Inverse Problems 10 (1994), no. 5, 995–1007. MR 1296358
  • 3. R.A. Albanese, J.W. Penn and R.L. Medina, Short-rise-time microwave pulse propagation through dispersive biological media, J. Optical Society of America A 6 (1989), 1441-1446.
  • 4. J.C. Anderson, Dielectrics, Chapman and Hall, London, 1967.
  • 5. H. T. Banks and Kathleen L. Bihari, Modelling and estimating uncertainty in parameter estimation, Inverse Problems 17 (2001), no. 1, 95–111. MR 1818494, https://doi.org/10.1088/0266-5611/17/1/308
  • 6. H. T. Banks and V. A. Bokil, A computational and statistical framework for multidimensional domain acoustooptic material interrogation, Quart. Appl. Math. 63 (2005), no. 1, 156–200. MR 2126573, https://doi.org/10.1090/S0033-569X-05-00949-0
  • 7. H.T. Banks, V.A. Bokil, D. Cioranescu, N.L. Gibson, G. Griso and B. Miara, Homogenization of periodically varying coefficients in electromagnetic materials, CRSC-TR05-05, January, 2005; J. Scientific Computing, to appear.
  • 8. H. T. Banks and Carlos Castillo-Chavez (eds.), Bioterrorism, Frontiers in Applied Mathematics, vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. Mathematical modeling applications in homeland security. MR 2036539
  • 9. H. T. Banks and D. M. Bortz, Inverse problems for a class of measure dependent dynamical systems, J. Inverse Ill-Posed Probl. 13 (2005), no. 2, 103–121. MR 2147300, https://doi.org/10.1163/1569394053978515
  • 10. H. T. Banks, D. M. Bortz, and S. E. Holte, Incorporation of variability into the modeling of viral delays in HIV infection dynamics, Math. Biosci. 183 (2003), no. 1, 63–91. MR 1965457, https://doi.org/10.1016/S0025-5564(02)00218-3
  • 11. H. T. Banks, M. W. Buksas, and T. Lin, Electromagnetic material interrogation using conductive interfaces and acoustic wavefronts, Frontiers in Applied Mathematics, vol. 21, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. MR 1787981
  • 12. H. T. Banks and N. L. Gibson, Well-posedness in Maxwell systems with distributions of polarization relaxation parameters, Appl. Math. Lett. 18 (2005), no. 4, 423–430. MR 2124300, https://doi.org/10.1016/j.aml.2004.02.008
  • 13. H. T. Banks and N. L. Gibson, Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters, CRSC-TR05-29, August, 2005.
  • 14. H. T. Banks, Nathan L. Gibson, and William P. Winfree, Gap detection with electromagnetic terahertz signals, Nonlinear Anal. Real World Appl. 6 (2005), no. 2, 381–416. MR 2111660, https://doi.org/10.1016/j.nonrwa.2004.09.004
  • 15. H. T. Banks and K. Kunisch, Estimation techniques for distributed parameter systems, Systems & Control: Foundations & Applications, vol. 1, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 1045629
  • 16. H.T. Banks, A.J. Kurdila and G. Webb, Identification of hysteretic control influence operators representing smart actuators, Part I: Formulation, Mathematical Problems in Engineering, 3 (1997), 287-328.
  • 17. H.T. Banks, A.J. Kurdila and G. Webb, Identification of hysteretic control influence operators representing smart actuators: Part II, Convergent approximations, J. of Intelligent Material Systems and Structures, 8 (1997), 536-550.
  • 18. H.T. Banks, N.G. Medhin and G.A. Pinter, Multiscale considerations in modeling of nonlinear elastomers, CRSC-TR03-42, October, 2003; J. Comp. Meth. Sci. and Engr., to appear.
  • 19. H. T. Banks, Negash G. Medhin, and Gabriella A. Pinter, Nonlinear reptation in molecular based hysteresis models for polymers, Quart. Appl. Math. 62 (2004), no. 4, 767–779. MR 2104273, https://doi.org/10.1090/qam/2104273
  • 20. H. T. Banks and Gabriella A. Pinter, Maxwell-systems with nonlinear polarization, Nonlinear Anal. Real World Appl. 4 (2003), no. 3, 483–501. MR 1956350, https://doi.org/10.1016/S1468-1218(02)00074-3
  • 21. H. T. Banks and Gabriella A. Pinter, High-frequency pulse propagation in nonlinear dielectric materials, Nonlinear Anal. Real World Appl. 5 (2004), no. 4, 597–612. MR 2079271, https://doi.org/10.1016/j.nonrwa.2003.10.002
  • 22. H. T. Banks and Gabriella A. Pinter, A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue, Multiscale Model. Simul. 3 (2005), no. 2, 395–412. MR 2122994, https://doi.org/10.1137/040603693
  • 23. J.G. Blaschak and J. Franzen, Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence, J. Optical Society of America A 12 (1995), 1501-1512.
  • 24. C. J. F. Böttcher and P. Bordewijk, Theory of Electric Polarization, Vol. II, Elsevier, New York, 1978.
  • 25. R.W. Boyd, Nonlinear Optics, Academic Press, Boston, 1992.
  • 26. M. Caputo, Rigorous time domain responses of polarizable media, Annali di Geofisica XL (1997), 423-434.
  • 27. W.T. Coffey, Y.P. Kalmykov and S.V. Titov, Inertial effects in anomalous dielectric relaxation, J. Molecular Liquids 114 (2004), 35-41.
  • 28. K.S. Cole and R.H. Cole, Dispersion and absorption in dielectrics, J. Chemical Phys. 9 (1941), 341-351.
  • 29. M. Davidian and D.M. Giltinan, Nonlinear Models for Repeated Measurement Data, Monographs on Statistics and Applied Probability, Vol. 62, Chapman and Hall, New York, 1995.
