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Reduced Basis Model Reduction ofTime-Harmonic Maxwell’s Equations
Using a Compliant Expanded FormulationPeter Benner, Martin Hess
MOR 4 MEMSNovember, 17-18, 2015
KIT, Karlsruhe
Max Planck Institute for Dynamics of Complex Technical SystemsComputational Methods in Systems and Control Theory
1. Motivation
2. Introduction to Reduced Basis Method
3. Electromagnetic Model
4. Numerical Results
5. Conclusions
Martin Hess RBM in Electromagnetics 2/19
Motivation
1. Motivation
2. Introduction to Reduced Basis Method
3. Electromagnetic Model
4. Numerical Results
5. Conclusions
Martin Hess RBM in Electromagnetics 3/19
MotivationFull wave simulations allow an accurate and reliable prediction of devicebehavior.
Figure: Printed circuit board (PCB, left) with 4 inputs and 4 outputs. Microstriplines (single and double) on the right.
Martin Hess RBM in Electromagnetics 3/19
Motivation
Figure: Two parallel microstrip lines in a 0.13µm CMOS process andcomputational mesh employed for solving Maxwell’s equations.
Figure: Optical lithography process (left) and phase shift mask (right).
Martin Hess RBM in Electromagnetics 4/19
Motivation
Figure: Two parallel microstrip lines in a 0.13µm CMOS process andcomputational mesh employed for solving Maxwell’s equations.
Figure: Optical lithography process (left) and phase shift mask (right).
Martin Hess RBM in Electromagnetics 4/19
Reduced Basis Concept
1. Motivation
2. Introduction to Reduced Basis Method
3. Electromagnetic Model
4. Numerical Results
5. Conclusions
Martin Hess RBM in Electromagnetics 5/19
Reduced Basis Concept
Model Problem, [Rozza et al., 2008]For ν ∈ D evaluate
s(ν) = `(u(ν); ν),
where u(ν) ∈ X satisfies
a(u(ν), v ; ν) = f (v ; ν), ∀v ∈ X
Find linear space approximating M = {u(ν)|ν ∈ D}
Offline-Online decomposition ⇒ computational efficiency
Martin Hess RBM in Electromagnetics 5/19
Reduced Basis Concept
Model Problem, [Rozza et al., 2008]For ν ∈ D evaluate
s(ν) = `(u(ν); ν),
where u(ν) ∈ X satisfies
a(u(ν), v ; ν) = f (v ; ν), ∀v ∈ X
Find linear space approximating M = {u(ν)|ν ∈ D}
Offline-Online decomposition ⇒ computational efficiency
Martin Hess RBM in Electromagnetics 5/19
Reduced Basis Concept
Model Problem, [Rozza et al., 2008]For ν ∈ D evaluate
s(ν) = `(u(ν); ν),
where u(ν) ∈ X satisfies
a(u(ν), v ; ν) = f (v ; ν), ∀v ∈ X
Find linear space approximating M = {u(ν)|ν ∈ D}
Offline-Online decomposition ⇒ computational efficiency
Martin Hess RBM in Electromagnetics 5/19
Reduced Basis Concept
Parameter-dependent linear systems
A(ν)x(ν) = b(ν)
with affine parameter dependence
A(ν) =∑Qa
q=1 Θqa(ν)Aq
b(ν) =∑Qb
q=1 Θqb(ν)bq
⇒ Offline-Online decomposition.
Martin Hess RBM in Electromagnetics 6/19
Reduced Basis Concept
Parameter-dependent linear systems
A(ν)x(ν) = b(ν)
with affine parameter dependence
A(ν) =∑Qa
q=1 Θqa(ν)Aq
b(ν) =∑Qb
q=1 Θqb(ν)bq
⇒ Offline-Online decomposition.
Martin Hess RBM in Electromagnetics 6/19
Sampling
Greedy Sampling
Let Ξ denote a finite sample of D.
Set S1 = {ν1} and V1 = span{u(ν1)}.For N = 2, ...,Nmax , find νN = arg maxν∈Ξ∆N−1(ν),
then set SN = SN−1 ∪ νN , VN = VN−1 + span{u(νN)}.
Projection onto low order space VN (Snapshot space)
AqN = V T
N AqVN , bqN = V TN bq.
Parameter-preserving model reduction Qa∑q=1
Θqa(ν)Aq
N
xN =
Qb∑q=1
Θqb(ν)bqN .
