A model reduction algorithm for parametrized
.
multiple particle electromagnetic configurations
M. Ganesh (Colorado School of Mines)
J. Hesthaven (Brown), B. Stamm (UC, Berkeley)
φrcs
φinc
Parameterized models : Offline Comp./Online Simulations
d˜rcs
pinc
•Basic parameter set D ⊂ R7:
?Incident wavenumber k ∈ R;
k, d˜inc
?Incident direction deinc ∈ R3;
?Incident polarization pinc ∈ R3.
z
y
x
Parameterized models : Offline Comp./Online Simulations
d˜rcs
pinc
•Basic parameter set D ⊂ R7:
?Incident wavenumber k ∈ R;
k, d˜inc
?Incident direction deinc ∈ R3;
?Incident polarization pinc ∈ R3.
z
•D is augmented with a larger set:
?Receiver direction dercs for RCS;
y
x
?Number of particles in config.;
?Shape of each particle in config.;
?Location and orientation of each.
Parametrized Multiple Electromagnetic (EM) Scattering in 3D
• Multiple particle configuration: Ω =
SJ
j=1 Ωj ,
• with disjoint perfect conductors Ω1, Ω2, . . . , ΩJ,
• located in a homogeneous medium with vanishing conductivity.
• The configuration is described using a large parameter set.
Parametrized Multiple Electromagnetic (EM) Scattering in 3D
• Multiple particle configuration: Ω =
SJ
j=1 Ωj ,
• with disjoint perfect conductors Ω1, Ω2, . . . , ΩJ,
• located in a homogeneous medium with vanishing conductivity.
• The configuration is described using a large parameter set.
• Let µ = (k, deinc, pinc) ∈ D := [k −, k +] × S 2 × P ⊂ R7
(P – a bounded polarization vector domain in R3).
Parametrized Multiple Electromagnetic (EM) Scattering in 3D
• Multiple particle configuration: Ω =
SJ
j=1 Ωj ,
• with disjoint perfect conductors Ω1, Ω2, . . . , ΩJ,
• located in a homogeneous medium with vanishing conductivity.
• The configuration is described using a large parameter set.
• Let µ = (k, deinc, pinc) ∈ D := [k −, k +] × S 2 × P ⊂ R7
(P – a bounded polarization vector domain in R3).
• Sufficient to focus on the electric field. (iωµ0H = curl E.)
p
• Let Z = µ0/0 — the intrinsic impedance of vacuum.
• Maxwell equations and the radiation condition:
curl curl E(x) − k 2E(x) = 0,
x ∈ R3 \ Ω,
lim [curl E(x) × x − ikZ|x|E(x)] = 0.
|x|→∞
• Each particle is a perfect conductor and hence the tangential components
of the total electric field E + E inc vanish on the ensemble surface.
A system of parametric variational form EFIEs
−1/2
• Variational form: find J surface currents wj ∈ Ht (div∂Ωj , ∂Ωj), j = 1, . . . , J
by solving the coupled surface scattering system
J n
o
X
−1/2
ij
i
(div∂Ωi , ∂Ωi),
[w
(µ),
v
;
µ]
=
f
[v
;
µ],
v
∈
H
aij
[w
(µ),
v
;
µ]
+
a
j
i
i
i
j
i
t
2
1
j=1
i = 1, . . . , J,
A system of parametric variational form EFIEs
−1/2
• Variational form: find J surface currents wj ∈ Ht (div∂Ωj , ∂Ωj), j = 1, . . . , J
by solving the coupled surface scattering system
J n
o
X
−1/2
ij
i
(div∂Ωi , ∂Ωi),
[w
(µ),
v
;
µ]
=
f
[v
;
µ],
v
∈
H
aij
[w
(µ),
v
;
µ]
+
a
j
i
i
i
j
i
t
2
1
j=1
i = 1, . . . , J,
−1/2
−1/2
ij
(div∂Ωj , ∂Ωj)×Ht (div∂Ωi , ∂Ωi) → C are sesquilinear
[·,
·;
µ],
a
• aij
2 [·, ·; µ] : Ht
1
−1/2
−1/2
forms, defined for ψ ∈ Ht (div∂Ωj , ∂Ωj), η ∈ Ht (div∂Ωi , ∂Ωi), as
Z Z
aij
Φk (x, y)ψ(y) · η(x) ds(y) ds(x),
1 [ψ, η; µ] = ikZ
Z∂Ωi Z∂Ωj
iZ
ij
a2 [ψ, η; µ] = −
Φk (x, y)divy ψ(y)divx η(x) ds(y) ds(x),
k ∂Ωi ∂Ωj
• and
f i[η; µ] = −
Z
∂Ωi
E inc(x; µ) · η(x) ds(x).
A BEM/MoM discrete model: Truth solution
−1/2
• For i = 1, . . . , J, replace Ht (div∂Ωi , ∂Ωi) in the variational form by a finite
dimensional subspace Vh,i, based on the discretization mesh parameter h.
