Electromagnetic Relaxation Time Distribution Inverse Problems in the Time-domain

Transcription

1 Electromagnetic Relaxation Time Distribution Inverse Problems in the Time-domain Prof Nathan L Gibson Department of Mathematics Joint Math Meeting Jan 9, 2011 Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

2 Acknowledgements Megan Armentrout, OSU Graduate Student Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

3 Acknowledgements Megan Armentrout, OSU Graduate Student REU 2010: REU 2009: REU 2008: Erin Bela Chapman University Erik Hortsch OSU Marie Milne Lawrence University Danielle Wedde Rochester University Karen Barrese OSU Neel Chugh Tufts University Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

4 Outline 1 Maxwell s Equations Description Simplifications Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

5 Outline 1 Maxwell s Equations Description Simplifications 2 Polarization Description Distributions Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

6 Outline 1 Maxwell s Equations Description Simplifications 2 Polarization Description Distributions 3 Inverse Problem for Distributions Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

7 Outline 1 Maxwell s Equations Description Simplifications 2 Polarization Description Distributions 3 Inverse Problem for Distributions 4 Forward Simulation Polynomial Chaos Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

8 Outline 1 Maxwell s Equations Description Simplifications 2 Polarization Description Distributions 3 Inverse Problem for Distributions 4 Forward Simulation Polynomial Chaos 5 Inverse Problem Numerical Results Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

9 Maxwell s Equations Description Maxwell s Equations Maxwell s Equations were formulated circa 1870 They represent a fundamental unification of electric and magnetic fields predicting electromagnetic wave phenomenon Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

10 Maxwell s Equations Description Maxwell s Equations D t + J = H (Ampere) B t = E (Faraday) D = ρ (Poisson) B = 0 (Gauss) E = H = ρ = Electric field vector Magnetic field vector Electric charge density D = B = J = Electric displacement Magnetic flux density Current density Note: Need initial conditions and boundary conditions Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

11 Maxwell s Equations Description Constitutive Laws Maxwell s equations are completed by constitutive laws that describe the response of the medium to the electromagnetic field D = ɛe + P B = µh + M J = σe + J s P = M = J s = Polarization Magnetization Source Current ɛ = Electric permittivity µ = Magnetic permeability σ = Electric Conductivity Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

12 Maxwell s Equations Simplifications Linear, Isotropic, Non-dispersive and Non-conductive media Assume no material dispersion, ie, speed of propagation is not frequency dependent D = ɛe B = µh ɛ = ɛ 0 ɛ r ɛ r = Relative Permittivity µ = µ 0 µ r µ r = Relative Permeability c = 1/ ɛµ Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

13 Maxwell s Equations x Simplifications x Maxwell s Equations in One Space Dimension The time evolution of the fields is thus completely specified by the curl equations ɛ E = H t µ H = E t Assuming that the electric field is polarized to oscillate only in the y direction, propagate in the x direction, and there is uniformity in the z direction: Equations involving E y and H z ɛ E y t µ H z t = H z x = E y x Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35 E y H z

14 Maxwell s Equations Sample Signal Propagation 100 time = e 10 s 100 time = e 10 s time = e 10 s 100 time = e 09 s Snapshots of a windowed electromagnetic pulse with f =10GHz for the interrogation problem Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

15 Polarization Description Dispersive Dielectrics Recall D = ɛe + P where P is the dielectric polarization We can generally define P in terms of a convolution P(t, x) = g E(t, x) = t 0 g(t s, x; q)e(s, x)ds, where g is a general dielectric response function (DRF), and q is some parameter set Debye model g(t, x) = ɛ 0 (ɛ s ɛ )/τ e t/τ or equivalently, τṗ + P = ɛ 0(ɛ s ɛ )E where q = {ɛ, ɛ s, τ} and, in particular, τ is called the relaxation time Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

16 Polarization Description Frequency Domain Converting to frequency domain via Fourier transforms D = ɛe + P becomes ˆD = ɛ(ω)ê where ɛ(ω) is called the complex permittivity Debye model gives ɛ(ω) = ɛ + ɛ s ɛ 1 + iωτ Cole-Cole model (heuristic generalization) ɛ(ω) = ɛ + ɛ s ɛ 1 + (iωτ) 1 α Unfortunately, the Cole-Cole model corresponds to a fractional order differential equation in the time domain, and simulation is not straight-forward Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

17 Polarization Dry Skin Data 10 3 True Data Debye Model Cole Cole Model ε f (Hz) Figure: Real part of ɛ(ω), ɛ, or the permittivity Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

18 Polarization Dry Skin Data σ True Data Debye Model Cole Cole Model f (Hz) Figure: Imaginary part of ɛ(ω), σ, or the conductivity Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