Martin Hess RBM in Electromagnetics 7/19
Sampling
Greedy Sampling
Let Ξ denote a finite sample of D.
Set S1 = {ν1} and V1 = span{u(ν1)}.For N = 2, ...,Nmax , find νN = arg maxν∈Ξ∆N−1(ν),
then set SN = SN−1 ∪ νN , VN = VN−1 + span{u(νN)}.
Projection onto low order space VN (Snapshot space)
AqN = V T
N AqVN , bqN = V TN bq.
Parameter-preserving model reduction Qa∑q=1
Θqa(ν)Aq
N
xN =
Qb∑q=1
Θqb(ν)bqN .
Martin Hess RBM in Electromagnetics 7/19
Sampling
Greedy Sampling
Let Ξ denote a finite sample of D.
Set S1 = {ν1} and V1 = span{u(ν1)}.For N = 2, ...,Nmax , find νN = arg maxν∈Ξ∆N−1(ν),
then set SN = SN−1 ∪ νN , VN = VN−1 + span{u(νN)}.
Projection onto low order space VN (Snapshot space)
AqN = V T
N AqVN , bqN = V TN bq.
Parameter-preserving model reduction Qa∑q=1
Θqa(ν)Aq
N
xN =
Qb∑q=1
Θqb(ν)bqN .
Martin Hess RBM in Electromagnetics 7/19
Error EstimatorsField error estimator
∆N(ν) =‖rpr (·; ν)‖X ′
βLB(ν).
Output error estimator
∆oN(ν) =
‖rpr (·; ν)‖X ′‖rdu(·; ν)‖X ′
βLB(ν).
Output error estimator (compliant case ` = f )
∆sN(ν) =
‖rpr (·; ν)‖2X ′
βLB(ν).
Martin Hess RBM in Electromagnetics 8/19
Error EstimatorsField error estimator
∆N(ν) =‖rpr (·; ν)‖X ′
βLB(ν).
Output error estimator
∆oN(ν) =
‖rpr (·; ν)‖X ′‖rdu(·; ν)‖X ′
βLB(ν).
Output error estimator (compliant case ` = f )
∆sN(ν) =
‖rpr (·; ν)‖2X ′
βLB(ν).
Martin Hess RBM in Electromagnetics 8/19
Error EstimatorsField error estimator
∆N(ν) =‖rpr (·; ν)‖X ′
βLB(ν).
Output error estimator
∆oN(ν) =
‖rpr (·; ν)‖X ′‖rdu(·; ν)‖X ′
βLB(ν).
Output error estimator (compliant case ` = f )
∆sN(ν) =
‖rpr (·; ν)‖2X ′
βLB(ν).
Martin Hess RBM in Electromagnetics 8/19
Coplanar Waveguide
1. Motivation
2. Introduction to Reduced Basis Method
3. Electromagnetic Model
4. Numerical Results
5. Conclusions
Martin Hess RBM in Electromagnetics 9/19
Coplanar Waveguide
Figure: Geometry of coplanar waveguide.
Martin Hess RBM in Electromagnetics 9/19
Maxwell’s Equations
Time-Harmonic Maxwell’s Equations, [Hiptmair, 2002]
µ−1(∇× E ,∇× v) + iωσ(E , v)− ω2ε(E , v) = iωJ ∀v ∈ X
E × n = 0 on ΓPEC ∪ Γconductor
∇× E × n = 0 on ΓPMC
Assemble matrices
Aµ ≡ µ−1(∇× E ,∇× v)
Aσ ≡ σ(E , v)
Aε ≡ ε(E , v)
⇒ (Aµ + iωAσ − ω2Aε)(xreal + iximag ) = ib
Martin Hess RBM in Electromagnetics 10/19
Maxwell’s Equations
Time-Harmonic Maxwell’s Equations, [Hiptmair, 2002]
µ−1(∇× E ,∇× v) + iωσ(E , v)− ω2ε(E , v) = iωJ ∀v ∈ X
E × n = 0 on ΓPEC ∪ Γconductor
∇× E × n = 0 on ΓPMC
Assemble matrices
Aµ ≡ µ−1(∇× E ,∇× v)
Aσ ≡ σ(E , v)
Aε ≡ ε(E , v)
⇒ (Aµ + iωAσ − ω2Aε)(xreal + iximag ) = ib
Martin Hess RBM in Electromagnetics 10/19
Real symmetric system
Real symmetric system, [H. and Benner, 2013][Aµ − ω2Aε −ωAσ
−ωAσ −Aµ + ω2Aε
] [xrealximag
]=
[0−b
]
A(ν) =Qa∑q=1
Θqa(ν)Aq = A1 + ωA2 + ω2A3