A BEM/MoM discrete model: Truth solution
−1/2
• For i = 1, . . . , J, replace Ht (div∂Ωi , ∂Ωi) in the variational form by a finite
dimensional subspace Vh,i, based on the discretization mesh parameter h.
• In our case we choose the lowest order complex Raviart-Thomas space
Vh,i = RT0(∂Ωh,i) spanned by the Rao-Wilton-Glisson (RWG) basis as an
approximation space defined on a mesh ∂Ωh,i on the surface ∂Ωi.
A BEM/MoM discrete model: Truth solution
−1/2
• For i = 1, . . . , J, replace Ht (div∂Ωi , ∂Ωi) in the variational form by a finite
dimensional subspace Vh,i, based on the discretization mesh parameter h.
• In our case we choose the lowest order complex Raviart-Thomas space
Vh,i = RT0(∂Ωh,i) spanned by the Rao-Wilton-Glisson (RWG) basis as an
approximation space defined on a mesh ∂Ωh,i on the surface ∂Ωi.
• Let Vh,i = span {φin : n = 1, . . . , Nh}.
A BEM/MoM discrete model: Truth solution
−1/2
• For i = 1, . . . , J, replace Ht (div∂Ωi , ∂Ωi) in the variational form by a finite
dimensional subspace Vh,i, based on the discretization mesh parameter h.
• In our case we choose the lowest order complex Raviart-Thomas space
Vh,i = RT0(∂Ωh,i) spanned by the Rao-Wilton-Glisson (RWG) basis as an
approximation space defined on a mesh ∂Ωh,i on the surface ∂Ωi.
• Let Vh,i = span {φin : n = 1, . . . , Nh}.
• For each fixed µ ∈ D, and j = 1, . . . , J, the BEM multiple scattering model:
find wh,j(µ) ∈ Vh,j solving the discrete variational problem
J
X
aij[wh,j(µ), vh,j; µ] = f i[vh,j; µ],
∀vh,j ∈ Vh,i,
j=1
• where,
ij
aij[·, · ; µ] = aij
[·,
·
;
µ]
+
a
1
2 [·, · ; µ].
i = 1, . . . , J,
Truth solution for dynamic configurations for all µ ∈ D
• Our interest is in the truth solution wh,j(µ),
for varying parameters µ ∈ D and j = 1, · · · , J.
Truth solution for dynamic configurations for all µ ∈ D
• Our interest is in the truth solution wh,j(µ),
for varying parameters µ ∈ D and j = 1, · · · , J.
• Nh is assumed to be large enough to guarantee that
the truth solution is accurate
to within a user specific error tolerance.
Truth solution for dynamic configurations for all µ ∈ D
• Our interest is in the truth solution wh,j(µ),
for varying parameters µ ∈ D and j = 1, · · · , J.
• Nh is assumed to be large enough to guarantee that
the truth solution is accurate
to within a user specific error tolerance.
• Offline computations will be carried out only
for reference geometries
using some greedy algorithm based selection of
N parameters to build associated N << Nh reduced basis.
Truth solution for dynamic configurations for all µ ∈ D
• Our interest is in the truth solution wh,j(µ),
for varying parameters µ ∈ D and j = 1, · · · , J.
• Nh is assumed to be large enough to guarantee that
the truth solution is accurate
to within a user specific error tolerance.
• Offline computations will be carried out only
for reference geometries
using some greedy algorithm based selection of
N parameters to build associated N << Nh reduced basis.
• Fast online simulations can be carried out for any parameter
(including the number of particle parameter J)
within the parameter space
used in the offline computations.
Truth solution for dynamic configurations for all µ ∈ D
• Our interest is in the truth solution wh,j(µ),
for varying parameters µ ∈ D and j = 1, · · · , J.
• Nh is assumed to be large enough to guarantee that
the truth solution is accurate
to within a user specific error tolerance.
• Offline computations will be carried out only
for reference geometries
using some greedy algorithm based selection of
N parameters to build associated N << Nh reduced basis.
• Fast online simulations can be carried out for any parameter
(including the number of particle parameter J)
within the parameter space
used in the offline computations.
References geometries leading to online dynamic configurations
• We assume that particles inSthe
dynamic configuration Ω = Jj=1 Ωj are
(i) stretch/shrink, (ii) rotation, and (iii) translation
of some reference geometries.
References geometries leading to online dynamic configurations
• We assume that particles inSthe
dynamic configuration Ω = Jj=1 Ωj are
(i) stretch/shrink, (ii) rotation, and (iii) translation
of some reference geometries.
• For notational convenience,
b
assume just one reference geometry D.
References geometries leading to online dynamic configurations
• We assume that particles inSthe
dynamic configuration Ω = Jj=1 Ωj are
(i) stretch/shrink, (ii) rotation, and (iii) translation
of some reference geometries.
