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
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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.
- 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, DOI https://doi.org/10.1088/0266-5611/18/6/330
- Richard A. Albanese, Richard L. Medina, and John W. Penn, Mathematics, medicine and microwaves, Inverse Problems 10 (1994), no. 5, 995–1007. MR 1296358
Albanese89 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.
Anderson1967 J.C. Anderson, Dielectrics, Chapman and Hall, London, 1967.
- H. T. Banks and Kathleen L. Bihari, Modelling and estimating uncertainty in parameter estimation, Inverse Problems 17 (2001), no. 1, 95–111. MR 1818494, DOI https://doi.org/10.1088/0266-5611/17/1/308
- 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, DOI https://doi.org/10.1090/S0033-569X-05-00949-0
homog 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.
- 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
- 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, DOI https://doi.org/10.1163/1569394053978515
- 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, DOI https://doi.org/10.1016/S0025-5564%2802%2900218-3
- 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
- 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, DOI https://doi.org/10.1016/j.aml.2004.02.008
bib:long H. T. Banks and N. L. Gibson, Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters, CRSC-TR05-29, August, 2005.
- 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, DOI https://doi.org/10.1016/j.nonrwa.2004.09.004
- 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
BKW1 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.
BKW2 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.
BMP1 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.
- 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, DOI https://doi.org/10.1090/qam/2104273
- 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, DOI https://doi.org/10.1016/S1468-1218%2802%2900074-3
- 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, DOI https://doi.org/10.1016/j.nonrwa.2003.10.002
- 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, DOI https://doi.org/10.1137/040603693
BF95 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.
BB78 C. J. F. Böttcher and P. Bordewijk, Theory of Electric Polarization, Vol. II, Elsevier, New York, 1978.
Bo R.W. Boyd, Nonlinear Optics, Academic Press, Boston, 1992.
Cap M. Caputo, Rigorous time domain responses of polarizable media, Annali di Geofisica XL (1997), 423–434.
Cof W.T. Coffey, Y.P. Kalmykov and S.V. Titov, Inertial effects in anomalous dielectric relaxation, J. Molecular Liquids 114 (2004), 35–41.
Cole-Cole K.S. Cole and R.H. Cole, Dispersion and absorption in dielectrics, J. Chemical Phys. 9 (1941), 341–351.
Davidian 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.
Debye P. Debye, Polar Molecules, Chemical Catalogue Co., New York, 1929.
Ell R.S. Elliott, Electromagnetics: History, Theory and Applications, IEEE Press, New York, 1993.
FosterSchwan 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.
Gabriel 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.
G1 C. Gabriel, S. Gabriel and E. Corthout, The dielectric properties of biological tissues: I. Literature survey, Phys. Med. Biol. 41 (1996), 2231–2249.
G2 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.
G3 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.
GGC 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.
NGthesis N. L. Gibson, Terahertz-Based Electromagnetic Interrogation Techniques for Damage Detection, Ph.D. Thesis, N. C. State University, Raleigh, 2004.
Hurt W.D. Hurt, Multiterm Debye dispersion relations for permittivity of muscle, IEEE Trans. Biomed. Engr. 32 (1985), 60–64.
- John David Jackson, Classical electrodynamics, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. MR 0436782
JT 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.
- M. A. Krasnosel′skiĭ and A. V. Pokrovskiĭ, Sistemy s gisterezisom, “Nauka”, Moscow, 1983 (Russian). MR 742931
- I. D. Mayergoyz, Mathematical models of hysteresis, Springer-Verlag, New York, 1991. MR 1083150
- E. J. McShane, Generalized curves, Duke Math. J. 6 (1940), 513–536. MR 2469
- E. J. McShane, Relaxed controls and variational problems, SIAM J. Control 5 (1967), 438–485. MR 0218949
Miura2003 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.