Matrices in the affine form will also be real symmetric ⇒RB computation in real arithmetics
real symmetric eigenvalue problem for β(ν), [H. et al., 2015]
BUT: system size doubles, here from 1′012 to 2′024
Martin Hess RBM in Electromagnetics 11/19
Real symmetric system
Real symmetric system, [H. and Benner, 2013][Aµ − ω2Aε −ωAσ
−ωAσ −Aµ + ω2Aε
] [xrealximag
]=
[0−b
]
A(ν) =Qa∑q=1
Θqa(ν)Aq = A1 + ωA2 + ω2A3
Matrices in the affine form will also be real symmetric ⇒RB computation in real arithmetics
real symmetric eigenvalue problem for β(ν), [H. et al., 2015]
BUT: system size doubles, here from 1′012 to 2′024
Martin Hess RBM in Electromagnetics 11/19
Real symmetric system
Real symmetric system, [H. and Benner, 2013][Aµ − ω2Aε −ωAσ
−ωAσ −Aµ + ω2Aε
] [xrealximag
]=
[0−b
]
A(ν) =Qa∑q=1
Θqa(ν)Aq = A1 + ωA2 + ω2A3
Matrices in the affine form will also be real symmetric ⇒RB computation in real arithmetics
real symmetric eigenvalue problem for β(ν), [H. et al., 2015]
BUT: system size doubles, here from 1′012 to 2′024
Martin Hess RBM in Electromagnetics 11/19
Quadratic OutputsOutput quantity s(ν) = |`(u)|Using the real form ⇒
s(ν) =√`1(u)2 + `2(u)2.
Define Q(·, ·) by `T1 `1 + `T2 `2 ⇒
s2(ν) = Q(u, u).
Expanded Formulation, [Sen, 2007]
A(ν) =
[2A(ν)− Q −Q−Q 2A(ν)− Q
], F =
[b
−b
].
For the parametric problem A(ν)x = F , it holds s2(ν) = Fx .BUT: system size doubles again, here from 2′024 to 4′048
Martin Hess RBM in Electromagnetics 12/19
Quadratic OutputsOutput quantity s(ν) = |`(u)|Using the real form ⇒
s(ν) =√`1(u)2 + `2(u)2.
Define Q(·, ·) by `T1 `1 + `T2 `2 ⇒
s2(ν) = Q(u, u).
Expanded Formulation, [Sen, 2007]
A(ν) =
[2A(ν)− Q −Q−Q 2A(ν)− Q
], F =
[b
−b
].
For the parametric problem A(ν)x = F , it holds s2(ν) = Fx .BUT: system size doubles again, here from 2′024 to 4′048
Martin Hess RBM in Electromagnetics 12/19
Quadratic OutputsOutput quantity s(ν) = |`(u)|Using the real form ⇒
s(ν) =√`1(u)2 + `2(u)2.
Define Q(·, ·) by `T1 `1 + `T2 `2 ⇒
s2(ν) = Q(u, u).
Expanded Formulation, [Sen, 2007]
A(ν) =
[2A(ν)− Q −Q−Q 2A(ν)− Q
], F =
[b
−b
].
For the parametric problem A(ν)x = F , it holds s2(ν) = Fx .
BUT: system size doubles again, here from 2′024 to 4′048
Martin Hess RBM in Electromagnetics 12/19
Quadratic OutputsOutput quantity s(ν) = |`(u)|Using the real form ⇒
s(ν) =√`1(u)2 + `2(u)2.
Define Q(·, ·) by `T1 `1 + `T2 `2 ⇒
s2(ν) = Q(u, u).
Expanded Formulation, [Sen, 2007]
A(ν) =
[2A(ν)− Q −Q−Q 2A(ν)− Q
], F =
[b
−b
].
For the parametric problem A(ν)x = F , it holds s2(ν) = Fx .BUT: system size doubles again, here from 2′024 to 4′048
Martin Hess RBM in Electromagnetics 12/19
Petrov-Galerkin RB
EM models contain resonances, i.e., A(ν) is singular.
A(ν) singular ⇐⇒ β(ν) = 0.