• For notational convenience,
b
assume just one reference geometry D.
b → Ωj
• For each j = 1, . . . , J. assume a transformation Tj : D
of the form Tj(x̂) = γjBjx̂ + bj where
γj ∈ R is a stretching/shrinking factor,
Bj ∈ R3×3 a rotation matrix
and bj ∈ R3 a translation vector.
• For j = 1, . . . , J, the Piola transformation
bj : H −1/2(div∂Ω , ∂Ωj) → H −1/2(div b , ∂ D)
b
P
t
t
j
∂D
bjψ)(x̂) = BTj ψ(Tj(x̂)),
(P
−1/2
ψ ∈ Ht
(div∂Ωj , ∂Ωj)
b
x̂ ∈ ∂ D,
b and vice-versa.
facilitates transfer of densities on ∂Ωj to that on ∂ D
Multiple scattering variational form on the reference shape
b h = RT0(∂ D
b h) be the BEM space for the reference shape.
• Let V
Multiple scattering variational form on the reference shape
b h = RT0(∂ D
b h) be the BEM space for the reference shape.
• Let V
• Reference truth solution: for each fixed µ ∈ D, and j = 1, . . . , J, find
b h such that
bh,j(µ) ∈ V
w
J
X
bh,j(µ), v
bh; µ] = fˆi[b
b
aij[w
vh; µ],
b h,
bh ∈ V
∀v
i = 1, . . . , J.
j=1
bj−1w
bh,j(µ), j = 1, . . . , J.
• Physical truth solution: wh,j(µ) = P
Multiple scattering variational form on the reference shape
b h = RT0(∂ D
b h) be the BEM space for the reference shape.
• Let V
• Reference truth solution: for each fixed µ ∈ D, and j = 1, . . . , J, find
b h such that
bh,j(µ) ∈ V
w
J
X
bh,j(µ), v
bh; µ] = fˆi[b
b
aij[w
vh; µ],
b h,
bh ∈ V
∀v
i = 1, . . . , J.
j=1
bj−1w
bh,j(µ), j = 1, . . . , J.
• Physical truth solution: wh,j(µ) = P
•
b η;
b µ] = ikZγiγj
b
aij[ψ,
Z
b
∂D
Z
b
∂D
b ij(x̂, ŷ)
Φ
k
b ŷ)) · (Biη(
b x̂))
(Bjψ(
−
•
b µ] = −γi
fˆi[η;
Z
b
∂D
1
b ŷ)divx̂ η(
b x̂)
divŷ ψ(
2
k γiγj
b x̂) ds(x̂).
BTi E inc(Ti(x̂); µ) · η(
ds(ŷ) ds(x̂).
The reduced basis method (RBM) : Offline step 1
• First step of the RBM for parametric multiple scattering model:
b N of the BEM space V
b h = RT0(∂ D
b h) that approxiDesign a subspace V
mates the infinite dimensional parametrized solution space
ch(µ) | ∀µ ∈ D}.
Mh = {w
b h are the BEM solutions of single scatterer EM model for D:
b
ch(µ) ∈ V
•w
The reduced basis method (RBM) : Offline step 1
• First step of the RBM for parametric multiple scattering model:
b N of the BEM space V
b h = RT0(∂ D
b h) that approxiDesign a subspace V
mates the infinite dimensional parametrized solution space
ch(µ) | ∀µ ∈ D}.
Mh = {w
b h are the BEM solutions of single scatterer EM model for D:
b
ch(µ) ∈ V
•w
bh(µ), v
bh; µ] = fˆ[b
b
a[ w
vh; µ],
b h,
bh ∈ V
∀v
The reduced basis method (RBM) : Offline step 1
• First step of the RBM for parametric multiple scattering model:
b N of the BEM space V
b h = RT0(∂ D
b h) that approxiDesign a subspace V
mates the infinite dimensional parametrized solution space
ch(µ) | ∀µ ∈ D}.
Mh = {w
b h are the BEM solutions of single scatterer EM model for D:
b
ch(µ) ∈ V
•w
bh(µ), v
bh; µ] = fˆ[b
b
a[ w
vh; µ],
b h,
bh ∈ V
∀v
•
Z
b η;
b µ] = ikZ
b
a[ψ,
b
∂D
Z
1
b y )divxb η(b
b
b x)) − 2 divyb ψ(b
b x)
Φk (b
x, yb) (ψ(b
y )) · (η(b
k
b
∂D
•
Z
b µ] = −
fb[η;
b
∂D
b x) ds(b
E inc(b
x; µ) · η(b
x),
ds(b
y ) ds
Finite dimensional online ansatz spaces for the RBM
b N used for the online part of the RBM is:
• The N -dimensional space V
b N = span{wh(µn) | n = 1, . . . , N }.
V
b N is an offline procedure, using N -iterative steps.
• Construction of V
• N “snapshot” parameters µn selection is based on a greedy algorithm.
• At each iterative step:
b h is solved to obtain a basis function.