PP P.G. Petropoulos, On the time-domain response of Cole-Cole dielectrics, IEEE Trans. Antennas and Propag., (2005), to appear.
- Olivier Pironneau, Optimal shape design for elliptic systems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 725856
- W. W. Schmaedeke, Optimal control theory for nonlinear vector differential equations containing measures, J. SIAM Control Ser. A 3 (1965), 231–280. MR 189870
- Martin H. Schultz, Spline analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR 0362832
- Larry L. Schumaker, Spline functions: basic theory, John Wiley & Sons, Inc., New York, 1981. Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 606200
Von Schweidler E.R Von Schweidler, Studien über anomalien im verhalten der dielektrika, Ann. Physik 24 (1907), 711–770.
Wagner K.W. Wagner, Zur theorie der unvollkommenen dielektrika, Ann. Physik 40 (1913), 817–855.
- J. Warga, Relaxed variational problems, J. Math. Anal. Appl. 4 (1962), 111–128. MR 142020, DOI https://doi.org/10.1016/0022-247X%2862%2990033-1
- J. Warga, Functions of relaxed controls, SIAM J. Control 5 (1967), 628–641. MR 0226474
- J. Warga, Optimal control of differential and functional equations, Academic Press, New York-London, 1972. MR 0372708
WF 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.
Y1 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.
Y2 L.C. Young, Necessary conditions in the calculus of variations, Acta Math. 69 (1938), 239–258.
Albanese2002 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), 1935–1958.
Albanese94 R.A. Albanese, R.L. Medina and J.W. Penn, Mathematics, medicine and microwaves, Inverse Problems 10 (1994), 995–1007.
Albanese89 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.
Anderson1967 J.C. Anderson, Dielectrics, Chapman and Hall, London, 1967.
BBi H.T. Banks and K.L. Bihari, Modelling and estimating uncertainty in parameter estimation, Inverse Problems 17 (2001), 95–111.
BBo H.T. Banks and V. A. Bokil, A computational and statistical framework for multidimensional domain acoustooptic material interrogation, Quarterly of Applied Mathematics 63 (2005), 156–200.
homog 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.
BBPP H.T. Banks, D. Bortz, G.A. Pinter and L.K. Potter, Modeling and imaging techniques with potential for application in bioterrorism, Chapter 6 in Bioterrorism: Mathematical Modeling Applications in Homeland Security, (H.T. Banks and C. Castillo-Chavez, eds.), Frontiers in Applied Mathematics, SIAM, Philadelphia, 2003, 129–154.
BBdelay H.T. Banks and D.M. Bortz, Inverse problems for a class of measure dependent dynamical systems, J. Inverse and Ill-posed Problems, 13 (2005), 103–121.
BBH 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), 63–91.
BBL00 H.T. Banks, M.W. Buksas and T. Lin, Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts, Frontiers in Applied Mathematics, Vol. FR21, SIAM, Philadelphia, PA, 2000.
BG05 H. T. Banks and N. L. Gibson, Well-posedness in Maxwell systems with distributions of polarization relaxation parameters, Applied Mathematics Letters 18 (2005), 423–430.
bib:long H. T. Banks and N. L. Gibson, Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters, CRSC-TR05-29, August, 2005.
BGW05 H.T. Banks, N.L. Gibson and W.P. Winfree, Gap detection with electromagnetic terahertz signals, Nonlinear Analysis: Real World Applications 6 (2005), 381–416.
BanksKunisch H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhäuser, Boston, 1989.
BKW1 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.
BKW2 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.
BMP1 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.
BMP2 H.T. Banks, N.G. Medhin and G.A. Pinter, Nonlinear reptation in molecular based hysteresis models for polymers, Quarterly Applied Math., 62 (2004), 767–779.
BPiNL1 H.T. Banks and G.A. Pinter, Maxwell systems with nonlinear polarization, Nonlinear Analysis: Real World Applications, 4 (2003), 483–501.
