![Computation of Electromagnetic Fields Scattered From Dielectric
Objects of Uncertain Shapes Using MLMC
A. Litvinenko1
, A. C. Yucel3
, H. Bagci2
, J. Oppelstrup4
, E. Michielssen5
, R. Tempone1,2
1
RWTH Aachen, 2
KAUST, 3
Nanyang Technological University in Singapore, 4
KTH Royal Institute of Technology, 5
University of Michigan
Center for Uncertainty
Quantification
Center
Quanti
Problem
Motivation: Tools for electromagnetic scattering from
objects with uncertain shapes are needed in various
applications.
Goal: Develop numerical methods for predicting radar
and scattering cross sections (RCS and SCS) of com-
plex targets.
How: To reduce cost of Monte Carlo (MC) we offer
modified multilevel MC (CMLMC) method.
CMLMC optimally balances statistical and discretiza-
tion errors. It requires very few samples on fine
meshes and more on coarse.
Excitation: Observation:
x
y
z
(r),0
0
,0
(,)
ˆu(,)
ˆuinc
(inc
,inc
)
(inc
,inc
)
-1
1
-0.5
1
0
0.5
0
1
0
-1 -1
1
0
-1-1
-0.5
0
0.5
1
0.5
0
-0.5
-1
1
1
0
-1
1
0
-1
1
0.5
0
-0.5
-1
-1.5
1. Frequency domain surface integral
equation
Electromagnetic scattering from dielectric objects is analyzed by
using the Poggio-Miller-Chan-Harrington-Wu-Tsai surface inte-
gral equation (PMCHWT-SIE) solver. The PMCHWT-SIE is dis-
cretized using the method of moments (MoM) and the iterative
solution of the resulting matrix system is accelerated using a
(parallelized) fast multipole method (FMM) - fast Fourier trans-
form (FFT) scheme (FMM-FFT).
2. Generation of random geometries
The perturbed shape is defined as
v(ϑm, ϕm) ≈ ˜v(ϑm, ϕm) +
K
k=1
akκk(ϑm, ϕm). (1)
where ϑm and ϕm are angular coordinates of node m, v(ϑm, ϕm)
is its (perturbed) radial coordinate, and ˜v(ϑm, ϕm) = 1 m is its
(unperturbed) radial coordinate on the unit sphere (all in spheri-
cal coordinate system). Here, κk(ϑ, ϕ) are obtained from spheri-
cal harmonics by re-scaling their arguments and ak, which satisfy
K
k=1 ak < 0.5, are uncorrelated random variables. Further k = 2,
κ1(ϑ, ϕ) = cos(α1ϑ) and κ2(ϑ, ϕ) = sin(α2ϑ) sin(α3ϕ), where α1, α2,
and α3 are positive integers.
Mesh P0, which is now after the application of (1), is also rotated
and scaled using the simple transformation
xm
ym
zm
= ¯L(lx, ly, lz) ¯Rx(ϕx) ¯Ry(ϕy) ¯Rz(ϕz)
xm
ym
zm
. (2)
Here, (xm, ym, zm), and (xm, ym, zm) are the coordinates of node
m before and after the transformation, matrices ¯Rx(ϕx), ¯Ry(ϕy),
and ¯Rz(ϕz) perform rotations around x, y, and z axes by angles
ϕx, ϕy, and ϕz, and matrix ¯L(lx, ly, lz) implements (down) scaling
along x, y, and z axes by lx, ly, and lz, respectively.
The random variables used in generating the coarsest perturbed
mesh P0 are the perturbation weights ak, k = 1, . . . , K, the rota-
tion angles ϕx, ϕy, and ϕz, and the scaling factors lx, ly, and lz.
Random parameter vector
ξ = (a1, . . . , aK, ϕx, ϕy, ϕz, lx, ly, lz) ∈ RK+6. (3)
The mesh P1 ( = 1) is generated by refining each triangle of the
perturbed P0 into four (by halving all three edges and connect-
ing mid-points). The mesh at level = 2, P2, is generated in the
same way from P1. All meshes P at all levels = 1, . . . , L are
nested discretizations of P0. This method of refinement results in
β = 2. Note that no uncertainties are added on meshes P , > 0;
the uncertainty is introduced only at level = 0.
