Computation of electromagnetic fields scattered from dielectric objects of uncertain shapes using MLMC

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
Workshop “Scattering by random heterogeneous media”,
13-15 Sept. 2021, Augsburg, Germany

Motivation
Efficient computation tools for characterizing scattering from objects
of uncertain shapes are needed in the fields of electromagnetics,
optics, and photonics.
How: Use CMLMC method (advanced version of Multi-Level Monte
Carlo).
CMLMC optimally balances statistical and discretization errors. It
requires very few samples on fine meshes and more on coarse.
taken from wiki, reddit.com, EMCoS
1/40

Plan:
1. Scattering problem setup
2. Deterministic solver
3. Generation of random shapes
4. Shape transformation
5. QoI on perturbed shape
6. Continuation Multi Level Monte Carlo (CMLMC)
7. Results (time, work vs. TOL, weak and strong convergences)
8. Conclusion
2/40

Scattering problem
Input: randomly perturbed shape
Output: radar and scattering cross sections, electric and magnetic
surface current densities
3/40

Previous works
I Monte Carlo (N. Gacia, Jandhyala, Michielssen,)
I surrogate methods (A. Yucel, H. Bagci, L. Gomez, L.H. Garcia)
I stochastic collocation ([C. Chauviere, J. Hesthaven, K. Wilcox’07],
[D. Xiu, J. Hesthaven’07], [Zh. Zeng, J. M. Jin’07])
4/40

Deterministic solver
Electromagnetic scattering from dielectric objects is analyzed by
using the Poggio-Miller-Chan-Harrington-Wu-Tsai surface integral
equation (PMCHWT-SIE) solver.
The PMCHWT-SIE is discretized 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 transform (FFT) scheme (FMM-FFT).
Input uncertainties: position, orientation, roughness and shape of
scatterers, internal and/or external excitation characteristics such as
the frequency, amplitude, and angle of arrival.
5/40

Generation of random shapes
Perturbed shape v(ϑm, ϕm) is defined as
v(ϑm, ϕm) ≈ ṽ(ϑm, ϕm) +
K
X
k=1
akκk(ϑm, ϕm). (1)
where ϑm and ϕm are angular coordinates of node m,
ṽ(ϑm, ϕm) = 1 m is unperturbed radial coordinate on the unit sphere.
κk(ϑ, ϕ) obtained from spherical harmonics by re-scaling their
arguments, κ1(ϑ, ϕ) = cos(α1ϑ), κ2(ϑ, ϕ) = sin(α2ϑ) sin(α3ϕ),
where α1, α2, α3 > 0.
-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
6/40

Some fun (un)realistic uncertain shapes
7/40

Mesh transformation
The perturbed mesh P0 is also rotated and scaled using the following
transformation


xm
ym
zm

:=L(lx, ly, lz)Rx(ϕx)Ry(ϕy)Rz(ϕz)


xm
ym
zm

 , (2)
matrices Rx(ϕx), Ry(ϕy), and Rz(ϕz) perform rotations around x, y,
and z axes by angles ϕx, ϕy, and ϕz,
matrix L(lx, ly, lz) implements scaling along x, y, and z axes by lx, ly,
and lz, respectively.
8/40

Random rotation, stretching and expanding
rotations around axes x, y, and z by angles ϕx, ϕy, and ϕz:
Rx(ϕx) =


1 0 0
0 cos ϕx − sin ϕx
0 sin ϕx cos ϕx


Ry(ϕy) =


cos ϕy 0 sin ϕy
0 1 0
− sin ϕy 0 cos ϕy


Rz(ϕz) =


cos ϕz − sin ϕz 0
sin ϕz cos ϕz 0
0 0 1

 .
L(lx, ly, lz) implements scaling along axes x,y,z by factors lx, ly, and lz:
L̄(lx, ly, lz) =


1/lx 0 0
0 1/ly 0
0 0 1/lz

 .
9/40

Input random vector
RVs used in generating the coarsest perturbed mesh P0 are:
1. perturbation weights ak, k = 1, . . . , K,
2. rotation angles ϕx, ϕy, and ϕz,
3. scaling factors lx, ly, and lz.
Thus, random input parameter vector:
ξ = (a1, . . . , aK , ϕx, ϕy, ϕz, lx, ly, lz) ∈ RK+6
(3)
defines the perturbed shape.
10/40

Mesh refinement
Mesh P0: the coarsest discretisation of the sphere (e.g., icosahedron)
Mesh P`=1 is generated by refining each triangle of the perturbed P0
into four (by halving all three edges and connecting mid-points).
Mesh P2 is generated in the same way from P1.
All meshes P` at all levels ` = 1, . . . , L are nested discretizations of
P0.
(!!!) No uncertainties are added on meshes P`, ` > 0;
the uncertainty is introduced only at level ` = 0.
11/40

