Polynomial Chaos Approach for Simulations in Dispersive Mediagibsonn/njit2010gibson.pdf · September 9, 2010 N. L. Gibson (OSU) Polynomial Chaos for Stochastic Polarization NJIT Sep

Polynomial Chaos Approach for Simulations in

Dispersive Media

N. L. GibsonV. A. Bokil

Department of MathematicsOregon State University

September 9, 2010

N. L. Gibson (OSU) Polynomial Chaos for Stochastic Polarization NJIT Sep 2010 1 / 41

Acknowledgments

Karen Barrese and Neel Chugh (REU 2008)

Anne Marie Milne and Danielle Wedde (REU 2009)

Erin Bela and Erik Hortsch (REU 2010)


Outline

1 Maxwell’s EquationsDescriptionPolarization ModelsDistribution of Relaxation Times

2 Polynomial ChaosStochastic PolarizationGalerkin Projection

3 DiscretizationThe Yee SchemeTime Discretization of PC SolutionStability AnalysisNumerical Simulations


Maxwell’s Equations Description

Maxwell’s Equations

∂D

∂t+ J = ∇× H (Ampere)

∂B

∂t= −∇× E (Faraday)

∇ · D = ρ (Poisson)

∇ · B = 0 (Gauss)

E = Electric field vector

H = Magnetic field vector

ρ = Electric charge density

D = Electric displacement

B = Magnetic flux density

J = Current density

With appropriate initial conditions and boundary conditions.N. L. Gibson (OSU) Polynomial Chaos for Stochastic Polarization NJIT Sep 2010 4 / 41

Maxwell’s Equations Description

Constitutive Laws

Maxwell’s equations are completed by constitutive laws that describethe response of the medium to the electromagnetic field.

D = ǫE + P

B = µH + M

J = σE + Js

P = Polarization

M = Magnetization

Js = Source Current

ǫ = Electric permittivity

µ = Magnetic permeability

σ = Electric Conductivity


Maxwell’s Equations Polarization Models

Complex permittivity

We can 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 the dielectric response function (DRF).

In the frequency domain D = ǫ0ǫ(ω)E, where ǫ(ω) is called thecomplex permittivity.

ǫ(ω) described by the polarization model (and conductivity)

We are interested in ultra-wide bandwidth electromagnetic pulseinterrogation of dispersive dielectrics, therefore we want anaccurate representation of ǫ(ω) over a broad range offrequencies.



Dispersive Media

0 0.2 0.4 0.6 0.8−1000

−500

0

500

1000

z

E

Time=5.0ns

0 0.2 0.4 0.6 0.8−80

−60

−40

−20

0

20

40

Time=10.0ns

E

z

Figure: Debye model simulations.



Dry skin data

102

104

106

108

1010

101

102

103

f (Hz)

ε

True DataDebye ModelCole−Cole Model

Figure: Real part of ǫ(ω), ǫ, or the permittivity [GLG96].



Dry skin data

102

104

106

108

1010

10−4

10−3

10−2

10−1

100

101

f (Hz)

σ

True DataDebye ModelCole−Cole Model

Figure: Imaginary part of ǫ(ω), σ, or the conductivity.



Sample models

Debye model [1929] q = [ǫd , τ ]

g(t, x) = ǫ0ǫd/τ e−t/τ

or τ P + P = ǫ0ǫdE

or ǫ(ω) = ǫ∞ +ǫd

1 + iωτ

with ǫd := ǫ0(ǫs − ǫ∞).



Sample models

Debye model [1929] q = [ǫd , τ ]

g(t, x) = ǫ0ǫd/τ e−t/τ

or τ P + P = ǫ0ǫdE

or ǫ(ω) = ǫ∞ +ǫd

1 + iωτ

with ǫd := ǫ0(ǫs − ǫ∞).

Cole-Cole model [1936] (heuristic generalization)q = [ǫd , τ, α]

ǫ(ω) = ǫ∞ +ǫd

1 + (iωτ)1−α


Maxwell’s Equations Distribution of Relaxation Times

Motivation

Broadband wave propagation suggests time-domain simulation.

The Cole-Cole model corresponds to a fractional order ODE inthe time-domain and is difficult to simulate.

Debye is efficient to simulate, but does not represent permittivitywell.

Better fits to data are obtained by taking linear combinations ofDebye models (discrete distributions), idea comes from theknown existence of multiple physical mechanisms.

An alternative approach is to consider the Debye model but witha (continuous) distribution of relaxation times [vonSchweidler1907].

Empirical measurements suggest a log-normal distribution[Wagner1913], but uniform is easier.