An example of a perturbed shape and its refinement
An example of four nested meshes with (a) 320, (b) 1280, (c)
5120, and (d) 20480 triangular elements, which are generated for
a perturbed shape with α1 = 2, α2 = 3, α3 = 2, a1 = 0.04 m,
a2 = 0.048 m, ϕx = 0.32 rad, ϕy = 0.88 rad, ϕz = 0.81 rad,
lx = 1.06, ly = 1.08, and lz = 1.07.
Electric J(r) and magnetic M(r) surface current densities and
radar cross sections (RCS) of the perturbed geometry is signif-
icantly different than that of the unit sphere:
Amplitudes of (a) J(r) and (b) M(r) induced on the unit sphere
under excitation by an ˆx-polarized plane wave propagating in −ˆz
direction at 300 MHz. Amplitudes of (c) J(r) and (d) M(r) induced
on the perturbed shape under excitation by the same plane wave.
For all figures, amplitudes are normalized to 1 and plotted in dB
scale.
3 /4 /2 /4 0 /4 /2 3 /4
(rad)
-10
-5
0
5
10
15
20
25
rcs
(dB)
Sphere
Perturbed surface
( a )
3 /4 /2 /4 0 /4 /2 3 /4
(rad)
-10
-5
0
5
10
15
20
25
rcs
(dB)
Sphere
Perturbed surface
( b )
RCS of the unit sphere and the perturbed shape computed on
(a) xz and (b) yz planes under excitation by an ˆx-polarized plane
wave propagating in −ˆz direction at 300 MHz. For (a) ϕ = 0 and
ϕ = π rad in the first and second halves of the horizontal axis,
respectively. For (b), ϕ = π/2 rad and ϕ = 3π/2 rad in the first
and second halves of the horizontal axis, respectively.
Multilevel Monte Carlo Algorithm:
Aim: to approximate the mean E (g(u)) of QoI g(u) to a given ac-
curacy ε := TOL, where u = u(ω) and ω - random perturbations
in the domain.
Idea: Balance discretization and statistical errors. Very few sam-
ples on a fine grid and more on coarse (denote by M ).
Assume: have a hierarchy of L + 1 meshes {h }L
=0, h := h0β−
for each realization of random domain.
Consider: E (gL) = L
=0 E (g (ω) − g −1(ω)) =: L
=0 E (G ) ≈
L
=0 E ˜G , where ˜G = M−1 M
m=0 G (ω ,m).
Finally obtain the MLMC estimator: A ≈ E (g(u)) ≈ L
=0
˜G .
Cost of generating one sample of ˜G : W ∝ h
−γ
= (h0β− )−γ.
Total work of estimation A: W = L
=0 M W .
Estimator A satisfies a tolerance with a prescribed failure proba-
bility 0 < ν ≤ 1, i.e.,
P[|E (g) − A| ≤ TOL] ≥ 1 − ν (4)
while minimizing W.
3. CMLMC numerical tests
The QoI is the SCS over a user-defined solid angle Ω =
[1/6, 11/36]π rad × [5/12, 19/36]π rad (i.e., a measure of far-field
scattered power in a cone).
CMLMC is executed for random variables a1, a2 ∼ U [−0.14, 0.14]
m, ϕx, ϕy, ϕz ∼ U [0.2, 3] rad, and lx, ly, lz ∼ U [0.9, 1.1];
here U [a, b] is the uniform distribution between a and
b. The CMLMC algorithm is run for TOL values rang-
ing from 0.2 to 0.008. At TOL ≈ 0.008, CMLMC requires
L = 5 meshes with {320, 1280, 5120, 20480, 81920} triangles.
10−3
10−2
10−1
100
TOL
104
105
106
107
108
AverageTime(s)
TOL−2
CMLMC
MC Estimate
10−3
10−2
10−1
100
TOL
101
102
103
104
105
106
Workestimate
TOL−2
CMLMC
MC Estimate
(a) Computation times required by the CMLMC and MC
methods vs. TOL. (b) Value of the computational cost estimate
W for CMLMC and MC vs. TOL.