? A full BEM model in V
? N is adaptively chosen using a posteriori error estimator of the error
between the RBM approx. wN (µ) and the “truth” solution wh(µ).
Finite dimensional online ansatz spaces for the RBM
b N used for the online part of the RBM is:
• The N -dimensional space V
b N = span{wh(µn) | n = 1, . . . , N }.
V
b N is an offline procedure, using N -iterative steps.
• Construction of V
• N “snapshot” parameters µn selection is based on a greedy algorithm.
• At each iterative step:
b h is solved to obtain a basis function.
? A full BEM model in V
? N is adaptively chosen using a posteriori error estimator of the error
between the RBM approx. wN (µ) and the “truth” solution wh(µ).
b N reducing the dimension of the BEM space V
bh
• A sample dimension of V
by over 90% for a benchmark dynamic configuration.
mesh size (h) Dim BEM (Nh) Dim Reduced (N ) Residual error
0.05
17280
918
9.94e-4
• BEM requires solutions of dense complex linear systems and setting up of
each entry requires evaluation of 4-dimensional singular surface integrals.
Essential requirements for efficient offline/offline simulations
• The offline greedy algorithm required for
b N in N steps
construction of V
is based on a fine-grid of the parameter space D.
Essential requirements for efficient offline/offline simulations
• The offline greedy algorithm required for
b N in N steps
construction of V
is based on a fine-grid of the parameter space D.
• At each step, the greedy algorithm based computations
should be independent of Nh.
Essential requirements for efficient offline/offline simulations
• The offline greedy algorithm required for
b N in N steps
construction of V
is based on a fine-grid of the parameter space D.
• At each step, the greedy algorithm based computations
should be independent of Nh.
and our algorithm requires solutions of only
`-dimensional systems for the `-th step of the iteration, ` = 1, . . . , N .
Essential requirements for efficient offline/offline simulations
• The offline greedy algorithm required for
b N in N steps
construction of V
is based on a fine-grid of the parameter space D.
• At each step, the greedy algorithm based computations
should be independent of Nh.
and our algorithm requires solutions of only
`-dimensional systems for the `-th step of the iteration, ` = 1, . . . , N .
• The online solves should be
independent of Nh (and just N -dim solves).
Essential requirements for efficient offline/offline simulations
• The offline greedy algorithm required for
b N in N steps
construction of V
is based on a fine-grid of the parameter space D.
• At each step, the greedy algorithm based computations
should be independent of Nh.
and our algorithm requires solutions of only
`-dimensional systems for the `-th step of the iteration, ` = 1, . . . , N .
• The online solves should be
independent of Nh (and just N -dim solves).
• The online simulation
should avoid even solutions of JN -dim systems.
Essential requirements for efficient offline/offline simulations
• The offline greedy algorithm required for
b N in N steps
construction of V
is based on a fine-grid of the parameter space D.
• At each step, the greedy algorithm based computations
should be independent of Nh.
and our algorithm requires solutions of only
`-dimensional systems for the `-th step of the iteration, ` = 1, . . . , N .
• The online solves should be
independent of Nh (and just N -dim solves).
• The online simulation
should avoid even solutions of JN -dim systems.
EIM based approximations of sesquilinear, input, output terms
• Using the empirical interpolation method (EIM),
approximate (to any required accuracy)
all individual, interactive sesquilinear forms, input and output functions
in the model by constructing decompositions of the form:
F(· ; µ) ≈ FM (· ; µ) =
M
X
βp(µ)F(· ; µp).
p=1
• The sample points {µp} in the EIM are chosen by a greedy algorithm.
• For each new parameter value µ,
the coefficients βp(µ) are obtained by
solving a lower triangular M dimensional linear system.
• General frame work for EIM:
[Y. Maday, N.C. Nguyen, A. T. Patera, G. S. H. Pau,
A general multipurpose interpolation procedure: The magic points.
Commun. Pure Appl. Anal. 8 (2009), 383-404. ]
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• Let µ1 ∈ D be arbitrary. Let n = 1.
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• Let µ1 ∈ D be arbitrary. Let n = 1.
o
n
bp : p = 1, . . . , Nh
ch := span φ
is the unique
• For r = 1, . . . , n, ξbh(µr ) ∈ V
solution of the Nh dimensional boundary element system
bq ; µr ] = fb[φ
bq ; µr ],
b
a[ξbh(µr ), φ
q = 1, . . . , Nh,
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• Let µ1 ∈ D be arbitrary. Let n = 1.
o
n
bp : p = 1, . . . , Nh
ch := span φ
is the unique
• For r = 1, . . . , n, ξbh(µr ) ∈ V
solution of the Nh dimensional boundary element system
bq ; µr ] = fb[φ
bq ; µr ],
b
a[ξbh(µr ), φ
q = 1, . . . , Nh,
n
o
• Let {ξbr : r = 1, . . . , n} be the G-S orthonormalization of ξbh(µ1), . . . , ξbh(µn) .
