Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (\CMLMC) method is used together with a surface integral equation solver. The \CMLMC method optimally balances statistical errors due to sampling of the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine. The number of realizations of finer discretizations can be kept low, with most samples computed on coarser discretizations to minimize computational cost. Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.

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
GAMM 2020 in Kassel, Germany

2. Motivation: Comput. 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 complex targets.
How: To reduce comp. cost we 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 1/33

3. 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
9. Best practices
10. Appendix (details of CMLMC)
2/33

4. Scattering problem
Excitation: Observation:
x
y
z
(r),0
0
,0
(,)
ˆu(,)
ˆuinc
(inc
,inc
)
(inc
,inc
)
Input: randomly perturbed shape
Output: radar and scattering cross sections, electric and magnetic
surface current densities 3/33

5. 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).
4/33

6. Generation of random shapes
Perturbed shape v(ϑm, ϕm) 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) = 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
5/33

7. Mesh transformation
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)
Node (xm, ym, zm) before and after transformation.
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.
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8. 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

 .
Matrix L(lx, ly, lz) implements scaling along axes x, y, z by factors lx,
ly, and lz:
¯L(lx, ly, lz) =


1/lx 0 0
0 1/ly 0
0 0 1/lz

 .
7/33

9. 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.
8/33

10. Mesh refinement
Mesh P1 ( = 1) is generated by refining each triangle of the
perturbed P0 into four (by halving all three edges and connecting
mid-points).
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.
(!!!) No uncertainties are added on meshes P , > 0;
the uncertainty is introduced only at level = 0.
9/33

11. Refinement of the perturbed shape
4 nested meshes with {320, 1280, 5120, 20480} triangular elements.
10/33

12. Electric (left) and magnetic (right) surface current densities
11/33

13. 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 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.
12/33

14. 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
(a) xz and (b) yz planes
under excitation by an ˆx-polarized
plane wave propagating in −ˆz di-
rection at 300 MHz.
(a) ϕ = 0 and ϕ = π rad in the
first and second halves of the hori-
zontal axis, respectively.
(b)ϕ = π/2 rad and ϕ = 3π/2 rad
in the first and second halves of the
horizontal axis.
13/33

15. 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) and ω - random perturbations
in the domain.
Idea: Balance discretization and statistical errors. Very few samples
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.
14/33

16. 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).
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];
U[a, b] is the uniform distribution between a and b.
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.
15/33

17. Average time vs. TOL
10−3
10−2
10−1
100
TOL
104
105
106
107
108
AverageTime(s)
TOL−2
CMLMC
MC Estimate
16/33

18. Work estimate vs. TOL
10−3
10−2
10−1
100
TOL
101
102
103
104
105
106
Workestimate
TOL−2
CMLMC
MC Estimate
17/33

19. Time required to compute G vs. .
0 1 2 3 4
101
102
103
104
105
Time(s)
22
G
18/33

20. 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).
19/33

21. 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).
20/33

22. Value of θ
10−3
10−2
10−1
100
TOL
0.0
0.2
0.4
0.6
0.8
1.0
θ
The experiment is repeated 15 times independently and the obtained
values are shown as error bars on the curves.
21/33

23. Prob. 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
(a) = 1 and (b) = {2, 3}.
22/33

24. Best practices for applying CMLMC method to CEM problems
Download CMLMC:
https://github.com/StochasticNumerics/mimclib.git (or use
MLMC from M. Giles)
Implement interface to couple CMLMC and your deterministic
solver
Generate a hierarchy of meshes (mimimum 3), nested are better
Generate 5-7 random shapes on first 3 meshes
Estimate the strong and weak convergence rates, q1, q2, (later they
will be corrected by CMLMC algorithm)
Run CMLMC solver and check visually automatically generated
plots
23/33

25. Conclusion
Used CMLMC method to characterize EM wave scattering 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.
To increase the efficiency further, each of the simulations is carried
out using the FMM-FFT accelerated PMCHWT-SIE solver.
Confirmed that the known advantages of the CMLMC algorithm
can be observed when it is applied to EM wave scattering:
non-intrusiveness, dimension independence, better convergence
rates compared to the classical MC method, and higher immunity
to irregularity w.r.t. uncertain parameters, than, for example,
sparse grid methods.
24/33

26. Conclusion
WARING: Some random perturbations may affect the convergence
rates in CMLMC. With difficult-to-predict convergence rates, it is
hard for the CMLMC algorithm to estimate the computational cost
W , the number of levels L, the number of samples on each level M ,
the computation time, and the parameter θ, and the variance in QoI.
All these may result in a sub-optimal performance.
25/33

27. Acknowledgements
SRI UQ at KAUST and Alexander von Humboldt
foundation.
Results are published:
1. A. Litvinenko, A. C. Yucel, H. Bagci, J. Oppelstrup,
E. Michielssen, R. Tempone,
Computation of Electromagnetic Fields Scattered From Objects With
Uncertain Shapes Using Multilevel Monte Carlo Method,
IEEE J. on Multiscale and Multiphysics Comput. Techniques,
pp 37-50, 2019.
2. A. Litvinenko, A. C. Yucel, H. Bagci, J. Oppelstrup,
E. Michielssen, R. Tempone,
Computation of Electromagnetic Fields Scattered From Objects With
Uncertain Shapes Using Multilevel Monte Carlo Method,
arXiv:1809.00362, 2018
26/33