0 1 2 3 4
101
102
103
104
105
Time(s)
22
G
0 1 2 3 4
10−6
10−5
10−4
10−3
10−2
10−1
100
E
2−3
G
0 1 2 3 4
10−7
10−6
10−5
10−4
10−3
10−2
10−1
V
2−5
G
10−3
10−2
10−1
100
TOL
0.0
0.2
0.4
0.6
0.8
1.0
θ
(a) Time required to compute G vs. . (b) E = E (G ) vs. and
assumed weak convergence curve 2−3 (q1 = 3). (c) V = varG
vs. and assumed strong convergence curve 2−5 (q2 = 5).(d)
value of θ. The experiment is repeated 15 times independently
and the obtained values are shown as error bars on the curves.
-1 0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
g1-g0
-0.2 -0.1 0 0.1 0.2
0
20
40
60
80
100
120
g2-g1
g3-g2
CMLMC pdfs of g − g −1 for (a) = 1 and (b) = {2, 3}.
4. Conclusion
• Used multi-level MC method to characterize EM wave scatter-
ing from dielectric objects with uncertain shapes.
• Researched how uncertainties in the shape propagate to the
solution.
• Demonstrated that the CMLMC algorithm can be 10 times
faster than MC.
Acknowledgements: SRI UQ at KAUST and Alexander von
Humboldt foundation.
Computation of Electromagnetic Fields Scattered From Objects With Uncer-
tain Shapes Using Multilevel Monte Carlo Method, A. Litvinenko, A. C. Yu-
cel, H. Bagci, J. Oppelstrup, E. Michielssen, R. Tempone, arXiv:1809.00362,
2018. Accepted to IEEE J. on Multiscale and Multiphysics Comput. Tech-
niques, 2019.
litvinenko@uq.rwth-aachen.de](https://image.slidesharecdn.com/computationofelectromagneticfieldsscatteredfromdielectricobjectsofuncertainshapesusingmlmclitv-190214110715/85/Computation-of-electromagnetic_fields_scattered_from_dielectric_objects_of_uncertain_shapes_using_mlmc_litv-1-320.jpg)
Computation of electromagnetic_fields_scattered_from_dielectric_objects_of_uncertain_shapes_using_mlmc_litv
- 1. Computation of Electromagnetic Fields Scattered From Dielectric Objects of Uncertain Shapes Using MLMC A. Litvinenko1 , A. C. Yucel3 , H. Bagci2 , J. Oppelstrup4 , E. Michielssen5 , R. Tempone1,2 1 RWTH Aachen, 2 KAUST, 3 Nanyang Technological University in Singapore, 4 KTH Royal Institute of Technology, 5 University of Michigan Center for Uncertainty Quantification Center Quanti Problem Motivation: Tools for electromagnetic scattering from objects with uncertain shapes are needed in various applications. Goal: Develop numerical methods for predicting radar and scattering cross sections (RCS and SCS) of com- plex targets. How: To reduce cost of Monte Carlo (MC) we offer modified multilevel MC (CMLMC) method. CMLMC optimally balances statistical and discretiza- tion errors. It requires very few samples on fine meshes and more on coarse. Excitation: Observation: x y z (r),0 0 ,0 (,) ˆu(,) ˆuinc (inc ,inc ) (inc ,inc ) -1 1 -0.5 1 0 0.5 0 1 0 -1 -1 1 0 -1-1 -0.5 0 0.5 1 0.5 0 -0.5 -1 1 1 0 -1 1 0 -1 1 0.5 0 -0.5 -1 -1.5 1. Frequency domain surface integral equation Electromagnetic scattering from dielectric objects is analyzed by using the Poggio-Miller-Chan-Harrington-Wu-Tsai surface inte- gral equation (PMCHWT-SIE) solver. The PMCHWT-SIE is dis- cretized using the method of moments (MoM) and the iterative solution of the resulting matrix system is accelerated using a (parallelized) fast multipole method (FMM) - fast Fourier trans- form (FFT) scheme (FMM-FFT). 