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• Let µ1 ∈ D be arbitrary. Let n = 1.
o
n
bp : p = 1, . . . , Nh
ch := span φ
is the unique
• For r = 1, . . . , n, ξbh(µr ) ∈ V
solution of the Nh dimensional boundary element system
bq ; µr ] = fb[φ
bq ; µr ],
b
a[ξbh(µr ), φ
q = 1, . . . , Nh,
n
o
• Let {ξbr : r = 1, . . . , n} be the G-S orthonormalization of ξbh(µ1), . . . , ξbh(µn) .
• We construct µn+1 as follows using a greedy algorithm and at the of the
n-th step of the iteration, the basis function ξbh(µn+1) is computed using
the above system with r = n + 1.
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• Let µ1 ∈ D be arbitrary. Let n = 1.
o
n
bp : p = 1, . . . , Nh
ch := span φ
is the unique
• For r = 1, . . . , n, ξbh(µr ) ∈ V
solution of the Nh dimensional boundary element system
bq ; µr ] = fb[φ
bq ; µr ],
b
a[ξbh(µr ), φ
q = 1, . . . , Nh,
n
o
• Let {ξbr : r = 1, . . . , n} be the G-S orthonormalization of ξbh(µ1), . . . , ξbh(µn) .
• We construct µn+1 as follows using a greedy algorithm and at the of the
n-th step of the iteration, the basis function ξbh(µn+1) is computed using
the above system with r = n + 1.
• Adaptively stop the offline algorithm in N steps when a computable
error indicator (µ) satisfies some error tolerance.
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• Let µ1 ∈ D be arbitrary. Let n = 1.
o
n
bp : p = 1, . . . , Nh
ch := span φ
is the unique
• For r = 1, . . . , n, ξbh(µr ) ∈ V
solution of the Nh dimensional boundary element system
bq ; µr ] = fb[φ
bq ; µr ],
b
a[ξbh(µr ), φ
q = 1, . . . , Nh,
n
o
• Let {ξbr : r = 1, . . . , n} be the G-S orthonormalization of ξbh(µ1), . . . , ξbh(µn) .
• We construct µn+1 as follows using a greedy algorithm and at the of the
n-th step of the iteration, the basis function ξbh(µn+1) is computed using
the above system with r = n + 1.
• Adaptively stop the offline algorithm in N steps when a computable
error indicator (µ) satisfies some error tolerance.
• The offline procedure includes computation and storage of individual,
interactive sesquilinear forms for the EIM sample parameters.
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• The n-th step of the iteration requires solutions of several n × n systems:
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• The n-th step of the iteration requires solutions of several n × n systems:
o
n
b n = span ξb1, . . . , ξbn .
• Let V
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• The n-th step of the iteration requires solutions of several n × n systems:
o
n
b n = span ξb1, . . . , ξbn .
• Let V
• Let D (n) ⊂ D be a fine mesh discretization of the parameter space D.
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• The n-th step of the iteration requires solutions of several n × n systems:
o
n
b n = span ξb1, . . . , ξbn .
• Let V
• Let D (n) ⊂ D be a fine mesh discretization of the parameter space D.
b n, by solving the n × n system
• For each µ ∈ D (n), compute ηbn(µ) ∈ V
b
aMM [ηbn(µ), ξbr ; µ] = fbMF [ξbr ; µ],
r = 1, . . . , n.
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• The n-th step of the iteration requires solutions of several n × n systems:
o
n
b n = span ξb1, . . . , ξbn .
• Let V
• Let D (n) ⊂ D be a fine mesh discretization of the parameter space D.
b n, by solving the n × n system
• For each µ ∈ D (n), compute ηbn(µ) ∈ V
b
aMM [ηbn(µ), ξbr ; µ] = fbMF [ξbr ; µ],
r = 1, . . . , n.
• Then, for each µ ∈ D (n), compute an approximation (µ) to
• the true error indicator kξbh(µ) − ηbn(µ)kH −1/2(div
t
b ,∂ D)
∂D
,
b
• without the need to compute the boundary element solution ξbh(µ):
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• Let
(µ) =
b
b
b
−
f(µ)
A(µ)
η
(µ)
2
n
l
b e l2
maxµ∈D
(n) kf(µ)k
e
,
µ ∈ D (n).
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• Let
(µ) =
b
b
b
−
f(µ)
A(µ)
η
(µ)
2
n
l
b e l2
maxµ∈D
(n) kf(µ)k
e
,
µ ∈ D (n).
ch,
• ηbn(µ) is the vectorial representation of ηbn(µ) expressed in the basis of V
ch,
b
• A(µ)
is the matrix representation of b
aMM [·, ·; µ] expressed in the basis V
• and efficiently computed using EIM sample point based the matrices
bp, φ
bq ; m],
cm]p,q = b
[A
a[ φ
p, q = 1, . . . , Nh,
m = 0, . . . , MM.
