International audienceIt is common to accelerate the boundary element method by compression methods (FMM, H-Matrix / ACA) that enable a more accurate solution or a solution in higher frequency. In this work, we present a compression method based on a transformation of the linear system into Tensor-Train format by the quantization technique. The method is applied to a scattering problem by a canonical object with a regular mesh and improves the performance obtained by the previous methods

Bellouch, Ayoub

Coulaud, Olivier

Kaya, Oguz

Poirier, Jean-René

English

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Fast BEM solution for scattering problems using
Quantized Tensor Train format
Jean-René Poirier, Ayoub Bellouch, Olivier Coulaud, Oguz Kaya
To cite this version:
Jean-René Poirier, Ayoub Bellouch, Olivier Coulaud, Oguz Kaya. Fast BEM solution for scattering
problems using Quantized Tensor Train format. COMPUMAG 2019 - 22nd International Conference
on the Computation of Electromagnetic Field, Jul 2019, Paris, France. ￿hal-02264277￿
Fast BEM solution for scattering problems using Quantized
Tensor-Train format
J.-R. Poirier1,2, A. Bellouch2, O. Coulaud2, and O. Kaya3
1LAPLACE, Universite´ de Toulouse, CNRS, INPT, UPS, Toulouse, France
2INRIA Bordeaux Sud-Ouest-HIEPACS
3Universite´ Paris Saclay
It is common to accelerate the boundary element method by compression methods (FMM, H-Matrix / ACA) that enable a more
accurate solution or a solution in higher frequency. In this work, we present a compression method based on a transformation of the
linear system into Tensor-Train format by the quantization technique. The method is applied to a scattering problem by a canonical
object with a regular mesh and improves the performance obtained by the previous methods.
Index Terms—Boundary integral equations, electromagnetic diffraction, Quantized Tensor-Train
I. Introduction
MANY applications in science and engineering are for-mulated in terms of boundary integral equations. This
is namely the case in electromagnetic scattering to avoid the
use of an artifical condition to truncate the domain of study.
From a numerical point of view, the main difficulty is the
solution of a full complex linear system
Ax = b.
Many compression and solution accelerations of this linear
system have been developed and allowed to increase the
maximum accessible sizes from a few thousand to several
millions. Continuing to improve performance and accelerate
solution remains a topical issue. In this work we propose the
use of a tensor format for solving the algebraic problem.
As numerical example, we will consider the scattering by
a perfectly conducting cylinder Γ in the E-polarization (u =
Ez). The involved boundary-value problem to solve is then the
Helmholtz equation with a Dirichlet boundary condition and a
radiation condition at infinity.
The BEM solution can be written with a single layer
potential [1],∫
Γ
G(x, xs) j(xs) dγ(xs) = −uinc(x), ∀x on Γ, (1)
where Γ is the boundary, uinc the incident electric field, j the
sought density current and G the Green’s function.
In free space, this Green function is usually given by
G(x, xs) = 14i H
(2)
0 (k|x − xs|) where i is the imaginary unit, k
the wave number and H(2)0 the Hankel function of second kind.
II. Solution with tensor techniques
For many years various types of problems have been for-
mulated using tensors instead of a classical linear matrix
algebra [3]. More recently, tools have been developed to treat
high order tensors (dimension higher than 3) based on the
implementation of low rank techniques to effectively reproduce
the algebraic structure of the system. In the literature, there are
two main formats in terms of tensors for an efficient and stable
”hollow” representation of a very large system. These are the
approximations by Tensor-Train (TT) and Hierarchical Tucker
decompositions. In this work we have chosen to apply the TT
format to formulate the integral equation problem for a fast
solution.
A. Tensor-Train Format
The TT format [2] consists of writing a d-dimensional tensor
T ∈ Rn×...×n as a chain of 3-dimensional tensors according to
the formula
T (i1, · · · , id) =
∑
α0,···,αd
G1(α0, i1, α1)G2(α1, i2, α2) · · ·G1(αd−1, id, αd)
which can be written in a compact form
T (i1, · · · , id) = G1[i1]︸︷︷︸ G2[i2]︸ ︷︷ ︸ · · ·G3[i3]︸︷︷︸
r0 × r1 r1 × r2 rd−1 × rd
where
• Gi : TT-cores (matrices of size rk−1× rk with r0 = rd = 1)
• ri : TT - rank
• r = max(ri), r maximal rank of cores.
The complexity of the storage is O(dnr2) for O(nd) elements.
If r is small the tensor is of low rank and the storage and as
a result the solver will be fast.
The same format is given for a linear operator in Rn×...×n
which is represented by a 2d-dimensional tensor A that couple
elements as (in, jn) for n = 1 . . . d resulting from the couple
(i,j). TT representation of this tensor has the compact form
A(i1, · · · , id; j1, · · · , jd) = M1[i1, j1]︸     ︷︷     ︸ M2[i2, j2]︸      ︷︷      ︸ · · ·M3[i3, j3]︸     ︷︷     ︸
where Mk[ik, jk] is a matrix of rank rk−1 × rk.