2. Generation of random geometries The perturbed shape is defined as v(ϑm, ϕm) ≈ ˜v(ϑm, ϕm) + K k=1 akκk(ϑm, ϕm). (1) where ϑm and ϕm are angular coordinates of node m, v(ϑm, ϕm) is its (perturbed) radial coordinate, and ˜v(ϑm, ϕm) = 1 m is its (unperturbed) radial coordinate on the unit sphere (all in spheri- cal coordinate system). Here, κk(ϑ, ϕ) are obtained from spheri- cal harmonics by re-scaling their arguments and ak, which satisfy K k=1 ak < 0.5, are uncorrelated random variables. Further k = 2, κ1(ϑ, ϕ) = cos(α1ϑ) and κ2(ϑ, ϕ) = sin(α2ϑ) sin(α3ϕ), where α1, α2, and α3 are positive integers. Mesh P0, which is now after the application of (1), is also rotated and scaled using the simple transformation xm ym zm = ¯L(lx, ly, lz) ¯Rx(ϕx) ¯Ry(ϕy) ¯Rz(ϕz) xm ym zm . (2) Here, (xm, ym, zm), and (xm, ym, zm) are the coordinates of node m before and after the transformation, matrices ¯Rx(ϕx), ¯Ry(ϕy), and ¯Rz(ϕz) perform rotations around x, y, and z axes by angles ϕx, ϕy, and ϕz, and matrix ¯L(lx, ly, lz) implements (down) scaling along x, y, and z axes by lx, ly, and lz, respectively. The random variables used in generating the coarsest perturbed mesh P0 are the perturbation weights ak, k = 1, . . . , K, the rota- tion angles ϕx, ϕy, and ϕz, and the scaling factors lx, ly, and lz. Random parameter vector ξ = (a1, . . . , aK, ϕx, ϕy, ϕz, lx, ly, lz) ∈ RK+6. (3) The mesh P1 ( = 1) is generated by refining each triangle of the perturbed P0 into four (by halving all three edges and connect- ing mid-points). The mesh at level = 2, P2, is generated in the same way from P1. All meshes P at all levels = 1, . . . , L are nested discretizations of P0. This method of refinement results in β = 2. Note that no uncertainties are added on meshes P , > 0; the uncertainty is introduced only at level = 0. An example of a perturbed shape and its refinement An example of four nested meshes with (a) 320, (b) 1280, (c) 5120, and (d) 20480 triangular elements, which are generated for a perturbed shape with α1 = 2, α2 = 3, α3 = 2, a1 = 0.04 m, a2 = 0.048 m, ϕx = 0.32 rad, ϕy = 0.88 rad, ϕz = 0.81 rad, lx = 1.06, ly = 1.08, and lz = 1.07. Electric J(r) and magnetic M(r) surface current densities and radar cross sections (RCS) of the perturbed geometry is signif- icantly different than that of the unit sphere: Amplitudes of (a) J(r) and (b) M(r) induced on the unit sphere under excitation by an ˆx-polarized plane wave propagating in −ˆz direction at 300 MHz. Amplitudes of (c) J(r) and (d) M(r) induced on the perturbed shape under excitation by the same plane wave. For all figures, amplitudes are normalized to 1 and plotted in dB scale. 