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• Let
(µ) =
b
b
b
−
f(µ)
A(µ)
η
(µ)
2
n
l
b e l2
maxµ∈D
(n) kf(µ)k
e
µ ∈ D (n).
,
ch,
• ηbn(µ) is the vectorial representation of ηbn(µ) expressed in the basis of V
ch,
b
• A(µ)
is the matrix representation of b
aMM [·, ·; µ] expressed in the basis V
• and efficiently computed using EIM sample point based the matrices
bp, φ
bq ; m],
cm]p,q = b
[A
a[ φ
p, q = 1, . . . , Nh,
m = 0, . . . , MM.
ch,
• bf(µ) is the vectorial representation of fbMF [·; µ] expressed in the basis of V
and computed using the vectors
m
b
f [φq ; m],
Fq = b
q = 1, . . . , Nh,
m = 1, . . . , MF .
b
Offline procedure: Snapshot parameters + reduced basis for ∂ D
• Let
(µ) =
b
b
b
−
f(µ)
A(µ)
η
(µ)
2
n
l
b e l2
maxµ∈D
(n) kf(µ)k
e
µ ∈ D (n).
,
ch,
• ηbn(µ) is the vectorial representation of ηbn(µ) expressed in the basis of V
ch,
b
• A(µ)
is the matrix representation of b
aMM [·, ·; µ] expressed in the basis V
• and efficiently computed using EIM sample point based the matrices
bp, φ
bq ; m],
cm]p,q = b
[A
a[ φ
p, q = 1, . . . , Nh,
m = 0, . . . , MM.
ch,
• bf(µ) is the vectorial representation of fbMF [·; µ] expressed in the basis of V
and computed using the vectors
m
b
f [φq ; m],
Fq = b
q = 1, . . . , Nh,
m = 1, . . . , MF .
• Choose the next snapshot parameter µn+1 ∈ D as
n
o
(n)
n+1
µ
= argmax (µ) : µ ∈ D
.
Online multiple 3D EM scattering algorithm: Any µ ∈ D
• Recall:
bh(µ) =
? Reference truth solution: find w
J
X
j=1
bh,j(µ), v
bh; µ] = fˆi[b
b
aij[w
vh; µ],
PJ
bh,j(µ)
j=1 w
b h,
bh ∈ V
∀v
b h such that
∈V
i = 1, . . . , J.
Online multiple 3D EM scattering algorithm: Any µ ∈ D
• Recall:
bh(µ) =
? Reference truth solution: find w
J
X
bh,j(µ), v
bh; µ] = fˆi[b
b
aij[w
vh; µ],
PJ
bh,j(µ)
j=1 w
b h,
bh ∈ V
∀v
b h such that
∈V
i = 1, . . . , J.
j=1
? This is a large JNh × JNh dense complex linear system, for each µ ∈ D.
Online multiple 3D EM scattering algorithm: Any µ ∈ D
• Recall:
bh(µ) =
? Reference truth solution: find w
J
X
bh,j(µ), v
bh; µ] = fˆi[b
b
aij[w
vh; µ],
PJ
bh,j(µ)
j=1 w
b h,
bh ∈ V
∀v
b h such that
∈V
i = 1, . . . , J.
j=1
? This is a large JNh × JNh dense complex linear system, for each µ ∈ D.
? Physical truth solution: wh(µ) =
PJ
j=1 wh,j (µ) =
bj−1w
bh,j(µ).
P
j=1
PJ
Online multiple 3D EM scattering algorithm: Any µ ∈ D
• Recall:
bh(µ) =
? Reference truth solution: find w
J
X
bh,j(µ), v
bh; µ] = fˆi[b
b
aij[w
vh; µ],
PJ
bh,j(µ)
j=1 w
b h,
bh ∈ V
∀v
b h such that
∈V
i = 1, . . . , J.
j=1
? This is a large JNh × JNh dense complex linear system, for each µ ∈ D.
? Physical truth solution: wh(µ) =
PJ
j=1 wh,j (µ) =
bj−1w
bh,j(µ).
P
j=1
PJ
bN,
bh(µ) with reduced basis based solution w
bN,L(µ) ∈ V
• We approximate w
Online multiple 3D EM scattering algorithm: Any µ ∈ D
• Recall:
bh(µ) =
? Reference truth solution: find w
J
X
bh,j(µ), v
bh; µ] = fˆi[b
b
aij[w
vh; µ],
PJ
bh,j(µ)
j=1 w
b h,
bh ∈ V
∀v
b h such that
∈V
i = 1, . . . , J.
j=1
? This is a large JNh × JNh dense complex linear system, for each µ ∈ D.
? Physical truth solution: wh(µ) =
PJ
j=1 wh,j (µ) =
bj−1w
bh,j(µ).