In TT format, we have the corresponding algebraic opera-
tions (addition, matrix-vector product, dot product, etc.) that
are fast because of the reduced storage of the tensor.
B. Quantization and QTT format
Although some physical problems can naturally be formu-
lated in terms of tensors and therefore solved by adapted
techniques, a key point for the application of a tensor com-
pression technique on integral equations is quantization [6],
which enables a transition from a vector or a matrix to a tensor
representation. For a vector x ∈ RI and I, d ∈ N such that
I = I1I2 . . . Id where Ik ∈ N and for k = 1, . . . , d
we can transform x to a d-dimensional tensor X as
xi = X(i1, ..., id) with ik = 0, 1, .., Ik − 1 for k = 1, . . . , d)
using the following flattenning of the multi-index (i1, . . . , id):
i = i1 + i2I1 + .. + idI1I2...Id−1.
An equivalent procedure can be applied to obtain a QTT
representation for A ∈ RI×I with quantization in both dimen-
sions.
III. Solving Boundary Element Method with Tensor-Train
Format
A. Compression with TT-cross algorithm
A tensor X in full format can be approximated by X˜ in the
low-rank Tensor-Train (TT) format with the TT-SVD algorithm
[5] such that ∥∥∥X˜−X∥∥∥F ≤ ε ‖X‖F
where ‖X‖F is the Frobenius norm and  is the desired
accuracy.
This procedure being too expensive, it is convenient to use
a TT-cross algorithm with TT-rounding [5] that is a general-
ization of the well-known cross algorithm used for matrices. It
build
X˜= f (A˜, B˜, ....)
where A˜, B˜ are cross aproximations of cores A,B from the
TT-SVD algorithm.
The numerical implementation is done with the use of the
Python TT-Toolbox from Osedelets. The considered numerical
example here is a metallic cylinder of radius 1m illuminated by
an electromagnetic wave at frequency 0.6GHz. It is described
with a regular mesh with N = 2d elements. Table I gives the
memory storage required by the matrix A.
d N Full Matrix TT-cross TT-rounding
6 64 0.0625 Mb 0.067 Mb 0.007 Mb
10 1024 16 Mb 0.33 Mb 0.048 Mb
14 16384 4 Gb 0.54 Mb 0.063 Mb
16 65536 64 Gb 0.7 Mb 0.073 Mb
18 262144 1 Tb 0.75 Mb 0.079 Mb
20 1048576 16 Tb 1 Mb 0.084 Mb
TABLE I
Memory Storage for the matrix A
These results show clearly a very good compression arising
from this method even for very small value of N.
B. Solution using AMEN solver
The solution of the linear system may be performed by a
Krylov method applying the fast vector matrix product. In this
work we had applied the AMEN method developed specifically
for this tensor format [7]. The results are given Table II for
 = 0.001.
d N full solution xre f Amen xTT Error ‖xTT − xre f ‖
6 64 0.66 1.6E-11
8 256 1.23 1.09E-3
10 1024 2.15 1.78E-3
12 4096 3.57 4.5E-3
14 16384 5.41
16 65536
18 262144
20 1048576
TABLE II
Computation times (in seconds) and error with reference solution
As expected the solver make it possible to obtain very good
performance due to the reduced storage and while keeping the
desired accuracy.
C. Comparison with the H-Matrix techniques
We give in Table III the compression obtained on the same
example with H-matrix format using an ACA technique. As
N QTT/TT-Rounding H-Matrix
64 10.76 % 100 %
256 2.88 % 37.25%
1024 0.28 % 13.6%
4096 0.0208% 4.34%
16384 1.4.10−3 % 1.2%
65536 1.06.10−4 % 0.30 %
TABLE III
Comparison of Compression rate 100 size(ATT )size(AFull) with H-Matrix
a result compression and performance of the tensor solution
are much higher than those obtained by H-matrix techniques.
The QTT technique is known to be efficient for regular meshes
and solution. The performance on more complex problems or
geometries can be less effcicient [4] and this tensor technique
will be used as an efficient preconditioner in those cases.
References
[1] R. F. Harrington, ”Field Compilation by Moment Methods” Macmillan,
New York, 1968.
[2] I. V. Oseledets. 2011. ”Tensor-Train Decomposition” SIAM J. Sci. Com-
put., vol 33, no 5 September 2011, pp 2295-2317.
[3] T. G. Kolda and B. W. Bader,”Tensor Decompositions and Applications,”
SIAM Review, vol. 51, no. 3, pp. 455–500, 2009.
[4] Eduardo Corona, Abtin Rahimian, Denis Zorin,”A Tensor-Train accel-
erated solver for integral equations in complex geometries” Journal of
Computational Physics Volume 334, April 2017, Pages 145–169
[5] Ivan Oseledets, Eugene Tyrtyshnikov,”TT-cross approximation for multidi-
mensional arrays” Linear Algebra and its Applications,Volume 432, Issue
1,2010, Pages 70-88.
[6] Khoromskij, B. ”O ( d log N )-Quantics Approximation of N - d Tensors in
High-DimensionalNumerical Modeling” Constructive Approximation,vol
34, 2009.
[7] Sergey V. Dolgov and Dmitry V. Savostyanov. ”Alternating Minimal
Energy Methods for Linear Systems in Higher Dimensions”. SIAM
Journal on Scientific Computing vol 36, no 5, pp. A2248-A2271, 2014.


Fast BEM solution for scattering problems using Quantized Tensor Train format

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Fast BEM solution for scattering problems using Quantized Tensor Train format

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