19 Polarization Motivation Motivation Broadband wave propogation suggests time-domain simulation The Cole-Cole model corresponds to a fractional order ODE in the time-domain and is difficult to simulate Debye is efficient to simulate, but does not represent permittivity well Better fits to data are obtained by taking linear combinations of Debye models (multi-pole Debye), idea comes from the known existence of multiple physical mechanisms An alternative approach is to consider the Debye model but with a (continuous) distribution of relaxation times Empirical measurements suggest a log-normal distribution Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

20 Polarization Distributions Distributions of Parameters To account for the effect of possible multiple parameter sets q, consider h(t, x; F ) = g(t, x; q)df (q), where Q is some admissible set and F P(Q) Then the polarization becomes: P(t, x) = t 0 Q h(t s, x; F )E(s, x)ds Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

21 Polarization Fit to Dry Skin Data 10 3 Data Debye (2779) Cole Cole (104) Model A (1360) Model B (1219) ε f (Hz) Figure: Real part of ɛ(ω), called simply ɛ, or the permittivity Model A refers to the Debye model with a uniform distribution on τ Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

22 Polarization Random Polarization Random Polarization We define the random polarization P(x, t; τ) to be the solution to τṗ + P = ɛ 0 (ɛ s ɛ )E where τ is a random variable with PDF f (τ), for example, f (τ) = 1 τ b τ a for a uniform distribution Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

23 Polarization Random Polarization Random Polarization We define the random polarization P(x, t; τ) to be the solution to τṗ + P = ɛ 0 (ɛ s ɛ )E where τ is a random variable with PDF f (τ), for example, f (τ) = 1 τ b τ a for a uniform distribution The electric field depends on the macroscopic polarization, which we take to be the expected value of the random polarization at each point (x, t) P(x, t; F ) = τb τ a P(x, t; τ)f (τ)dτ Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

24 Polarization Random Polarization Well-Posedness of Forward Problem Existence and uniqueness of solutions to weak formulation of the forward problem follows as a special case of work in [BBL00] Continuous dependence of (E, Ė) on F in the Prohorov metric shown in [BG05] Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

25 Inverse Problem for Distributions Time-domain Inverse Problem Given data {Ê} j we seek to determine a probability measure F, such that F = min J (F ), F P(Q) where, for example, J (F ) = j ( E(t j ; F ) Êj) 2 Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

26 Inverse Problem for Distributions Time-domain Inverse Problem Given data {Ê} j we seek to determine a probability measure F, such that F = min J (F ), F P(Q) where, for example, J (F ) = j ( E(t j ; F ) Êj) 2 Continuity of F (E, Ė) = continuity of F J (F ) Compactness of Q = compactness of P(Q) with respect to the Prohorov metric Therefore, a minimum of J (F ) over P(Q) exists [BG05] Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

27 Inverse Problem for Distributions Numerical Approximation of Random Polarization To solve the inverse problem for the distribution of relaxation times, we need a method of accurately and efficiently simulating P(x, t; F ) Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

28 Inverse Problem for Distributions Numerical Approximation of Random Polarization To solve the inverse problem for the distribution of relaxation times, we need a method of accurately and efficiently simulating P(x, t; F ) Could apply a quadrature rule to the integral in the expected value Results in a linear combination of individual Debye solves Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

29 Inverse Problem for Distributions Numerical Approximation of Random Polarization To solve the inverse problem for the distribution of relaxation times, we need a method of accurately and efficiently simulating P(x, t; F ) Could apply a quadrature rule to the integral in the expected value Results in a linear combination of individual Debye solves Alternatively, we can use a method which separates the time derivative from the randomness and applies a truncated expansion in random space, called Polynomial Chaos Results in a linear system Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

30 Forward Simulation Polynomial Chaos Polynomial Chaos: Simple example Consider the first order, constant coefficient, linear ODE ẏ = ky, k = k(ξ) = ξ, ξ N (0, 1) We apply a Polynomial Chaos expansion in terms of orthogonal Hermite polynomials H j to the solution y: y(t, ξ) = α j (t)φ j (ξ), j=0 φ j (ξ) = H j (ξ) then the ODE becomes j=0 α j (t)φ j (ξ) = α j (t)ξφ j (ξ), j=0 Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

31 Forward Simulation Polynomial Chaos Triple recursion formula α j (t)φ j (ξ) = α j (t)ξφ j (ξ), j=0 We can eliminate the explicit dependence on ξ by using the triple recursion formula for Hermite polynomials j=0 ξh j = jh j 1 + H j+1 Thus j=0 α j (t)φ j + α j (t)(jφ j 1 + φ j+1 ) = 0 Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

32 Forward Simulation Polynomial Chaos Galerkin Projection onto span({φ i } p i=0 ) Taking the weighted inner product with each basis gives j=0 α j (t) φ j, φ i W + α j (t)(j φ j 1, φ i W + φ j+1, φ i W ) = 0, i = 0,, p Where f (ξ), g(ξ) W = f (ξ)g(ξ)w (ξ)dξ Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