  • 30. P. Debye, Polar Molecules, Chemical Catalogue Co., New York, 1929.
  • 31. R.S. Elliott, Electromagnetics: History, Theory and Applications, IEEE Press, New York, 1993.
  • 32. K.R. Foster and H.P. Schwan, Dielectric properties of tissues and biological materials: A critical review, Critical Rev. in Biomed. Engr. 17 (1989), 25-104.
  • 33. C. Gabriel, Compilation of the dielectric properties of body tissues at RF and microwave frequencies, Technical Report AL/OE-TR-1996-0037, USAF Armstrong Laboratory, Brooks AFB, TX, 1996.
  • 34. C. Gabriel, S. Gabriel and E. Corthout, The dielectric properties of biological tissues: I. Literature survey, Phys. Med. Biol. 41 (1996), 2231-2249.
  • 35. S. Gabriel, R.W. Lau and C. Gabriel, The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz, Phys. Med. Biol. 41 (1996), 2251-2269.
  • 36. S. Gabriel, R.W. Lau and C. Gabriel, The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues, Phys. Med. Biol. 41 (1996), 2271-2293.
  • 37. O.P. Gandhi, B.Q. Gao and J.Y. Chen, A frequency-dependent finite-difference time-domain formulation for general dispersive media, IEEE Trans. Microwave Theory and Tech., 41 (1993), 658-665.
  • 38. N. L. Gibson, Terahertz-Based Electromagnetic Interrogation Techniques for Damage Detection, Ph.D. Thesis, N. C. State University, Raleigh, 2004.
  • 39. W.D. Hurt, Multiterm Debye dispersion relations for permittivity of muscle, IEEE Trans. Biomed. Engr. 32 (1985), 60-64.
  • 40. John David Jackson, Classical electrodynamics, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. MR 0436782
  • 41. R.M. Joseph and A. Taflove, FDTD Maxwell's equations models for nonlinear electrodynamics and optics, IEEE Trans. Antennas and Propag., 45 (1997), 364-374.
  • 42. M.A. Krasnosel'skii and A.V. Pokrovskii, Systems with Hysteresis, Nauka, Moscow, 1983; translated from the Russian, Springer-Verlag, Berlin, 1989. MR 0742931 (86e:93005)
  • 43. I. D. Mayergoyz, Mathematical models of hysteresis, Springer-Verlag, New York, 1991. MR 1083150
  • 44. E. J. McShane, Generalized curves, Duke Math. J. 6 (1940), 513–536. MR 0002469
  • 45. E. J. McShane, Relaxed controls and variational problems, SIAM J. Control 5 (1967), 438–485. MR 0218949
  • 46. N. Miura, S. Yahihara, and S. Mashimo, Microwave dielectric properties of solid and liquid foods investigated by time-domain reflectometry, J. Food Science 68 (2003), 1396-1403.
  • 47. P.G. Petropoulos, On the time-domain response of Cole-Cole dielectrics, IEEE Trans. Antennas and Propag., (2005), to appear.
  • 48. O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, 1984. MR 0725856 (86e:49003)
  • 49. W. W. Schmaedeke, Optimal control theory for nonlinear vector differential equations containing measures, J. Soc. Indust. Appl. Math. Ser. A Control 3 (1965), 231–280. MR 0189870
  • 50. Martin H. Schultz, Spline analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR 0362832
  • 51. L. Schumaker, Spline Functions: Basic Theory, J. Wiley & Sons, New York, 1981. MR 0606200 (82j:41001)
  • 52. E.R Von Schweidler, Studien über anomalien im verhalten der dielektrika, Ann. Physik 24 (1907), 711-770.
  • 53. K.W. Wagner, Zur theorie der unvollkommenen dielektrika, Ann. Physik 40 (1913), 817-855.
  • 54. J. Warga, Relaxed variational problems, J. Math. Anal. Appl. 4 (1962), 111–128. MR 0142020, https://doi.org/10.1016/0022-247X(62)90033-1
  • 55. J. Warga, Functions of relaxed controls, SIAM J. Control 5 (1967), 628–641. MR 0226474
  • 56. J. Warga, Optimal control of differential and functional equations, Academic Press, New York-London, 1972. MR 0372708
  • 57. M.L. Williams and J.D. Ferry, Second approximation calculations of mechanical and electrical relaxation and retardation distributions, J. Poly. Sci. 11 (1953), 169-175.
  • 58. L.C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, C. R. Soc. Sci. et Lettres, Varsovie, Cl. III, 30 (1937), 212-234.
  • 59. L.C. Young, Necessary conditions in the calculus of variations, Acta Math. 69 (1938), 239-258.

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35Q60, 93A30, 49N45

Retrieve articles in all journals with MSC (2000): 35Q60, 93A30, 49N45


Additional Information

H. T. Banks
Affiliation: Center for Research in Scientific Computation, North Carolina State University, Raleigh, North Carolina 27695-8205
Email: htbanks@ncsu.edu

N. L. Gibson
Affiliation: Center for Research in Scientific Computation, North Carolina State University, Raleigh, North Carolina 27695-8205
Email: ngibson@ncsu.edu

DOI: https://doi.org/10.1090/S0033-569X-06-01036-X
Keywords: Electromagnetic interrogation, pulsed antenna source microwaves, inverse problems, complex dielectric materials, distributions of relaxation parameters and mechanisms.
Received by editor(s): April 13, 2006
Published electronically: September 14, 2006
Article copyright: © Copyright 2006 Brown University

American Mathematical Society