Maxwell’s Equations Fit to dry skin data with uniform distribution

102

104

106

108

1010

102

103

f (Hz)

ε

True DataDebye (27.79)Cole−Cole (10.4)Uniform (13.60)

Figure: Real part of ǫ(ω), ǫ, or the permittivity [REU2008].



102

104

106

108

1010

10−3

10−2

10−1

100

101

f (Hz)

σ

True DataDebye (27.79)Cole−Cole (10.4)Uniform (13.60)

Figure: Imaginary part of ǫ(ω)/ω, σ, or the conductivity [REU2008].



Distributions of Parameters

To account for the effect of possible multiple parameter sets q,consider

h(t, x; F ) =

∫

Q

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

h(t − s, x; F )E(s, x)ds.

The inverse problem for F given time domain electric field data iswell-posed [BG05, BG06].


Maxwell’s Equations Distribution of Relaxation Times

We define the stochastic polarization P(t, x; τ) to be the solution to

τ P + P = ǫ0ǫdE

where τ is a random variable with PDF f (τ), for example,

f (τ) =1

τb − τa

for a uniform distribution.The electric field depends on the macroscopic polarization, which wetake to be the expected value of the stochastic polarization at eachpoint (t, x)

P(t, x) =

∫ τb

τa

P(t, x; τ)f (τ)dτ.


Polynomial Chaos Stochastic Polarization

We can apply the generalized Polynomial Chaos method [XK03] tothe random ordinary differential equation (at each point in space andeach dimension independently)

τ P + P = ǫ0ǫdE , τ = τ(ξ) = τσξ + τµ

where ξ ∼ U(−1, 1), for example.We apply a Polynomial Chaos expansion in terms of orthogonalpolynomials φj(ξ) to the solution P:

P(t, ξ) =

∞∑

j=0

αj(t)φj(ξ).

The RODE becomes

(τσξ + τµ)∞

∑

j=0

αj(t)φj(ξ) +∞

∑

j=0

αj(t)φj(ξ) = ǫdE .


Polynomial Chaos Triple recursion formula

(τσξ + τµ)

∞∑

j=0

αj(t)φj(ξ) +

∞∑

j=0

αj(t)φj(ξ) = ǫdE

We can eliminate the explicit dependence on ξ by using the triplerecursion formula for orthogonal polynomials

ξφj = ajφj−1 + bjφj + cjφj+1

(with φ−1 = 0), for example, for Legendre polynomials

(2j + 1)ξφj = jφj−1 + (j + 1)φj+1.

In general, the RODE becomes

τσ

∞∑

j=0

αj(t)(ajφj−1 + bjφj + cjφj+1) + τµ

∞∑

j=0

αj(t)φj

+

∞∑

j=0

αj(t)φj = ǫdE .


Polynomial Chaos Galerkin Projection onto span({φj}pj=0

)

We take the weighted inner product with each basis {φj}pj=0 for a

fixed p resulting in the approximating system

(τσM + τµI )~α + ~α = ǫdE ~e1,

where ~α = [α0(t), . . . , αp(t)]T and

M =

b0 a1

c0 b1 a2

. . .. . .

. . .. . .

. . . ap

cp−1 bp

,

or, more simply,A~α + ~α = ~g .

The macroscopic polarization is taken to be the expected value of thestochastic polarization at each point (t, x), for each dimension

P(t, x) = E [P(t, x)] ≈ α0(t, x).



)

Exponential convergence

Any set of orthogonal polynomials can be used in the truncatedexpansion, but there may be an optimal choice.

If the polynomials are orthogonal with respect to weightingfunction f (ξ), and k has PDF f (k), then it is known that thePC solution to the ODE converges exponentially in terms of p.

In practice, approximately 4 are generally sufficient.



)

Generalized Polynomial Chaos

Table: Popular distributions and corresponding orthogonal polynomials.

Distribution Polynomial SupportGaussian Hermite (−∞,∞)gamma Laguerre [0,∞)beta Jacobi [a, b]

uniform Legendre [a, b]

Note: log-normal random variables may be handled as a non-linearfunction (e.g., Taylor expansion) of a normal random variable.



)

Existence of PC Solutions

Theorem (REU2010)

For the beta-Jacobi chaos (including uniform-Legendre), there existssolutions to the system

A~α + ~α = ~g

for any p.

Proof.Relies on the fact that the invertibility of A requires τµ > τσ. This isphysically reasonable as to disallow negative relaxation times.


Discretization Maxwell’s Equations in One Dimension

Assume uniformity in the x-direction.