Refinement of the perturbed shape
4 nested meshes with {320, 1280, 5120, 20480} triangular elements.12/40

Electric (left) and magnetic (right) surface current densities
Amplitudes: a) J(r); b) M(r) (sphere); c) J(r); d) M(r) (perturbed shape).
13/40

Electric (left) and magnetic (right) surface current densities
Amplitudes of (a) J(r) and (b) M(r) induced on the unit sphere
under excitation by an x̂-polarized plane wave propagating in −ẑ
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.
14/40

QoI: RCS and SCS
To compute RCS and SCS, the scatterer is excited by a plane wave
Einc
(r).
σrcs
(ϑ, ϕ) =

2 , (4)
F(ϑ, ϕ) is the scattered electric field pattern in the far field.
The SCS Csca
(Ω) is obtained by integrating σrcs
(ϑ, ϕ) over the angle
Ω:
Csca
(Ω) =
1
4π
Z
Ω
σrcs
(ϑ, ϕ) sin ϑdϑdϕ. (5)
15/40

RCS of unit sphere and perturbed shape
3 /4 /2 /4 0 /4 /2 3 /4
(rad)
-10
-5
0
5
10
15
20
25
rcs
(dB)
Sphere
Perturbed surface
3 /4 /2 /4 0 /4 /2 3 /4
(rad)
-10
-5
0
5
10
15
20
25
rcs
(dB)
Sphere
Perturbed surface
RCS is computed on
(top) xz
(bottom) yz planes
under excitation by an x̂-polarized plane
wave propagating in −ẑ direction at
300 MHz.
(top) ϕ = 0 and ϕ = π rad in the first
and second halves of the horizontal axis,
respectively.
(bottom) ϕ = π/2 rad and ϕ =
3π/2 rad in the first and second halves
of the horizontal axis.
16/40

Multilevel Monte Carlo Algorithm
Aim: to approximate the mean E[g(u)] of QoI g(u) to a given
accuracy ε := TOL, where u = u(ω) randoom shape.
Input: a hierarchy of L + 1 meshes {h`}L
`=0, h` := h0β−`
for each
realization of random domain.
Compute:
E[gL] =
PL
`=0 E[g`(ω) − g`−1(ω)] =:
PL
`=0 E[G`] ≈
PL
`=0 E
h
G̃`
i
,
where G̃` = M−1
`
PM`
m=0 G`(ω`,m).
Output: A ≈ E[g(u)] ≈
PL
`=0 G̃`.
Cost of one sample of G̃`: W` ∝ h−γ
` = (h0β−`
)−γ
.
Total work of estimation A: W =
PL
`=0 M`W`.
Estimator A satisfies a tolerance with a prescribed failure probability
0 < ν ≤ 1, i.e.,
P[|E[g] − A| ≤ TOL] ≥ 1 − ν (6)
while minimizing W .
17/40

Main idea of (C)MLMC method
Let {P`}L
`=0 be sequences of meshes with h` = h0β−`
, β > 1. Let
g`(ξ) represent the approximation to g(ξ) computed using mesh P`.
E[gL] =
L
X
`=0
E[G`] (7)
where G` is defined as
G` =
(
g0 if ` = 0
g` − g`−1 if ` > 0
. (8)
Note that g` and g`−1 are computed using the same input random
parameter ξ.
18/40

E[G`] ≈
∼
G` = M−1
`
PM`
m=1 G`,m,
E[g − g`] ≈ QW hq1
` (9a)
Var[g` − g`−1] ≈ QShq2
`−1 (9b)
for QW 6= 0, QS > 0, q1 > 0, and 0 < q2 ≤ 2q1.
QoI A =
PL
`=0
∼
G`.
Let the average cost of generating one sample of G` (cost of one
deterministic simulation for one random realization) be
W` ∝ h−dγ
` = h−dγ
0 β`dγ
(10)
The computational cost and memory requirement of the FMM-FFT
accelerated iterative solution scale as O(Niter
N4/3
log2/3
N) and
O(N4/3
log2/3
N).
19/40

The total CMLMC computational cost is
W =
L
X
`=0
M`W`. (11)
The estimator A satisfies a tolerance with a prescribed failure
probability 0 < ν ≤ 1, i.e.,
P[|E[g] − A| ≤ TOL] ≥ 1 − ν (12)
while minimizing W . The total error is split into bias and statistical
error,
|E[g] − A| ≤ |E[g − A]|
| {z }
Bias
+ |E[A] − A|
| {z }
Statistical error
20/40

Let θ ∈ (0, 1) be a splitting parameter, so that
TOL = (1 − θ)TOL
| {z }
Bias tolerance
+ θTOL
| {z }
Statistical error tolerance
. (13)
The CMLMC algorithm bounds the bias, B = |E[g − A]|, and the
statistical error as
B = |E[g − A]| ≤ (1 − θ)TOL (14)
|E[A] − A| ≤ θTOL (15)
where the latter bound holds with probability 1 − ν.
To satisfy condition in (15) we require:
Var[A] ≤

