Quarterly of Applied Mathematics

Electromagnetic inverse problems involving distributions of dielectric mechanisms and parameters

Author(s): H. T. Banks; N. L. Gibson
Journal: Quart. Appl. Math. 64 (2006), 749-795.
MSC (2000): Primary 35Q60, 93A30, 49N45
Posted: September 14, 2006
Retrieve article in: PDF DVI PostScript

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.

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), 1935-1958. MR 1955927 (2003k:78021)
2.: R.A. Albanese, R.L. Medina and J.W. Penn, Mathematics, medicine and microwaves, Inverse Problems 10 (1994), 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 K.L. Bihari, Modelling and estimating uncertainty in parameter estimation, Inverse Problems 17 (2001), 95-111. MR 1818494 (2002d:35215)
6.: 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. MR 2126573 (2005j:78032)
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, 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. MR 2036539 (2004k:92003)
9.: 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. MR 2147300 (2006a:34225)
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), 63-91. MR 1965457 (2004b:92032)
11.: 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. MR 1787981 (2001k:78036)
12.: 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. MR 2124300 (2005i:35253)
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, N.L. Gibson and W.P. Winfree, Gap detection with electromagnetic terahertz signals, Nonlinear Analysis: Real World Applications 6 (2005), 381-416. MR 2111660 (2005m:78012)
15.: H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Birkhäuser, Boston, 1989. MR 1045629 (91b:93085)
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, N.G. Medhin and G.A. Pinter, Nonlinear reptation in molecular based hysteresis models for polymers, Quarterly Applied Math., 62 (2004), 767-779. MR 2104273 (2005f:74022)
20.: H.T. Banks and G.A. Pinter, Maxwell systems with nonlinear polarization, Nonlinear Analysis: Real World Applications, 4 (2003), 483-501. MR 1956350 (2003k:78026)
21.: H.T. Banks and G.A. Pinter, High-frequency pulse propagation in nonlinear dielectric materials, Nonlinear Analysis: Real World Applications, 5 (2004), 597-612. MR 2079271 (2005h:78019)
22.: 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. MR 2122994 (2005k:74058)
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.: J.D. Jackson, Classical Electrodynamics, John Wiley and Sons, New York, 1975. MR 0436782 (55:9721)
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 (92b:78001)
44.: E.J. McShane, Generalized curves, Duke Math J. 6 (1940), 513-536. MR 0002469 (2:59d)
45.: E.J. McShane, Relaxed controls and variational problems, SIAM J. Control 5 (1967), 438-485. MR 0218949 (36:2033)
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. SIAM Ser. A Control 3 (1965), 231-280. MR 0189870 (32:7288)
50.: M.H. Schultz, Spline Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1973. MR 0362832 (50:15270)
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 (25:5415a)
55.: J. Warga, Functions of relaxed controls, SIAM J. Control 5 (1967), 628-641. MR 0226474 (37:2064a)
56.: J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. MR 0372708 (51:8915)
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.

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

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

PII: 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
Posted: September 14, 2006
Copyright of article: Copyright 2006, Brown University

		Journals Home Search Author Info Subscribe Tech Support Help
Online ISSN: 1552-4485 Print ISSN: 0033-569X

Previous issue \| Table of contents \| Next issue Recently posted articles \| Most recent issue \| All issues