Assume that the electric field is polarized to oscillate only in they direction.

Evolution equations involvingE , H, D, B , P and J :

∂D

∂t+ J =

∂H

∂z∂B

∂t=

∂E

∂z

Constitutive laws:

B = µH

D = ǫE + P

J = σE + Js

x

y

x

Ey

H z

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Discretization The Yee Scheme

Applying the central difference approximation, based on the Yeescheme, our one dimensional equations

ǫ∂E

∂t= −

∂H

∂z− σE −

∂P

∂t

and

µ∂H

∂t= −

∂E

∂zbecome

En+ 1

2k − E

n− 12

k

∆t= −

1

ǫ

Hn

k+ 12

− Hn

k− 12

∆z−

σ

ǫ

En+ 1

2k + E

n− 12

k

2−

1

ǫ

Pn+ 1

2k − P

n− 12

k

∆t

andHn+1

k+ 12

− Hn

k+ 12

∆t= −

1

µ

En+ 1

2k+1 − E

n+ 12

k

∆z.

Note that while the electric field and magnetic field are staggered intime, the electric field updates simultaneously with polarization.


Discretization Time Discretization of PC Solution

We discretize the PC system

A~α + ~α = ~g

by applying central differences to ~α = ~α(zk) for arbitrary zk

A~αn+ 1

2 − ~αn− 12

∆t+

~αn+ 12 + ~αn− 1

2

2=

~g n+ 12 + ~g n− 1

2

2.

Combining like terms gives

(2A + ∆tI )~αn+ 12 = (2A − ∆tI )~αn− 1

2 + ∆t(

~g n+ 12 + ~g n− 1

2

)

Note that we may first solve the discrete electric field equation for

En+ 1

2k and plug into ~g n+ 1

2 to define a decoupled update step for ~α.


Discretization Stability Analysis

Stability Analysis

We look for plane wave solutions and assume spatial dependence ofthe form

Hn+1j+ 1

2

= Hn+1(k)eik(j+ 12)∆z

En+ 1

2j = E n+ 1

2 (k)eikj∆z

αn+ 1

20, j = α

n+ 12

0 (k)eikj∆z

...

αn+ 1

2p, j = α

n+ 12

p (k)eikj∆z

where k is the wave number.



Substituting the above into our numerical method we obtain

BUn+1 = CUn

whereUn := [Hn, E n+ 1

2 , α0n+ 1

2 , . . . , αpn+ 1

2 ]

B :=

[

B11 BT12

B21 2A + ∆tI

]

B11 :=

[

1 γ/µ

0 θ+

]

B12 :=

0 20 0...

...0 0

B21 :=

0 −∆t ǫd

0 0...

...0 0

θ+ := 2ǫ + σ∆t γ :=2i∆t

∆zsin

(

k∆z

2

)



Continuing:BUn+1 = CUn

where

B :=

[

B11 BT12

B21 2A + ∆tI

]

B11 :=

[

1 γ/µ

0 θ+

]

C :=

[

C11 BT12

−B21 2A − ∆tI

]

C11 :=

[

1 0

−2γ θ−

]

θ+ := 2ǫ + σ∆t θ− := 2ǫ − σ∆t

γ :=2i∆t

∆zsin

(

k∆z

2

)

Note: for p = 0, A = τµ and we recover single Debye equations.



Stability of uniform-Legendre Chaos system

Theorem (REU2010)

The numerical polynomial chaos scheme is stable for Legendrepolynomials with p = 1 if and only if the following conditions hold

ν ≤ 1

ǫs ≥ ǫ∞

τµ ≥ 0.

Proof.Direct application of Routh-Horwitz criteria

The last condition again disallows negative relaxation times.



Numerical Stability Analysis

If the modulus of all the generalized (complex, time) eigenvaluesof (B , C ) are less than one, the method is stable.

The stability polynomial given by det(C −λB) is of degree p +3.

The roots depend on material and discretization parametersincluding: k∆z (quantifies ppw), h := ∆t/τµ (temporalresolution), ν (relates ∆z and ∆t), as well as τσ (quantifiesvariance).

We plot the largest modulus of λ as a function of k∆z in thefollowing with all other parameters fixed.