Problem: βN(ν) = 0 while β(ν) > 0.Solution: Supremizing operators T ν
T ν : X → X : (T νw , ·)X = a(w , ·; ν),
VN = span{u(ν1), u(ν2), ..., u(νN)},W ν
N = span{T νu(ν1),T νu(ν2), ...,T νu(νN)},
=⇒ βN(ν) ≥ β(ν)
Martin Hess RBM in Electromagnetics 13/19
Petrov-Galerkin RB
EM models contain resonances, i.e., A(ν) is singular.A(ν) singular ⇐⇒ β(ν) = 0.
Problem: βN(ν) = 0 while β(ν) > 0.Solution: Supremizing operators T ν
T ν : X → X : (T νw , ·)X = a(w , ·; ν),
VN = span{u(ν1), u(ν2), ..., u(νN)},W ν
N = span{T νu(ν1),T νu(ν2), ...,T νu(νN)},
=⇒ βN(ν) ≥ β(ν)
Martin Hess RBM in Electromagnetics 13/19
Petrov-Galerkin RB
EM models contain resonances, i.e., A(ν) is singular.A(ν) singular ⇐⇒ β(ν) = 0.
Problem: βN(ν) = 0 while β(ν) > 0.
Solution: Supremizing operators T ν
T ν : X → X : (T νw , ·)X = a(w , ·; ν),
VN = span{u(ν1), u(ν2), ..., u(νN)},W ν
N = span{T νu(ν1),T νu(ν2), ...,T νu(νN)},
=⇒ βN(ν) ≥ β(ν)
Martin Hess RBM in Electromagnetics 13/19
Petrov-Galerkin RB
EM models contain resonances, i.e., A(ν) is singular.A(ν) singular ⇐⇒ β(ν) = 0.
Problem: βN(ν) = 0 while β(ν) > 0.Solution: Supremizing operators T ν
T ν : X → X : (T νw , ·)X = a(w , ·; ν),
VN = span{u(ν1), u(ν2), ..., u(νN)},W ν
N = span{T νu(ν1),T νu(ν2), ...,T νu(νN)},
=⇒ βN(ν) ≥ β(ν)
Martin Hess RBM in Electromagnetics 13/19
PG RB - Quadratic OutputFull system
A(ν)x(ν) = b(ν), s2(ν) =(`T1 x(ν)
)2+(`T2 x(ν)
)2,
is projected as (W νT
N A(ν)VN
)x(ν) = W νT
N b(ν),
s2N(ν) =
(V TN `
T1 x(ν)
)2 +
(V TN `
T2 x(ν)
)2.
A(ν) =Qa∑q=1
Θqa(ν)Aq, W ν
N =Qa∑q=1
Θqa(ν)W q
N ,
AN(ν) = W νT
N A(ν)VN =Qa∑q=1
Qa∑q′=1
Θqa(ν)Θq′
a (ν)Wq′T
N AqVN.
Martin Hess RBM in Electromagnetics 14/19
PG RB - Quadratic OutputFull system
A(ν)x(ν) = b(ν), s2(ν) =(`T1 x(ν)
)2+(`T2 x(ν)
)2,
is projected as (W νT
N A(ν)VN
)x(ν) = W νT
N b(ν),
s2N(ν) =
(V TN `
T1 x(ν)
)2 +
(V TN `
T2 x(ν)
)2.
A(ν) =Qa∑q=1
Θqa(ν)Aq, W ν
N =Qa∑q=1
Θqa(ν)W q
N ,
AN(ν) = W νT
N A(ν)VN =Qa∑q=1
Qa∑q′=1
Θqa(ν)Θq′
a (ν)Wq′T
N AqVN.
Martin Hess RBM in Electromagnetics 14/19
PG RB - Quadratic OutputFull system
A(ν)x(ν) = b(ν), s2(ν) =(`T1 x(ν)
)2+(`T2 x(ν)
)2,
is projected as (W νT
N A(ν)VN
)x(ν) = W νT
N b(ν),
s2N(ν) =
(V TN `
T1 x(ν)
)2 +
(V TN `
T2 x(ν)
)2.
A(ν) =Qa∑q=1
Θqa(ν)Aq, W ν
N =Qa∑q=1
Θqa(ν)W q
N ,
AN(ν) = W νT
N A(ν)VN =Qa∑q=1
Qa∑q′=1
Θqa(ν)Θq′
a (ν)Wq′T
N AqVN.