P
j=1
PJ
bN,
bh(µ) with reduced basis based solution w
bN,L(µ) ∈ V
• We approximate w
? computed using L-iterations with each iteration requiring
? solutions of only J uncoupled N × N dense complex linear systems
Online multiple 3D EM scattering algorithm: Any µ ∈ D
• Recall:
bh(µ) =
? Reference truth solution: find w
J
X
bh,j(µ), v
bh; µ] = fˆi[b
b
aij[w
vh; µ],
PJ
bh,j(µ)
j=1 w
b h,
bh ∈ V
∀v
b h such that
∈V
i = 1, . . . , J.
j=1
? This is a large JNh × JNh dense complex linear system, for each µ ∈ D.
? Physical truth solution: wh(µ) =
PJ
j=1 wh,j (µ) =
bj−1w
bh,j(µ).
P
j=1
PJ
bN,
bh(µ) with reduced basis based solution w
bN,L(µ) ∈ V
• We approximate w
? computed using L-iterations with each iteration requiring
? solutions of only J uncoupled N × N dense complex linear systems
• For a chosen online parameter µ ∈ D,
? setting up of the iterative reduced systems require
? solutions of several lower triangular systems
? to compute the EIM coefficients, before start of the iterative procedure
Online multiple 3D EM scattering algorithm: Any µ ∈ D
•
bN,L(· ; µ) =
w
J X
L
X
(`)
bN,i(· ; µ)
w
i=1 `=1
(`)
bN,i(· ; µ) =
w
N
X
n=1
• L is such that for a prescribed tolerance level tol,
(L)
(L−1)
max max wn,i (µ) − wn,i (µ) < tol.
1≤i≤J
1≤n≤N
(`)
wn,i(µ)ξbni ,
Online multiple 3D EM scattering algorithm: Any µ ∈ D
•
bN,L(· ; µ) =
w
J X
L
X
(`)
bN,i(· ; µ)
w
(`)
bN,i(· ; µ) =
w
i=1 `=1
N
X
(`)
wn,i(µ)ξbni ,
n=1
• L is such that for a prescribed tolerance level tol,
(L)
(L−1)
max max wn,i (µ) − wn,i (µ) < tol.
1≤i≤J
1≤n≤N
(`)
b N , solutions of individual obstacle N × N systems:
• For i = 1, . . . , J, wN,i ∈ V
? Initial guess: without reflections
(1) b
bi [ξbr ; µ],
b
b
aii
[
w
,
ξ
;
µ]
=
f
r
MF
MM
N,i
r = 1, . . . , N.
Online multiple 3D EM scattering algorithm: Any µ ∈ D
•
bN,L(· ; µ) =
w
J X
L
X
(`)
bN,i(· ; µ)
w
(`)
bN,i(· ; µ) =
w
N
X
(`)
wn,i(µ)ξbni ,
n=1
i=1 `=1
• L is such that for a prescribed tolerance level tol,
(L)
(L−1)
max max wn,i (µ) − wn,i (µ) < tol.
1≤i≤J
1≤n≤N
(`)
b N , solutions of individual obstacle N × N systems:
• For i = 1, . . . , J, wN,i ∈ V
? Initial guess: without reflections
(1) b
bi [ξbr ; µ],
b
b
aii
[
w
,
ξ
;
µ]
=
f
r
MF
MM
N,i
r = 1, . . . , N.
? For ` = 1, . . . , L − 1: include reflections
(`+1)
bN,i , ξbr ; µ] = −
b
aii
MM [w
J
X
j=1
j 6= i
(`) b
b
b
aij
[
w
MG
N,i , ξr ; µ]
r = 1, . . . , N,
Online multiple 3D EM scattering algorithm: Any µ ∈ D
•
bN,L(· ; µ) =
w
J X
L
X
(`)
bN,i(· ; µ)
w
(`)
bN,i(· ; µ) =
w
N
X
(`)
wn,i(µ)ξbni ,
n=1
i=1 `=1
• L is such that for a prescribed tolerance level tol,
(L)
(L−1)
max max wn,i (µ) − wn,i (µ) < tol.
1≤i≤J
1≤n≤N
(`)
b N , solutions of individual obstacle N × N systems:
• For i = 1, . . . , J, wN,i ∈ V
? Initial guess: without reflections
(1) b
bi [ξbr ; µ],
b
b
aii
[
w
,
ξ
;
µ]
=
f
r
MF
MM
N,i
r = 1, . . . , N.
? For ` = 1, . . . , L − 1: include reflections
(`+1)
bN,i , ξbr ; µ] = −
b
aii
MM [w
J
X
(`) b
b
b
aij
[
w
MG
N,i , ξr ; µ]
r = 1, . . . , N,
j=1
j 6= i
bN,L(· ; µ) and solve lower triangular systems for EIM coef• Output: Use w
ficients to efficiently compute the RCS for any µ ∈ D.
Numerical Experiments
Example 1. (Two metallic unit spheres with varying separation.)
• Measurement/Experimental results are known for k = 7.41 and k = 11.048.
ka = kb = 7.41
Er.
w
2,-
-.