0 0.5 1 1.5 2 2.5 30.8

0.85

0.9

0.95

1

k ∆ z

Polynomial Chaos Debye dissipation with ν=1 and h=0.1m

ax |ζ

|

τσ=0

τσ=0.25 τµ

τσ=0.5 τµ

τσ=1 τµ

τσ=1.1 τµ

Figure: Using parameters of dry skin data and p = 2



0 0.5 1 1.5 2 2.5 3

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

k ∆ z

Single−pole Debye dissipation with ν=1 and h=0.1m

ax |ζ

|

τ=0.25 τµτ=0.5 τµτ=1 τµτ=2 τµτ=4 τµ




0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.98

0.985

0.99

0.995

1

k ∆ z

Polynomial Chaos Debye dissipation with ν=1 and h=0.01m

ax |ζ

|

τσ=0

τσ=0.25 τµ

τσ=0.5 τµ

τσ=1 τµ

τσ=1.1 τµ




0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.9

0.92

0.94

0.96

0.98

1

k ∆ z

Single−pole Debye dissipation with ν=1 and h=0.01m

ax |ζ

|

τ=0.25 τµτ=0.5 τµτ=1 τµτ=2 τµτ=4 τµ



Discretization Numerical Simulations

Numerical Simulations

Windowed 1010 Hz signal propagation in a stochastic Debyedielectric model of water.

Time trace measured at 0.015 m inside material.

Let hτ := ∆t/τµ, where τµ = 8.13 × 10−12 is the measureddeterministic value.

We use Uniform-Legendre chaos expansions with, for example,τ ∼ U[.75τµ, 1.25τµ].



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10

−9

−100

−80

−60

−40

−20

0

20

40

60

80

100

t

E

[.75,1.25]τµ, htau=0.025

p=0p=1p=2

Figure: Using parameters of dry skin data with τ ∼ U[.75τµ, 1.25τµ], andusing p = 0, 1, 2 polynomials. Shows significant convergence after justp = 1.



0 1 2 3 4 5 6 7 8 9 1010

−12

10−10

10−8

10−6

10−4

10−2

100

102

Maximum Difference Calculated for different values of p and r

p

Max

imum

Err

or

r = 1.00τ

r = 0.75τ

r = 0.50τ

r = 0.25τ

Figure: Maximum Error for various values of p and r .



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10

−9

−100

−80

−60

−40

−20

0

20

40

60

80

100

t

E

p=0, htau=0.025

τ=0.75*τµ

τ=1*τµ

τ=1.25*τµ

Figure: Using parameters of dry skin data with deterministicτ ∈ [.75τµ, 1.25τµ]. Shows suggests that stochastic polarization will haveslightly higher amplitude if considered as an average of these simulations.



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10

−9

−100

−80

−60

−40

−20

0

20

40

60

80

100

t

E

p=0, [.75,1.25]τµ

htau=2.500000e−01htau=1.000000e−01htau=5.000000e−02htau=2.500000e−02

Figure: Using parameters of dry skin data and p = 0. Shows hτ = 0.01required for accuracy.



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10

−9

−100

−80

−60

−40

−20

0

20

40

60

80

100

t

E

p=1, [.75,1.25]τµ


Figure: Using parameters of dry skin data and p = 1. Shows hτ = 0.005required for accuracy. Non-zero variance implies smaller relaxation timesare present.



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10

−9

−100

−80

−60

−40

−20

0

20

40

60

80

100

t

E

p=2, [.75,1.25]τµ


Figure: Using parameters of dry skin data and p = 2. Shows hτ = 0.005required for accuracy. As expected, including more polynomials does notreduce temporal resolution errors.



Conclusions

Stochastic Polarization well suited for modeling realisticdielectric materials

Distributions of parameters avoids fractional order derivativemodels

Polynomial Chaos is a simple-to-use method for efficientlysimulating stochastic polarization

Stability properties of the numerical method are similar todeterministic case

Stochastic polarization exhibits less dissipation for comparablediscretization parameters


References

H. T. Banks and N. L. Gibson.Well-posedness in Maxwell systems with distributions ofpolarization relaxation parameters.Applied Math Letters, 18(4):423–430, 2005.

HT Banks and NL Gibson.Electromagnetic inverse problems involving distributions ofdielectric mechanisms and parameters.Quarterly of Applied Mathematics, 64(4):749, 2006.

S. Gabriel, RW Lau, and C. Gabriel.The dielectric properties of biological tissues: III.Phys. Med. Biol, 41(11):2271–2293, 1996.

D. Xiu and G. E. Karniadakis.The Wiener-Askey polynomial chaos for stochastic differentialequations.SIAM Journal on Scientific Computing, 24(2):619–644, 2003.


Documents

Polynomial Chaos Approach for Simulations in Dispersive Mediagibsonn/njit2010gibson.pdf · September 9, 2010 N. L. Gibson (OSU) Polynomial Chaos for Stochastic Polarization NJIT Sep