3 /4 /2 /4 0 /4 /2 3 /4 (rad) -10 -5 0 5 10 15 20 25 rcs (dB) Sphere Perturbed surface ( a ) 3 /4 /2 /4 0 /4 /2 3 /4 (rad) -10 -5 0 5 10 15 20 25 rcs (dB) Sphere Perturbed surface ( b ) RCS of the unit sphere and the perturbed shape computed on (a) xz and (b) yz planes under excitation by an ˆx-polarized plane wave propagating in −ˆz direction at 300 MHz. For (a) ϕ = 0 and ϕ = π rad in the first and second halves of the horizontal axis, respectively. For (b), ϕ = π/2 rad and ϕ = 3π/2 rad in the first and second halves of the horizontal axis, respectively. Multilevel Monte Carlo Algorithm: Aim: to approximate the mean E (g(u)) of QoI g(u) to a given ac- curacy ε := TOL, where u = u(ω) and ω - random perturbations in the domain. Idea: Balance discretization and statistical errors. Very few sam- ples on a fine grid and more on coarse (denote by M ). Assume: have a hierarchy of L + 1 meshes {h }L =0, h := h0β− for each realization of random domain. Consider: E (gL) = L =0 E (g (ω) − g −1(ω)) =: L =0 E (G ) ≈ L =0 E ˜G , where ˜G = M−1 M m=0 G (ω ,m). Finally obtain the MLMC estimator: A ≈ E (g(u)) ≈ L =0 ˜G . Cost of generating one sample of ˜G : W ∝ h −γ = (h0β− )−γ. Total work of estimation A: W = L =0 M W . Estimator A satisfies a tolerance with a prescribed failure proba- bility 0 < ν ≤ 1, i.e., P[|E (g) − A| ≤ TOL] ≥ 1 − ν (4) while minimizing W. 3. CMLMC numerical tests The QoI is the SCS over a user-defined solid angle Ω = [1/6, 11/36]π rad × [5/12, 19/36]π rad (i.e., a measure of far-field scattered power in a cone). CMLMC is executed for random variables a1, a2 ∼ U [−0.14, 0.14] m, ϕx, ϕy, ϕz ∼ U [0.2, 3] rad, and lx, ly, lz ∼ U [0.9, 1.1]; here U [a, b] is the uniform distribution between a and b. The CMLMC algorithm is run for TOL values rang- ing from 0.2 to 0.008. At TOL ≈ 0.008, CMLMC requires L = 5 meshes with {320, 1280, 5120, 20480, 81920} triangles. 10−3 10−2 10−1 100 TOL 104 105 106 107 108 AverageTime(s) TOL−2 CMLMC MC Estimate 10−3 10−2 10−1 100 TOL 101 102 103 104 105 106 Workestimate TOL−2 CMLMC MC Estimate (a) Computation times required by the CMLMC and MC methods vs. TOL. (b) Value of the computational cost estimate W for CMLMC and MC vs. TOL. 0 1 2 3 4 101 102 103 104 105 Time(s) 22 G 0 1 2 3 4 10−6 10−5 10−4 10−3 10−2 10−1 100 E 2−3 G 0 1 2 3 4 10−7 10−6 10−5 10−4 10−3 10−2 10−1 V 2−5 G 10−3 10−2 10−1 100 TOL 0.0 0.2 0.4 0.6 0.8 1.0 θ (a) Time required to compute G vs. . (b) E = E (G ) vs. and assumed weak convergence curve 2−3 (q1 = 3). (c) V = varG vs. and assumed strong convergence curve 2−5 (q2 = 5).(d) value of θ. The experiment is repeated 15 times independently and the obtained values are shown as error bars on the curves. -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 g1-g0 -0.2 -0.1 0 0.1 0.2 0 20 40 60 80 100 120 g2-g1 g3-g2 CMLMC pdfs of g − g −1 for (a) = 1 and (b) = {2, 3}. 4. Conclusion • Used multi-level MC method to characterize EM wave scatter- ing from dielectric objects with uncertain shapes. • Researched how uncertainties in the shape propagate to the solution. • Demonstrated that the CMLMC algorithm can be 10 times faster than MC. Acknowledgements: SRI UQ at KAUST and Alexander von Humboldt foundation. Computation of Electromagnetic Fields Scattered From Objects With Uncer- tain Shapes Using Multilevel Monte Carlo Method, A. Litvinenko, A. C. Yu- cel, H. Bagci, J. Oppelstrup, E. Michielssen, R. Tempone, arXiv:1809.00362, 2018. Accepted to IEEE J. on Multiscale and Multiphysics Comput. Tech- niques, 2019. litvinenko@uq.rwth-aachen.de