BPiNL2 H.T. Banks and G.A. Pinter, High-frequency pulse propagation in nonlinear dielectric materials, Nonlinear Analysis: Real World Applications, 5 (2004), 597–612.
BPi H.T. Banks and G.A. Pinter, A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue, SIAM J. Multiscale Modeling and Simulation 3 (2005), 395–412.
BF95 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.
BB78 C. J. F. Böttcher and P. Bordewijk, Theory of Electric Polarization, Vol. II, Elsevier, New York, 1978.
Bo R.W. Boyd, Nonlinear Optics, Academic Press, Boston, 1992.
Cap M. Caputo, Rigorous time domain responses of polarizable media, Annali di Geofisica XL (1997), 423–434.
Cof W.T. Coffey, Y.P. Kalmykov and S.V. Titov, Inertial effects in anomalous dielectric relaxation, J. Molecular Liquids 114 (2004), 35–41.
Cole-Cole K.S. Cole and R.H. Cole, Dispersion and absorption in dielectrics, J. Chemical Phys. 9 (1941), 341–351.
Davidian 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.
Debye P. Debye, Polar Molecules, Chemical Catalogue Co., New York, 1929.
Ell R.S. Elliott, Electromagnetics: History, Theory and Applications, IEEE Press, New York, 1993.
FosterSchwan 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.
Gabriel 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.
G1 C. Gabriel, S. Gabriel and E. Corthout, The dielectric properties of biological tissues: I. Literature survey, Phys. Med. Biol. 41 (1996), 2231–2249.
G2 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.
G3 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.
GGC 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.
NGthesis N. L. Gibson, Terahertz-Based Electromagnetic Interrogation Techniques for Damage Detection, Ph.D. Thesis, N. C. State University, Raleigh, 2004.
Hurt W.D. Hurt, Multiterm Debye dispersion relations for permittivity of muscle, IEEE Trans. Biomed. Engr. 32 (1985), 60–64.
Jackson J.D. Jackson, Classical Electrodynamics, John Wiley and Sons, New York, 1975.
JT 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.
KP M.A. Krasnosel’skii and A.V. Pokrovskii, Systems with Hysteresis, Nauka, Moscow, 1983; translated from the Russian, Springer-Verlag, Berlin, 1989.
May I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer-Verlag, New York, 1991.
Mc1 E.J. McShane, Generalized curves, Duke Math J. 6 (1940), 513–536.
Mc E.J. McShane, Relaxed controls and variational problems, SIAM J. Control 5 (1967), 438–485.
Miura2003 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.
PP P.G. Petropoulos, On the time-domain response of Cole-Cole dielectrics, IEEE Trans. Antennas and Propag., (2005), to appear.
Pir83 O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York, 1984.
S W.W. Schmaedeke, Optimal control theory for nonlinear vector differential equations containing measures, J. SIAM Ser. A Control 3 (1965), 231–280.
Sl M.H. Schultz, Spline Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1973.
Sm L. Schumaker, Spline Functions: Basic Theory, J. Wiley & Sons, New York, 1981.
Von Schweidler E.R Von Schweidler, Studien über anomalien im verhalten der dielektrika, Ann. Physik 24 (1907), 711–770.
Wagner K.W. Wagner, Zur theorie der unvollkommenen dielektrika, Ann. Physik 40 (1913), 817–855.
Wa1 J. Warga, Relaxed variational problems, J. Math. Anal. Appl. 4 (1962), 111–128.
Wa2 J. Warga, Functions of relaxed controls, SIAM J. Control 5 (1967), 628–641.
Wa J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
WF 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.
Y1 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.
Y2 L.C. Young, Necessary conditions in the calculus of variations, Acta Math. 69 (1938), 239–258.
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Additional Information
H. T. Banks
Affiliation:
Center for Research in Scientific Computation, North Carolina State University, Raleigh, North Carolina 27695-8205
MR Author ID:
194993
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
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