33 Forward Simulation Polynomial Chaos Galerkin Projection onto span({φ i } p i=0 ) Taking the weighted inner product with each basis gives j=0 α j (t) φ j, φ i W + α j (t)(j φ j 1, φ i W + φ j+1, φ i W ) = 0, i = 0,, p Where f (ξ), g(ξ) W = f (ξ)g(ξ)w (ξ)dξ Using orthogonality, φ j, φ i W = φ i, φ i W δ ij, we have α i φ i, φ i W + (i + 1)α i+1 φ i, φ i W + α i 1 φ i, φ i W = 0, i = 0,, p, Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

34 Forward Simulation Polynomial Chaos Deterministic ODE system Letting α represent the vector containing α 0 (t),, α p (t) (and assuming α p+1 (t), etc are identically zero) the system of ODEs can be written with M = The mean value of y(t, ξ) is α 0 (t) α + M α = 0, p 1 0 Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

35 Forward Simulation Polynomial Chaos Generalizations For any choice of family of orthogonal polynomials, there exists a triple recursion formula Given the arbitrary relation ξφ j = a j φ j 1 + b j φ j + c j φ j+1 (with φ 1 = 0) then the matrix above becomes b 0 a 1 c 0 b 1 a 2 M = ap c p 1 b p Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

36 Forward Simulation Polynomial Chaos Generalizations Consider the non-homogeneous ODE ẏ + ky = g(t), k = k(ξ) = σξ + µ, ξ N (0, 1) then α i + σ [(i + 1)α i+1 + α i 1 ] + µα i = g(t)δ 0i, i = 0,, p, or the deterministic ODE system α + (σm + µi ) α = g(t) e 1 Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

37 Forward Simulation Polynomial Chaos Exponential convergence Any set of orthogonal polynomials can be used in the truncated expansion, but there may be an optimal choice If the polynomials are orthogonal with respect to weighting function f (ξ), and k has PDF f (k), then it is known that the PC solution converges exponentially in terms of p In practice, approximately 4 are generally sufficient Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

38 Forward Simulation Polynomial Chaos Generalized Polynomial Chaos Table: Popular distributions and corresponding orthogonal polynomials Distribution Polynomial Support Gaussian Hermite (, ) gamma Laguerre [0, ) beta Jacobi [a, b] uniform Legendre [a, b] Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

39 Forward Simulation Polynomial Chaos 25 x 1010 Beta log Normal Figure: Shape of Beta distribution can mimic log-normal, but with finite support Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

40 Forward Simulation Random Polarization Random Polarization We can apply Polynomial Chaos method to our random polarization τṗ + P = ɛ 0 (ɛ s ɛ )E, τ = τ(ξ) = rξ + m with, eg, ξ Beta(a, b), resulting in (rm + mi ) α + α = ɛ 0 (ɛ s ɛ )E e 1 =: g or A α + α = g Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

41 Forward Simulation Random Polarization Random Polarization We can apply Polynomial Chaos method to our random polarization τṗ + P = ɛ 0 (ɛ s ɛ )E, τ = τ(ξ) = rξ + m with, eg, ξ Beta(a, b), resulting in or (rm + mi ) α + α = ɛ 0 (ɛ s ɛ )E e 1 =: g A α + α = g The macroscopic polarization, the expected value of the random polarization at each point (t, x), is simply P(t, x; F ) = α 0 (t, x) Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

42 Inverse Problem Numerical Results Comparison, noise = 01, refinement = 1, perturb = 08 Data Initial J= Optimal J= Actual J= E t x 10 9 Comparison of simulations to data Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

43 Inverse Problem Numerical Results 6 x 1011 Distributions, noise = 01, refinement = 1, perturb = 08 5 Initial J= Optimal J= Actual J= Comparison of initial to final distribution Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

44 Inverse Problem Numerical Results 105 x The Confidence Intervals for µ g (813e 12) refinement =1, perturb = µ g Noise Level Estimates for the mean of the relaxation time distribution (RTD) Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

45 Inverse Problem Numerical Results 15 The Confidence Inervals for b (2), refinement = 1, perturb = b Noise Level Estimates for (a function of) the variance of the RTD Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35

46 Conclusions Comments on Time-domain Inverse Problems for Distributions Previous work showed that estimation methods worked well for discrete distributions and continuous uniform distribution and Gaussian distributions (using quadrature) We are able to accurately determine the mean in the Beta distributions with confidence in spite of noise Variance information is highly sensitive to noise and may be unreliable in practice with current data Need to test with very broad bandwith signal Next step is to combine multiple polarization poles (mixtures of distributions) Goal is to distinguish dry skin from wet skin and possibly determine moisture content from reflection data using broad band (THz-range) pulse modelled as a log-normal distribution of frequencies Prof Gibson (OSU) Inverse Problems for Distributions JMM / 35