Martin Hess RBM in Electromagnetics 14/19
PG RB - Expanded Form
Full system
A(ν)x(ν) = F(ν), s2(ν) = FT x(ν),
is projected as (W νT
N A(ν)VN
)x(ν) = W νT
N F(ν),
s2N(ν) = V T
N FT x(ν).
It is a compliant system ⇒ fast error decay expected
Martin Hess RBM in Electromagnetics 15/19
PG RB - Expanded Form
Full system
A(ν)x(ν) = F(ν), s2(ν) = FT x(ν),
is projected as (W νT
N A(ν)VN
)x(ν) = W νT
N F(ν),
s2N(ν) = V T
N FT x(ν).
It is a compliant system ⇒ fast error decay expected
Martin Hess RBM in Electromagnetics 15/19
Transfer function
1. Motivation
2. Introduction to Reduced Basis Method
3. Electromagnetic Model
4. Numerical Results
5. Conclusions
Martin Hess RBM in Electromagnetics 16/19
Transfer function
2 4 6 8 102
3
4
5
6
ω in GHz
‖H(iω
)‖in
dB
Figure: Transfer function over frequency range [0.6, 10] GHz.
Martin Hess RBM in Electromagnetics 16/19
Mean error in the output
50 100 150 200 250 300
10−3
10−2
10−1
100
Reduced order N
Rel
ativ
eap
prox
imat
ion
erro
r
Figure: Mean relative error over sampled grid. Field estimator (blue), outputestimator using expanded form (green), heuristic optimum (red).
Martin Hess RBM in Electromagnetics 17/19
Max error in the output
50 100 150 200 250 30010−3
10−2
10−1
100
101
102
Reduced order N
Rel
ativ
eap
prox
imat
ion
erro
r
Figure: Maximum relative error over sampled grid. Field estimator (blue), outputestimator using expanded form (green), heuristic optimum (red).
Martin Hess RBM in Electromagnetics 18/19
Conclusions
1. Motivation
2. Introduction to Reduced Basis Method
3. Electromagnetic Model
4. Numerical Results
5. Conclusions
Martin Hess RBM in Electromagnetics 19/19
Conclusions
expanded form improves approximation quality significantly
increased offline cost ⇒ applicable only to small or medium sizedmodels
sometimes even better than the heuristic optimum
actual optimum is infeasible to compute
Thank you for your attention!
Martin Hess RBM in Electromagnetics 19/19
Conclusions
expanded form improves approximation quality significantly
increased offline cost ⇒ applicable only to small or medium sizedmodels
sometimes even better than the heuristic optimum
actual optimum is infeasible to compute
Thank you for your attention!
Martin Hess RBM in Electromagnetics 19/19
Conclusions
expanded form improves approximation quality significantly
increased offline cost ⇒ applicable only to small or medium sizedmodels
sometimes even better than the heuristic optimum
actual optimum is infeasible to compute
Thank you for your attention!
Martin Hess RBM in Electromagnetics 19/19
Conclusions
expanded form improves approximation quality significantly
increased offline cost ⇒ applicable only to small or medium sizedmodels
sometimes even better than the heuristic optimum
actual optimum is infeasible to compute
Thank you for your attention!
Martin Hess RBM in Electromagnetics 19/19
Conclusions
expanded form improves approximation quality significantly
increased offline cost ⇒ applicable only to small or medium sizedmodels
sometimes even better than the heuristic optimum
actual optimum is infeasible to compute
Thank you for your attention!
Martin Hess RBM in Electromagnetics 19/19
References
H., M. W. and Benner, P. (2013).
Fast Evaluation of Time-Harmonic Maxwell’s Equations Using the Reduced Basis Method.IEEE Transactions on Microwave Theory and Techniques, 61:2265 – 2274.
H., M. W., Grundel, S., and Benner, P. (2015).
Estimating the Inf-Sup Constant in Reduced Basis Methods for Time-Harmonic Maxwell’s Equations.to be published: IEEE Transactions on Microwave Theory and Techniques.
Hiptmair, R. (2002).
Finite Elements in Computational Electromagnetism.Acta Numerica, pages 237 – 339.
Rozza, G., Huynh, D. B. P., and Patera, A. T. (2008).
Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive PartialDifferential Equations.Archives of Computational Methods in Engineering, 15:229 – 275.
Sen, S. (2007).
Reduced Basis Approximation and A Posteriori Error Estimation for Non-Coercive Elliptic Problems: Application toAcoustics.PhD thesis, Massachusetts Institute of Technology.
Martin Hess RBM in Electromagnetics 20/19
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