θTOL
Cν
2
(16)
for some given confidence parameter, Cν, such that Φ(Cν) = 1 − ν
2,
Φ is the cdf of a standard normal random variable.
21/40

By construction of the MLMC estimator, E[A] = E[gL], and by
independence Var[A] =
PL
`=0 V`M−1
` , where V` = Var[G`].
Given L, TOL, and 0 θ 1, and by minimizing W obtain the
following optimal number of samples per level `:
M` =

Cν
θTOL
2
s
V`
W`
L
X
`=0
p
V`W`
!
. (17)
Summing the optimal numbers of samples over all levels yields:
W (TOL, L) =

Cν
θTOL
2 L
X
`=0
p
V`W`
!2
. (18)
(this is the total optimal computational cost in terms of TOL)
22/40

CMLMC numerical tests
General settings: the scatterer resides in free space (vacuum) with
µ0 = 4π × 10−7
H/m and ε0 = c2
0/µ0 F/m, where c0 is the speed of
light in vacuum, and the scatterer’s permittivity and permeability are
µ1 = µ0 and ε1 = 4ε0, respectively.
For the plane wave excitation, E0 = x̂, ûinc
= −ẑ, and ω = 2πf with
f = 300 MHz. Parameters of the quasi-spherical harmonics are
α1 = 2, α2 = 3, and α3 = 2.
15 independent CMLMC runs are executed for each experiment to
statistically characterize the performance of the method.
23/40

CMLMC numerical tests
Computing electric (J(r)) and magnetic (M(r)) surface
current densities:
The intermediate matrix system is solved iteratively using the
transpose-free quasi-minimal residual (TFQMR) method with a
tolerance (residual) of ` = 0β−2`
, where ` = 0, 1, 2, 3, 4 represent
mesh levels, 0 = 6.0 × 10−4
, and β = 2.
The intermediate integrals were computed by the Gaussian
quadrature rules.
For all levels of meshes, the FMM-FFT scheme ensures six digits of
accuracy.
All simulations were done on Intel(R) Xeon(R) CPU E5-2680 v2 @
2.80 GHz DELL workstations with 40 cores and 128 GB RAM.
24/40

Test 1
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).
Uniform RVs are:
a1, a2 ∼ U[−0.14, 0.14] m,
ϕx, ϕy, ϕz ∼ U[0.2, 3] rad,
lx, ly, lz ∼ U[0.9, 1.1];
CMLMC runs for TOL ranging from 0.2 to 0.008.
At TOL ≈ 0.008, CMLMC requires L = 5 meshes with
{320, 1280, 5120, 20480, 81920} triangles.
25/40

Probability density functions of (g` − g`−1)
-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
(left) ` = 1, i.e., pdf of g1 − g0;
(right) ` = {2, 3}, i.e., pdfs of g2 − g1 and g3 − g2.
We observe that the variance is decreasing with increasing the level `.
26/40

Average time vs. TOL
10−3
10−2
10−1
100
TOL
104
105
106
107
108
Average
Time
(s)
TOL−2
CMLMC
MC Estimate
The experiment is repeated 15 times independently and the obtained
values are shown as error bars on the curves. 27/40

Work estimate vs. TOL
10−3
10−2
10−1
100
TOL
101
102
103
104
105
106
Work
estimate
TOL−2
CMLMC
MC Estimate
28/40

Time required to compute G` vs. `.
0 1 2 3 4
`
101
102
103
104
105
Time
(s)
22`
G`
29/40

E` = E (G`) vs. ` (weak convergence)
0 1 2 3 4
`
10−6
10−5
10−4
10−3
10−2
10−1
100
E
`
2−3`
G`
assumed weak convergence curve 2−3`
(q1 = 3).
30/40

V` = Var[G`] vs. ` (strong convergence)
0 1 2 3 4
`
10−7
10−6
10−5
10−4
10−3
10−2
10−1
V
`
2−5`
G`
assumed strong convergence curve 2−5`
(q2 = 5).
31/40

Value of θ
10−3
10−2
10−1
100
TOL
0.0
0.2
0.4
0.6
0.8
1.0
θ
32/40

Numerical Test 2: with lower stoch. variability
a1, a2 ∼ U[0.014, 0.49] m,
ϕx, ϕy, ϕz ∼ U[0.2, 1.0] rad, and lx, ly, lz ∼ U[0.8, 1.2].
CMLMC algorithm requires 4 mesh levels, i.e., P`, ` = 0, 1, 2, 3.
10−2
10−1
100
TOL
104
105
106
107
Average
Time
(s)
TOL−2
CMLMC
MC Estimate
Computation time of CMLMC and MC methods as function of TOL.
33/40

Computation of electromagnetic fields scattered from dielectric objects of uncertain shapes using MLMC

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