28. Appendix
Let us repeat the main idea of (C)MLMC method
27/33

29. CMLMC repeating
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
=0
E[G ] (4)
where G is defined as
G =
g0 if = 0
g − g −1 if > 0
. (5)
Note that g and g −1 are computed using the same input random
parameter ξ.
28/33

30. CMLMC repeating
E[G ] ≈
∼
G = M−1 M
m=1 G ,m,
E[g − g ] ≈ QW hq1
(6a)
Var[g − g −1] ≈ QShq2
−1 (6b)
for QW = 0, QS > 0, q1 > 0, and 0 < q2 ≤ 2q1.
QoI A = L
=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
(7)
29/33

31. CMLMC repeating
The total CMLMC computational cost is
W =
L
=0
M W . (8)
The estimator A satisfies a tolerance with a prescribed failure
probability 0 < ν ≤ 1, i.e.,
P[|E[g] − A| ≤ TOL] ≥ 1 − ν (9)
while minimizing W . The total error is split into bias and statistical
error,
|E[g] − A| ≤ |E[g − A]|
Bias
+ |E[A] − A|
Statistical error
30/33

32. CMLMC repeating
Let θ ∈ (0, 1) be a splitting parameter, so that
TOL = (1 − θ)TOL
Bias tolerance
+ θTOL
Statistical error tolerance
. (10)
The CMLMC algorithm bounds the bias, B = |E[g − A]|, and the
statistical error as
B = |E[g − A]| ≤ (1 − θ)TOL (11)
|E[A] − A| ≤ θTOL (12)
where the latter bound holds with probability 1 − ν.
To satisfy condition in (12) we require:
Var[A] ≤
θTOL
Cν
2
(13)
for some given confidence parameter, Cν, such that Φ(Cν) = 1 − ν
2,
Φ is the cdf of a standard normal random variable. 31/33

33. CMLMC repeating
By construction of the MLMC estimator, E[A] = E[gL], and by
independence, Var[A] = L
=0 V M−1
, where V = Var[G ]. Given L,
TOL, and 0 < θ < 1, and by minimizing W subject to the statistical
constraint (13) for {M }L
=0 gives the following optimal number of
samples per level (apply ceiling function to M if necessary):
M =
Cν
θTOL
2
V
W
L
=0
V W . (14)
Summing the optimal numbers of samples over all levels yields the
following expression for the total optimal computational cost in terms
of TOL:
W (TOL, L) =
Cν
θTOL
2 L
=0
V W
2
. (15)
32/33

34. Literature
1. Collier, N., Haji-Ali, A., Nobile, F. et al. A continuation multilevel Monte Carlo algorithm. Bit Numer Math 55, 399–432 (2015).
https://doi.org/10.1007/s10543-014-0511-3
2. D Liu, A Litvinenko, C Schillings, V Schulz, Quantification of Airfoil Geometry-Induced Aerodynamic Uncertainties—Comparison of Approaches, SIAM/ASA
Journal on Uncertainty Quantification 5 (1), 334-352, 2017
3. Litvinenko A., Matthies H.G., El-Moselhy T.A. (2013) Sampling and Low-Rank Tensor Approximation of the Response Surface. In: Dick J., Kuo F., Peters
G., Sloan I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelber
4. A. Litvinenko, Application of hierarchical matrices for solving multiscale problems, Dissertation, Leipzig University, Germany,
http://publications.rwth-aachen.de/record/754296/files/754296.pdf
5. A Litvinenko, R Kriemann, MG Genton, Y Sun, DE Keyes, HLIBCov: Parallel hierarchical matrix approximation of large covariance matrices and likelihoods
with applications in parameter identification MethodsX 7, 100600, 2020
6. A Litvinenko, D Logashenko, R Tempone, G Wittum, D Keyes, Solution of the 3D density-driven groundwater flow problem with uncertain porosity and
permeability. Int J Geomath 11, 10 (2020). https://doi.org/10.1007/s13137-020-0147-1
7. A Litvinenko, Y Sun, MG Genton, DE Keyes, Likelihood approximation with hierarchical matrices for large spatial datasets, Computational Statistics & Data
Analysis 137, 115-132, 2019
8. S. Dolgov, A. Litvinenko, D.Liu, KRIGING IN TENSOR TRAIN DATA FORMAT Conf. Proceedings, 3rd International Conference on Uncertainty
Quantification in Computational Sciences and Engineering, https://files.eccomasproceedia.org/papers/e-books/uncecomp_2019.pdf pp 309-329,
2019
9. A. Litvinenko, D. Keyes, V. Khoromskaia, B.N. Khoromskij, H. G. Matthies, Tucker tensor analysis of Mat´ern functions in spatial statistics, J.
Computational Methods in Applied Mathematics, Vol. 19, Issue 1, pp 101-122, 2019, De Gruyter
10. HG Matthies, E Zander, BV Rosic, A Litvinenko, Parameter estimation via conditional expectation: a Bayesian inversion, Advanced modeling and simulation
in engineering sciences 3 (1), 1-21, 2016
11. M Espig, W Hackbusch, A Litvinenko, HG Matthies, E Zander, Post-Processing of High-Dimensional Data, arXiv:1906.05669, 2019
33/33

35. Thank you!