1.0919
Numerical Experiments
Example 1. (Two metallic unit spheres with varying separation. k = 11.048)
Lo = kb = I1048
$=I0355
6, =
kd
m
b
Accuracy of RBM basis compared to BEM basis for ∂ D
b : Unit sphere (fixed wavenumber)
• ∂D
k
mesh size (h) Dim BEM (Nh) Dim Reduced (N ) Residual error
7.41
0.826e-1
7680
563
9.73e-4
11.048
0.551e-1
17280
918
9.94e-4
b
Accuracy of RBM basis compared to BEM basis for ∂ D
b : Unit sphere (fixed wavenumber)
• ∂D
k
mesh size (h) Dim BEM (Nh) Dim Reduced (N ) Residual error
7.41
0.826e-1
7680
563
9.73e-4
11.048
0.551e-1
17280
918
9.94e-4
b : Unit sphere (variable wavenumbers)
• ∂D
k ∈ mesh size (h) Dim BEM (Nh) Dim Reduced (N ) Residual error
[1, 3]
0.110
4320
290
2.23e-4
[3, 5]
0.110
4320
509
2.64e-4
b
Accuracy of RBM basis compared to BEM basis for ∂ D
b : Unit sphere (fixed wavenumber)
• ∂D
k
mesh size (h) Dim BEM (Nh) Dim Reduced (N ) Residual error
7.41
0.826e-1
7680
563
9.73e-4
11.048
0.551e-1
17280
918
9.94e-4
b : Unit sphere (variable wavenumbers)
• ∂D
k ∈ mesh size (h) Dim BEM (Nh) Dim Reduced (N ) Residual error
[1, 3]
0.110
4320
290
2.23e-4
[3, 5]
0.110
4320
509
2.64e-4
b : Open cavity/box (variable wavenumbers)
• ∂D
k ∈ mesh size (h) Dim BEM (Nh) Dim Reduced (N ) Residual error
[1, 3]
0.117
1795
239
7.144e-4
Convergence of the iterative RBM multiple EM algorithm
Example 2. Two spheres with variable wavenumbers and distance
kd
• Relative error of the RBM surface current compared to truth solution
Iteration count for the RBM multiple EM algorithm
Example 2. Two spheres with variable wavenumbers and distance
kd
• The total number of online multiple scattering iterations (L)
Numerical experiments: Variable electric size
Example 3. Two spheres with variable electric size
θrcs
• Validation of the reduced basis RCS for arbitrary shrink parameters
(γ1, γ2) = (0.549, 0.982)
Numerical experiments: 36 unit spheres
Example 4. Variable wavenumbers, sources, and observations
Numerical experiments: 36 unit spheres
Example 4. k = 5 snapshot: variable sources, and observations
φinc
φrcs
• The RCS, measured in the directions θrcs = π2 , φrcs ∈ [0, 2π], for an incident
plane wave of with angles θinc = π2 , φinc ∈ [0, 2π].
Numerical experiments: 36 unit spheres
Example 4. Variable wavenumbers in [1, 4.972], sources, and observations. Known: Three resonant wavenumbers for k ∈ [1, 4.972], k1 = 2.743,
k2 = 3.870, k3 = 4.493. The fourth resonant wavenum. in [1, 5], is k4 = 4.973.
k
φrcs
• The RCS, measured in the directions θrcs = π2 , φrcs ∈ [0, 2π], for an incident
plane wave of with angles θinc = 0 = φinc
Numerical experiments: Metallic open cavities
Example 5. (Two metallic open cavities with varying rotations).
d=4
z
y
α2
α1
x
Figure 1: (Example 5.) Setup for scattering by two open metallic cavities.
RCS
φrcs
• Accuracy of the reduced basis RCS for individually rotated random parameters α1 = 1.5396, α2 = 1.1299, and k = 2.491. The RCS, measured at
θrcs = π2 , φrcs ∈ [0, 2π], for an incident plane wave with θinc = 0 = φinc.
Numerical experiments: A stochastic multiple configuration
Example 6. (An uncertain configuration with 9 open cavities.)
√
• Model expected value (E) and standard deviation ( V ) of the RCS of a
stochastic configuration: Vertical location of the center cavity is specified
by a continuous uniformly distributed random variable in [−1, 1].
Numerical experiments: A stochastic configuration
Example 6. (An uncertain configuration with 9 open cavities. )
M ean
RCS
φrcs
√
√
• Expected value of monostatic RCS and the interval [E − V , E + V ] for
a stochastic 9 cavity configuration with incidence angle θinc = π3 , φinc = 0.
Numerical experiments: Several reference shapes
Example 7. (Configurations with different reference shapes.)
z
z
y
y
x
(a)
x
A config.: open and closed obs.
(b)
A Perturbed version of the config.
φ
φ
φrcs
(c)
RCS of the unperturbed config.
φrcs
(d)
RCS of the perturbed config.